Category: Physics

Introduction

Everything in the universe consists of tiny particles called molecules, which are classified based on the distance and the attractive force between their constituents. Different substances have distinct properties such as mass, temperature, density, volume, pressure, etc. 

The process of transfer of heat is achieved through three different processes in matter, namely, conduction, convection, and radiation. It is studied under thermodynamics, which deals with thermal energy and its relationship to heat, work, temperature, energy, entropy, and other physical properties of matter. 

Isotherms and adiabats

Thermodynamic system

A thermodynamic system is defined as a collection of matter confined within specific boundaries. Within the system, transformations can take place internally. At the same time, interaction with the outside world can occur through the boundaries, which can have a definite permeability. 

Based on the type of the boundary of the thermodynamic system, it can be classified into three categories: isolated system, closed system, and open system. 

• In an isolated system, no energy or matter is exchanged across its boundaries, and no work is performed. An example is items packed in a vacuum bag.
• A closed system allows for energy transformation, but matter cannot cross its boundaries, as seen in a cylinder closed by a valve. 
• And finally, an open system allows for the free exchange of energy and matter across its boundaries, such as a pool filled with water.

Thermodynamic processes

Thermodynamic processes involve an energy transfer between two systems or between systems and their surroundings, resulting in changes in volume, pressure, or temperature. Depending on the type of process, the process can change the energy of the system and perform some work on or by the system.

To classify thermodynamic processes, we look at how they are performed and what conditions they are performed in. 

• An isothermal process is one where the temperature of the system remains constant while energy is exchanged in or out.
• An isobaric process keeps the pressure constant.
• An isochoric process occurs when there is no change in the volume of the system.
• Finally, in an adiabatic process, no heat is exchanged by the systems. We’ll discuss adiabatic processes in detail here.

What is an adiabatic process?

An adiabatic process refers to a thermodynamic process in which, no heat energy is exchanged between the system and the surroundings. To be considered adiabatic, the system must satisfy two conditions.

1. It must be completely isolated. 
2. The process must occur over a short enough time frame, which makes it impossible for heat transfer to take place. 

Work done in adiabatic process

We start with a cylinder with walls made up of an insulating material. We assume that it has a frictionless piston attached to it also made up of an insulating material, effectively preventing heat from escaping. 

Reversible adiabatic process

An adiabatic process is considered reversible if the system can go back to its original state with no alterations. However, this can’t be achieved practically since a reversible adiabatic process does not really exist in nature. But theoretically, such a process is known as an isentropic process and one example of such a process is the adiabatic expansion of a real gas.

Irreversible adiabatic process

If the system cannot return to its original state after the process has occurred, the process is termed irreversible and is accompanied by a change in the entropy of the system. One example would be heat transfer.

Application of adiabatic process

Adiabatic processes occur in the following scenarios:

1. The principle is used in refrigerators.
2. Some processes in a thermal engine are adiabatic in nature. 
3. Compressors and turbines also work on adiabatic principles. 
4. Igloos and thermos flasks are isolated systems and they utilise adiabatic principles.
5. Quantum harmonic oscillators work adiabatically.

Carnot cycle

Summary

This tutorial covered the topic of thermodynamic systems and processes, including the work done in adiabatic processes. The discussion also included the differences between reversible and irreversible adiabatic processes and the various applications of adiabatic processes.

Frequently Asked Questions

1. Give some examples for all thermodynamic processes.

There are four types of thermodynamic processes: isothermal, adiabatic, isobaric, and isochoric. Some examples of these processes include boiling water, refrigeration, the Carnot engine, heat pumps, the freezing of water into ice, pressure cooking, and vertical atmospheric airflow.

2. Differentiate isothermal and adiabatic processes.

3. What is the specific heat capacity?

The specific heat capacity refers to the amount of energy that must be supplied to one mole of a substance to increase its temperature by one degree.

4. What do adiabatic compression and expansion do?

Adiabatic compression leads to an increase in system temperature and adiabatic expansion causes a decrease in temperature.

5. What is the first law of thermodynamics?

The first law relates the internal energy of the system with the heat exchange the work done. According to the first law, internal energy is the difference of the latter two quantities.

Also Read: Adiabatic Processes Derivation

Introduction

In thermodynamics, we explore various variables, including temperature, pressure, volume, entropy, heat, etc. These variables form the foundation for thermodynamic processes, which may be classified into isothermal, isobaric, adiabatic processes, etc. 

In an adiabatic process, thermodynamic variables can change value so long as no exchange of heat occurs. There are two critical requirements for such a process to occur:

1. The walls of the piston must be sealed to prevent heat exchange between the system and the environment. 
2. The process of compression or expansion must be completed rapidly.

 

Adiabatic processes

Derivation of adiabatic process formula

Adiabatic Relations between P, V, and T

The first law of thermodynamics gives rise to the conservation of energy, forbidding it from being created or destroyed.

 

Adiabatic relation between P and V

As per the first law, the change in heat energy is equal to the sum of the change in internal energy and the work done. That is,

Adiabatic relation between P and T

We again start with a mole of an ideal gas. Then, from the ideal gas equation,

 Adiabatic relation between V and T:

Examples of Adiabatic Process

1. An ice-box prevents heat from entering the system and thus, is an adiabatic system.
2. Hot water kept in a thermal flask is an example of an adiabatic system.
3. When warm air rises from the Earth’s surface, it expands adiabatically. As a result, the water vapour cools and condenses into water droplets forming a cloud.
4. A gas being compressed or undergoing rapid expansion is an adiabatic process since there isn’t enough time for heat exchange to occur.

Adiabatic Expansion

Adiabatic expansion is defined as a perfect condition for a confined system, where the pressure remains constant, and the temperature decreases.

Work done in an adiabatic expansion process

To derive the formula for work done, we start with a closed cylinder that contains n moles of an ideal gas. If P represents the pressure of this gas, and the piston in the cylinder moves up by a distance x, the work done by the gas may be written as:

Adiabatic work done

Description: The work done in an adiabatic process is the area under the curve

Here, A represents the cross-sectional area of the piston and dV = Adx is the increase in volume due to the piston’s movement. Suppose the initial and final stages of the system are \(\;({{\bf{P}}_1},{{\bf{V}}_1},{{\bf{T}}_1}){\bf{and}}({{\bf{P}}_2},{{\bf{V}}_2},{{\bf{T}}_3})\), respectively. Then the total work done is:

Shown above is the equation for the work done in an adiabatic process for a system consisting of n moles of an ideal gas. When the gas does work during adiabatic expansion, the work done is positive and the temperature of the gas decreases\(\left( {{T_2} < {T_1}} \right)\). Conversely, when work is done on the gas during adiabatic compression, the work done is negative and the temperature of the gas increases \(\left( {{T_2} > {T_1}} \right)\)

.

Adiabatic Compression

Adiabatic compression is a process wherein, a system undergoes compression without exchange of heat with its surroundings. It leads to an increase in temperature and a decrease in volume.

Adiabatic-reversible and irreversible process

Reversible adiabatic process: An adiabatic process is said to be reversible if the system can revert back to its original state after the process has taken place. Such a process isn’t possible in nature.

Irreversible process: An irreversible adiabatic process is one that cannot be reversed. It is accompanied by an increase in entropy and all real adiabatic processes are irreversible.

Conclusion

An adiabatic process is characterized by changes in the pressure, volume, and temperature of a system, but with no exchange of heat taking place between the system and its surroundings. For an adiabatic process to occur, certain conditions must be met: the walls of the container must be insulated, and the speed of compression or expansion must be rapid enough to prevent heat exchange.

For an adiabatic process, the relation between pressure and volume is \(P{V^\gamma } = \kappa \). This equation may be rewritten in terms of pressure and temperature as\({P^{1 – \gamma }}{T^\gamma } = {\rm{constant}}\), or in terms of volume and temperature as \({\rm{T}}{{\rm{V}}^{\gamma  – 1}} = {\rm{constant}}\)

The work done for an adiabatic process is given by the following expression:

Wadia = nR-1 T1 – T2

 

Frequently Asked Questions

1. The adiabatic process occurs at rapid rate. Explain why?

If the process were to occur slowly, it would give the system time to exchange heat, which would make the process non-adiabatic.

2. During the Adiabatic expansion, the temperature of gas gets lowered. why?

During adiabatic expansion, work is done by the system, causing a decrease in its internal energy. This lowers its temperature.

3.Difference between the Adiabatic and Isothermal processes?

4. Is the work done in an Adiabatic process change depending on which physical quantity?

5.  In adiabatic expansion, how the internal energy is affected? 

 

 

 

Introduction

Acoustics is a branch of physics that deals with the study of sound, including its generation, transmission, and effects. It encompasses a variety of topics, including mechanical waves in different mediums and the study of various sounds such as vibrations, noises, ultrasounds, and infrasounds. There are various sub-disciplines of acoustics, including aeroacoustics, bioacoustics, and the acoustics of vibration. Acoustics has many technological and scientific applications and is a broad field that extends far beyond just the study of sound.

What is Acoustics?

The field of acoustics involves the study of sounds, their properties and how they interact with the environment. This includes exploring different types of sound waves, such as infrasonic and ultrasonic, and examining their behaviors and effects.

Acoustic Physics

An understanding of mechanical waves is crucial before understanding what acoustics is about. This is because sound itself is a mechanical wave. Broadly, all mechanical waves are classified into two categories:

1. Longitudinal waves
2. Transverse waves

.

Transverse and longitudinal waves

Sound Acoustics

In acoustics, we examine the transfer of energy through sound waves in a medium and how it interacts with its surroundings. Sound is a type of energy that travels as longitudinal waves through compression and rarefaction. Key attributes of sound waves include amplitude, frequency, wavelength, and velocity. The field of sound acoustics delves into the production and propagation of sound waves in a material.

Acoustic energy

The energy used to transmit vibrations from a source to a recipient using acoustic waves is referred to as acoustic energy. This energy was discovered in the sixth century during an experiment studying the characteristics of vibrating strings.

Noise

In our daily lives, we encounter different types of sounds with varying levels of loudness and frequency. The term “noise” refers to a loud, disturbing sound that can be uncomfortable to the ears. Noise can cause discomfort and annoyance to both individuals and their surroundings.

Importance of Acoustics

In our daily life, sound plays an important role and is present in every aspect of our existence. Hearing and producing sound is crucial for our survival and sets humans apart as the most advanced species on Earth with their ability to communicate through speech. Even beyond speaking and hearing, we humans have harnessed the knowledge of sound waves and found various uses for it in industries such as music, architecture, technology, and medicine. Hence, the study of sound, or in other words, acoustics, holds great significance as it has a broad range of applications in multiple domains.

The applications of acoustics

Application of acoustics

Acoustics is an area of science that has numerous practical applications, some of which include:

• Medicine: Ultrasonography and other high-frequency sound-based diagnostic techniques are crucial to our medicinal processes. Sound waves are also used to treat various psychological and physical disorders.
• Controlling noise pollution: Understanding sound sources is crucial in developing methods to reduce negative effects of sound in the form of noise.
• SONAR: SONAR technology uses sound waves to locate underwater objects and determine the sea’s depth.
• Acoustic warfare: SONAR is also being utilised for target identification and location in combat.
• Music: Acoustics play a vital role in producing and synthesising musical instruments, setting up orchestras, and in producing sound tracks, etc.
• Acoustic signal processing: This involves electronic modification of sound and is used in hearing aids, noise-cancelling devices, and other processes like perceptual coding.

Summary

The field of acoustics in physics focuses on the study of mechanical waves, particularly sound waves, and their various applications. It is a constantly evolving discipline and is being researched across multiple areas. Acoustics play a role in almost every aspect of human culture and over time, our understanding of the topic has continued to grow, leading to advancements in the technologies that are available to us. The increasing popularity and applicability of acoustics in various fields is also generating new job opportunities for acousticians and acoustic engineers globally.

Frequently Asked Questions

1. What is acoustics?

Acoustics, as a field of study, encompasses the investigation of sound, from its production to its propagation and eventual effects. It encompasses a range of subfields, including aeroacoustics, environmental acoustics, ultrasonics, and more.

2. What is SONAR?

SONAR is a technique that uses sound waves to locate and communicate with underwater objects. It stands for Sound Navigation and Ranging and is used to determine the position and distance of objects below the surface of the sea.

3. What are mechanical waves?

A mechanical wave is a type of wave that requires a medium to propagate through space, and sound waves are the most commonly known form of mechanical waves.

4. Why are sound waves considered mechanical waves?

Sound waves require a medium to travel through and thus, they are classified as mechanical waves.

Introduction

Acceleration represents the change in velocity of an object. It can either be positive or negative, depending on whether the final velocity is greater or less than the initial velocity. Thus, if the velocity of an object is decreasing with time, it is said to possess negative acceleration or retardation. 

Acceleration is a vector quantity which has both magnitude as well as direction. During circular or rotational motion, the acceleration encountered is referred to as rotational acceleration. Note that it is possible for an object to have zero acceleration if it is moving with a constant velocity.

What is Acceleration? 

In simplest of terms, acceleration is the rate of change of an object’s velocity with respect to time. Hence, its SI unit is given as \(m/{s^2}\).

General Formula of Acceleration

Generally, acceleration can be calculated via the following formula:

Acceleration dimensional Formula in Physics

Acceleration Unit

This is the SI unit of acceleration and other derived units can be found in different systems.

Acceleration Types

Most generally, acceleration can be classified into the following types:

1. Uniform acceleration
2. Non-uniform acceleration

Uniform acceleration: An object whose velocity is changing at a constant rate is said to possess uniform acceleration.

Example: Suppose a car’s speed increases steadily by 30 m/s in 10s throughout a journey. That would mean that the car has a constant acceleration \(3m/{s^2}\).

Non-uniform acceleration: Non-uniform acceleration is acceleration which itself does not remain steady in time.

Example: Suppose that during the first 2 hours of a journey, a car travels with a velocity of 15 km/hr. In the next 3 hours, the velocity changes to 45 km/hr. The change in velocity occurs over unequal intervals of time and thus, over the course of the journey, the car has non-uniform acceleration.

A few other types of acceleration may be stated as follows:

Acceleration due to gravity: This is simply the acceleration experienced by a body due to the gravitational force. Mathematically, it is equal to the gravitational force per unit mass experienced by the object.

Centripetal acceleration: A body in rotational motion experiences this type of acceleration and it is given as the centripetal force per unit mass experienced by a body undergoing circular motion.

Radial acceleration: Acceleration directed along the radius for a body in circular motion is called radial acceleration.

Angular acceleration: Angular acceleration is also experienced by a body in circular motion and its effect is to change the angular velocity of an object.

Coriolis acceleration: Coriolis force comes into the picture when the frame of reference we are considering is itself rotating with a given velocity. The coriolis acceleration is the acceleration encountered due to this coriolis force.

Average acceleration

For a non-uniform motion, we can find out the average acceleration of the object in question to get an overall sense of how it may have moved over the course of its journey. It is defined mathematically as follows.

Instantaneous acceleration

The acceleration experienced by an object at a particular instant in time is referred to as instantaneous acceleration and it is given as the limiting value of the rate of change of velocity of an object when the time interval tends to zero.

Where a= acceleration of the body, s=displacement of the body, and t= time. Thus, instantaneous acceleration is the second derivative of displacement.

Negative acceleration or retardation: Acceleration that slows down the motion of an object is referred to as retardation. For such a phenomenon to occur, the acceleration applied must be negative.

Velocity-Time Graph

Acceleration can also be derived from the velocity-time graph of an object. The slope of the graph gives us acceleration as a function of time and if we find the value of this slope at a particular point, i.e., the slope of tangent at a point, we arrive at instantaneous acceleration. 

Graph for average acceleration

Graph for instantaneous acceleration

Graph for uniform acceleration

Graph for non-uniform acceleration

Examples of Acceleration

Example 1: A body moving with \(20m/{s^2}\) comes to a halt after 5.2 secs. Find the nature of acceleration and its value.

Solution: 

Example 2: If a car is moving with a velocity of 45 m/s and after 10 s, its velocity becomes constant at 60 m/s, find acceleration.

Solution:

Difference between Acceleration and Velocity

Summary

Acceleration measures the change in velocity of an object. Its SI unit is \(m/{s^2}\) and it is a vector quantity with both magnitude and direction. It can be determined as the first-order derivative of velocity or the second-order derivative of position vector.  An object moving with a constant velocity has zero acceleration since the derivative of a constant is equal to zero. 

Frequently Asked Questions

1. Give two examples of retardation.

Two examples of retardation are as follows:

1. A train that reaches a halt will slow down and thus, experience retardation.
2. A ball thrown upwards experiences retardation due to gravity.

2. What is the SI and CGS unit of acceleration?

In SI units, acceleration is measured in \(m/{s^2}\)  and in CGS units, in \(cm/{s^2}\).

3. What is gravitational acceleration?

Gravity is the force by which the Earth attracts a body towards its center. This force generates acceleration in a vertical motion, known as gravitational acceleration. The motion of an object falling solely under the effect of gravity is termed as free-fall.

4. What is the value of gravitational acceleration?

The approximate value of acceleration due to gravity is \(9.8m/{s^2}\).

5. What is angular acceleration?

Angular acceleration measures the time rate of change of angular velocity of an object and thus, is measured in \(rad/{s^2}\).

6. What is a Coriolis force?

Coriolis force is experienced by objects which are moving in a frame of reference that is itself rotating with a given angular velocity. It is responsible for wind in certain regions of the Earth.

Introduction

Various optical systems, including lenses, mirrors, and spherical mirrors, reflect or diverge light to create various kinds of images. Rules of reflection and laws of refraction are the two fundamental laws that govern picture generation. In the creation of images, these laws are crucial. Different-sized pictures are created at various positions when light is reflected by lenses or mirrors. It can vary in size from very little to extremely large to the same size as the thing, and it can also become blurry at times. Consequently, the creation of the hazy image is referred to as an abbreviation. A hazy image is created when light rays diverge after passing through a lens in a particular spot. This is known as an aberration of the lens. 

Lenses and curved mirrors are two objects that can induce optical aberration. This phenomenon causes a deviation in the light rays. Not all of the light’s beams concentrate on the focal point. By virtue of optical aberration, images are essentially blurry.

Types of Aberration

For a single wavelength light, there are five types of aberration.

1. Spherical aberration
2. Coma
3. Astigmatism
4. Curvature of field
5. Distortion

Let our expert guru Mr. Mayur be your guide toward improving your understanding of this chapter better. Watch the related video of this chapter in Class 7th Science lesson no-15

1. Spherical aberration

It is a defect of the lens due to which the light beam does not concentrate at a single point, as a result, the exterior part of the image gets blurred. This aberration is caused in lenses or mirrors when light beams fall on the exterior part and concentrate at different points.

Correction of the spherical aberration

• Spherical aberration can be corrected by using a mirror of different non-spherical shapes. A parabolic mirror or an ellipsoidal mirror can replace a spherical mirror.
• Spherical aberration can be removed by blocking the marginal rays. A circular annular mask on a lens can remove the marginal rays.
• By using suitable radii of curvature of the two surfaces of a lens the spherical aberration can be minimised.
• A combination of concave and convex lenses can also reduce the spherical aberration.

2. Coma

The coma is a type of spherical aberration. It occurs due to the variation in the magnification of the image. The rays that are coming from an off-axial object and the rays pass through different circular zones of the lens. Due to this the image will appear in a comet-like shape. So, this aberration of the lens is known as a coma.

Types of coma

The coma varies due to the magnification of different zones of the lens. There are two types of coma

1. Positive coma

When the lateral magnification of the outer zone is greater than the central zone, then the coma is said to be a positive coma.

2. Negative coma

When the lateral magnification of the outer zone is smaller than the central zone, then the coma is said to be a positive coma.

Correction of coma aberration:

• By bending the lens the coma can be corrected but not completely processed will only work in the case of a single lens.
• The coma can be corrected by the combination of lenses that are symmetrical and in the application of curvature type of lenses surface

3. Astigmatism

It is the most common defect that occurs in the eyes. In this defect, the front part of the lens or cornea has an irregular curve. So, when the light refracts by the retina it forms a blurry image. Astigmatism is caused when eyelids put pressure on the cornea. mostly these defects passed over generations from their parents.

Types of Astigmatism

There are two types of astigmatism. 

1. Positive astigmatism: This is an abbreviation for which a person suffered from a farsightedness defect of the eye. For convergent lenses, the transverse focal is greater than the meridional focal length. This type of astigmatism is known as positive astigmatism.

2. Negative astigmatism: This is an abbreviation for which a person suffered from a near sightedness defect of the eye. For divergent lenses, the sagittal focal is smaller than the meridional focal length. This type of astigmatism is known as negative astigmatism.

Correction for astigmatism aberration:

• The combination of concave and convex lenses can also reduce astigmatism.
• Using lenses with different radii of curvature helps to reduce astigmatism.

4. Curvature of Field

When the central part of the image is flat but the exterior part of the image is curved concerning the object then this type of aberration in the image is called curvature of field. The curvature of the field arises when the paraxial focal length is greater than the marginal focal length.

Correction for the curvature of field:

• The condition for no curvature is known as the Petal condition. When the image of the plane object projects on the Petal surface the aberration gets reduced.

5. Distortion

The type of aberration in which the shape of the image gets distorted in the transverse plane as a whole is known as distortion. It arises due to the variation in the lateral magnification with the lateral distance of an object from the lens distance. 

Types of distortion:

There are two types of distortion

1. Barrel distortion: If magnification decreases with axial distance then the image of a square takes the form resembling a barrel. This type of distortion is known as barrel distortion.

2. Pincushion distortion: If the magnification increases with lateral distance, then a square’s image takes a shape like a pin-cushion. This type of distortion is known as pin-cushion distortion.

Correction of the distortion:

• A combination of two lenses can help us to avoid distortion.
• Using both barrel distortion and pincushion distortion combination to cancel out each other’s effect.

6. Chromatic aberration

When the images are formed by the refraction of white light, As a result, the image becomes colored. This type of defect in the image is known as chromatic aberration. This phenomenon occurs due to the dispersion of the white light.

Types of chromatic aberration:

There are two types of chromatic aberration

1. Lateral chromatic aberration: In this aberration, all the light rays concentrate at the same plane but different points.

2. Longitudinal chromatic aberration: In this aberration, the light rays of different wavelengths concentrate at different points along the principal axis (horizontal line) of the lens. 

 Correction of the chromatic aberration:

• Lateral chromatic aberration gets reduced when the object is placed at infinity.
• The combination of two lenses also reduces chromatic aberration.
• Achromatic lenses can be used to reduce chromatic aberration.

Axial color

Axial color is a phenomenon of aberration in which the position of the image shifts as the light wavelengths differ. Color fringing occurs due to color aberration. This aberration is caused by the light rays having different wavelengths to focus at various points.

Summary 

For non-paraxial rays, the actual image differs from the result of the paraxial approximation. This defect of the image is known as an aberration. For monochromatic light or single-wavelength light, there are five types of aberration. It is impossible to eliminate all kinds of aberrations by using a single lens. We can use different lens combinations to eliminate them.

Frequently Asked Questions

Q1. What is a spherical aberration?

Ans: The spherical aberration is the defect of the image in which central and peripheral incident rays from images at different points on the axis, are known as spherical aberration.

Q2. How to correct the coma aberration?

Ans: Coma can be corrected by bending the lens. This process will only work in the case of a single lens.

Q3. What is the one difference between chromatic aberration and spherical aberration?

Ans: Chromatic aberration is shown by white-colored light but the spherical aberration is shown by monochromatic lights.

Q4. What is the principal axis?

Ans: The principal axis is the axis that passes through the focus and center of curvature of the lens.

Q5. What is a positive coma?

Ans: When the lateral magnification of the outer zone is greater than the central zone, then the coma is said to be a positive coma.

Q6: White light is the polychromatic source of light. Explain why?

Ans: white light is the polychromatic source of light because it constitutes seven colors and each of them has a different wavelength.

Introduction

Let’s start with a fascinating experiment that you may have witnessed in your daily life. Insert a pencil or pen into a beaker of water. When you look at the pencil from outside the beaker, what do you notice? It will show up where it shouldn’t be, as you’ll discover. As a result of light refraction, this has happened.

It’s interesting to note that this occurrence is not just in water. You would see something similar if you were to use oil or any other liquid. A light ray changes direction whenever it crosses the boundary between two materials with different refractive indices. Remember that a material’s refractive index measures how quickly light passes through it in comparison.

It’s also fascinating to notice that refraction can occur even in a single medium. This is due to the fact that a substance’s refractive index alters with its density. The refractive index of a material increases with the proximity of its constituent parts. As a result, for instance, air density varies with altitude, causing light to bend as it travels through the atmosphere, a process known as atmospheric refraction.

Definition of Refraction

Refraction is the shift in a wave’s direction of propagation brought on by an inhomogeneous material. Refraction is most frequently seen in light waves. This indicates that refraction occurs anytime it comes into contact with an inhomogeneity along the way. Light can therefore “bend” when passing through the interface of two different media or when propagating across a non-uniform medium.

You can see in the accompanying diagram that the light, which would have normally followed the blue line, became distorted and ended up following the red line. The initial light ray, shown in grey, is referred to as the incident ray, while the light ray in red is referred to as the refracted ray. Which way does light bend next? is a question that might come to mind. The solution is a generalisation: light bends toward the surface normal if the second material’s refractive index is higher. If not, it refracts differently from the norm. The difference in the media’s refractive indices determines the degree of bending.

Definition of Atmospheric Refraction

The phenomenon of atmospheric refraction occurs when light from space that enters the atmosphere travels in a curved path rather than a straight line. This intriguing finding has a simple explanation for its origin. It is well known that air does not have a constant density. It alters with height instead. As a result, the refractive index is lower near the top of the atmosphere, where air molecules are further apart. The refractive index rises when density falls, and vice versa.

Atmospheric Refraction

Light continuously bends while propagating through our atmosphere as a result of this gradient-like characteristic, leading to a curved path rather than the expected straight line. You can see in the image above how the blue light rays take a curved course as they move through the various layers of air.

The explanation for Advanced Sunrise

Not all of the sunlight that strikes the earth’s atmosphere is absorbed by it. It is somewhat reflected in space. Think of viewing Earth from the international space station. You’ll notice that it seems brilliant and sparkling as a result of the earth’s atmosphere reflecting sunlight. Atmospheric reflection is the term for this phenomenon. Let’s talk about two typical consequences of atmospheric refraction: an earlier sunrise and a later sunset.

In the summer, sunrise in New Delhi, the capital of India, happens about 05:30 AM. On the other hand, if you were to stand on a rooftop and face east, you would notice that the sun rises roughly two minutes earlier. Why does that happen? Spend a moment trying to guess the solution.

The explanation is quite straightforward. The sun must be at a certain height above the horizon for light beams from it to reach your eyes, which can only happen when the sun is visible. because air’s refractive index varies, which causes the light to travel in a curved path.

Since sunlight falling on the atmosphere continuously bends as it reaches the earth, you may see the sun even when it is just below the horizon. The sun is not actually in the position it appears to be in when you view a sunrise. Simply put, it just appears a little higher due to the curve in the path of the sun.

The explanation for Delayed Sunset

The sun has set, which means that you can no longer see it. It has descended so low on the horizon, in other words, that light cannot enter your eyes anymore. However, even after the sun has passed the horizon due to atmospheric refraction, sunlight is still visible for a few minutes. We refer to this as a delayed sunset. As with advanced sunrises, the curved path of sunlight in the atmosphere is the cause of delayed sunsets as well.

Summary

Refraction is defined as the change in direction of propagation of light when it encounters a change in refractive index. It is observed whenever there is some sort of inhomogeneity in the medium or the path of light. When light moves in the direction of an area with a higher refractive index, it bends in the direction of the normal. Our atmosphere’s refractive index is higher close to the earth’s surface than it is farther away because it also contains a density gradient. Thus, as light travels to the earth’s surface, it continuously curves into a path. The term for this is atmospheric refraction. The sunrise appears to happen a few minutes earlier than usual due to atmospheric refraction. Similar to this, the sun is still visible after it has actually crossed the horizon, delaying sunset. Both of these events take place because atmospheric refraction makes the sun appear to be a little bit higher in the sky than it actually is.

Frequently Asked Questions

1. State the laws of refraction.

Ans. When light passes from one medium to another medium with different refractive indices, two laws of refraction are followed:

1. The incident ray, the refracted ray, and the surface normal all together lie in the same plane.

2. The ratio of the sines of angles of incidence and refraction is equal to the ratio of refractive indices of the materials.

2. By what amount does light bend due to refraction?

Ans. According to the second law of refraction also named as Snell’s law, we can write a relation between the incidence and the refraction angles, such that,

$$
\frac{\sin \theta_i}{\sin \theta_r}=\frac{n_2}{n_1}
$$

Therefore, if we know the angle of incidence and refraction, and refractive indices of the two different mediums, we can calculate the angle of refraction. 

3. What other phenomena occur due to atmospheric refraction?

Ans. The phenomenon of mirage and twinkling of stars are the two very common natural examples of the effects of atmospheric refraction. 

4. Does the amount of refraction depend upon the colour of light?

Ans. Yes, the refraction of light depends upon the wavelength, and the colour of light depends upon the frequency. The refractive index that a beam of light “sees”, or “experiences” depends upon its wavelength. 

5. Which colour of light bends the most?

Ans. The Cauchy’s equation, refractive index $$
n=A+\frac{B}{\lambda^2}+\frac{C}{\lambda^4}+\ldots,
$$

where λ is the wavelength of light, and A, B, and C are constants. Thus, from the above formula we can speculate that the light of smaller wavelength, or higher frequency will bend more. This corresponds to violet light in the visible region, which bends the most.

Introduction

An accelerometer is a tool used to measure the acceleration of a body. We are able to monitor and analyse both linear and angular acceleration using the accelerometer’s sensor. This function is used by us in many aspects of daily life and is a requirement for many fundamental systems and gadgets. An accelerometer, which measures numbers in one, two, or three planes, can be used to measure the acceleration force, or “g.” When the threshold is reached, a routine may start. Capacitive accelerometers, piezoelectric accelerometers, and piezoresistance accelerometers are the three basic types of accelerometers. Both inertial navigation and guidance systems make extensive use of them.

What is an Accelerometer?

An accelerometer is a sensor that allows us to track both linear and angular acceleration. This function is used by us in many aspects of daily life and is a requirement for many fundamental systems and gadgets. Static forces and dynamic forces are the two categories into which acceleration forces are divided. Static forces are those that can be constantly applied to an object, such as friction or gravity. Dynamic forces are those that can be described as “moving” and are applied to the object at various rates (for instance, vibration).

How does an Accelerometer work?

• The accelerometer operates on a very straightforward principle. An accelerometer, which measures numbers in one, two, or three planes, can be used to measure the acceleration force, or “g.”
• If the threshold is exceeded, it can trigger a routine. 
• The three-axis accelerometer, which features a system made up of three different accelerometers, is the most often used type of accelerometer. Each one calculates the acceleration in the X, Y, and Z planes in a separate direction.
• One of the popular models of a 3 axis accelerometer is the OKYSTAR OKY3230 type. The accelerometer will only estimate the force of gravity as the standard when it is in a stable position, which is when there is no external acceleration experienced by the accelerometer.
• Now, if we take a look at a three axes accelerometer and put it in a position where the sensor on the X axis points in the left direction, the Y axis points down, and the Z axis points forward, then the accelerometer will present the reading that is as follows:

$X=0 g, Y=1 g, Z=0 g$

Purpose of accelerometer

An accelerometer is a device used in vehicles to measure the acceleration motion brought on by either motion or gravity. The accelerometer’s primary function is to transform mechanical motion into electrical impulses.

Capacitive accelerometer

A capacitive accelerometer measures an object’s acceleration by monitoring changes in electrical capacitance. The most popular accelerometer is this one.

• Comparing the capacitive accelerometers to the other two types of accelerometers, they are the smallest and least expensive.
• Its name implies that it is a micro-electro mechanical system, and its parts range in size from 1 to 100 micrometres.
• The primary mechanism by which the capacitive accelerometer operates relies on the movement of a known mass suspended on springs. The mass is attached to one end of the spring, and the capacitor is attached to the other.
• When the sensor is subjected to a force, the mass moves, changing the capacitance of the capacitors and, in turn, the distance between their plates.
• However, for a variety of high amplitude signals and frequencies, this accelerometer’s accuracy is lower than that of other types of accelerometers.

Piezoelectric accelerometer

Using the piezoelectric effect, piezoelectric accelerometers can identify changes in acceleration.

• The working principle is similar to the piezoresistive one.
• When a material, often PZT, is subjected to acceleration, a deformation occurs that results in a change. But in this instance, it’s electric charge rather than resistance.
• Piezoelectric accelerometers have a very high sensitivity and accuracy, making them suitable for widespread application.
• They can be used for crash and impact tests in challenging conditions as well as for obtaining particularly sophisticated and accurate seismic estimation.

Piezoresistance accelerometers

As the pressure exerted on the piezoresistance accelerators increases, so does their resistance.

• As compared to piezoelectric accelerometers, these are far less sensitive.
• These accelerometers make use of the piezoresistive effect, a phenomenon that occurs when mechanical stress is applied and causes a change in a semiconductor or metal’s electrical resistance. An electrical signal is then produced by the accelerometer from the change.

Applications of accelerometer

An accelerometer is a piece of equipment used to measure a body’s acceleration. Both inertial navigation and guidance systems make extensive use of it. Other typical applications for accelerometers include:

• It plays a crucial role in the airbag deployment system of contemporary cars.
• It can be used to gauge seismic activity, vehicle velocity, or even inclination.
• It can be used to gauge how deeply to compress the chest during CPR.
• It can also be used to rotate the screen of a digital camera or a smartphone while displaying the images on the screen in an upright position.
• In a variety of modern electronics, accelerometers are crucial and active components.

Summary

An accelerometer is a sensor that allows us to track both linear and angular acceleration. The accelerometer operates on a pretty straightforward principle. An accelerometer, which measures numbers in one, two, or three planes, can be used to measure the acceleration force, or “g.” Capacitive accelerometers, piezoelectric accelerometers, and piezoresistance accelerometers are the three basic types of accelerometers. Accelerometers are used in a wide variety of scientific applications. The inertial navigation systems are most frequently used for aircraft or missiles.

Frequently Asked Questions

1. What is a gravimeter?

Ans: An accelerometer is known as a gravimeter when it has been expressly designed to be used in gravimetry, or to measure gravity.

2. What is the most common use of accelerometers?

Ans: Accelerometers are used in a wide variety of scientific applications. The inertial navigation systems are most frequently used for aircraft or missiles.

3. What are MEMS accelerometers?

Ans: Micro-electromechanical systems, or MEMS, are able to sense vibrations even at extremely small scales. To track changes in these electrical devices’ positions, they are mostly employed in a variety of portable gadgets.

4. How can an accelerometer be used for hard drive protection?

Ans: When a device experiences excessive acceleration or vibration, an accelerometer can identify it. It safeguards the hard disc in this case by emptying the reading heads to prevent contact with the pattern.

5. What are the key characteristics we need to keep in mind while selecting an accelerometer?

Ans: The sensor bandwidth, sensitivity, frequency response, and dynamic range of an accelerometer are some of the important properties we must take into account for a particular application.

Introduction

A conductor’s ability to resist the flow of electric current through it is known as resistance. Resistors are parts that are used to stop the flow of electrons. Due to the attraction between positive particles and negative electrons, the positive conductor particles obstruct the passage of electrons. The flow of electricity is resistant as a result of this obstruction. Ohms are the units used to measure resistance. There are two categories of electrical resistance: static resistance and dynamic resistance. The length of the conductor, cross-section area, temperature, material, etc. are the parameters that affect or depend on resistance.

Resistance of Conductor

Resistance is defined as a conductor’s ability to obstruct the passage of current. The conductor’s resistance is expressed mathematically as the relationship between the current flowing through it and the potential difference along its length.  The movement of electrons across a conductor is known as electric current. Because of their attraction to one another, positive conductor particles obstruct the flow of electrons, which results in resistance to the movement of electricity. Resistance can be used to disperse voltage in a current as well as control the flow of electrons.

The resistivity of the conductor depends on

The ability of a material to resist electrical conduction is known as its resistivity. Resistivity is utilised to offset the effects of size on resistance. It is a non-size dependent material attribute. The resistivity of a conductor is influenced by elements such as temperature, alloying, cold work, age hardening, and mechanical stress. For most materials, resistance rises with temperature. Semiconductors are an exception, as their resistance increases with temperature.

Resistance also depends on the temperature of the conductor. As the temperature increases the resistivity increases.

\[R{\rm{_T}} = {\rm{ }}R{\rm{_0}}(1{\rm{ }} + \alpha \Delta T)\]

R = final resistance, \(R_0\) = initial resistance, and α = temperature coefficient 

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Resistance depends on temperature

The thermal energy of electrons rises with the temperature of metal conductors, which also raises the frequency of collisions between free electrons. As a result, they begin to move randomly, making it challenging to drift for conduction in a specific direction. The conductor’s resistance increases as a result. As a result, resistance rises as temperature rises for a metal conductor. Increasing current frequently results in temperature rise.

Conductor resistance formula

The resistance of a conductor is directly proportional to the length of the conductor (l). Thus, on doubling its length, resistance will be double, while on halving its length, resistance will be half. Also, the resistance of a conductor is inversely proportional to its cross-section area (A).

Since, \[R\propto L\]

And, \[R\propto \frac{1}{A}\]

Hence, \[R\propto \frac{L}{A}\]

Or, \[R = \rho \frac{L}{A}\]

Where R is resistance in ohm, L is length or conductor in meter, A is cross-section area in square meter and ρ is the resistivity constant in ohm per meter.

The bigger the value of resistance, the more it opposes the current flow. The value of resistance is given in Ω.

Temperature effect on resistance

A material’s resistivity changes with temperature. Resistance varies depending on the conductor, semiconductor, and insulator’s temperature. Resistance is influenced by temperature in two different ways: for metal conductors, it rises as the temperature rises, and for insulators, it falls as the temperature rises. At high temperatures, semiconductors have great conductivity.

Problem 1: What is the resistance of the circuit having length 10 cm and area 100 cm$^2$ having resistivity of 1.8 Ω.m?

Solution:

Length of circuit = 10 cm

Area of circuit = 100 cm$^2$

Resistivity = 1.8 Ω.m

The Formula used,

\(R = \rho \frac{L}{A}\)

\(\Rightarrow R = 1.8~\Omega.cm \frac{10~cm}{100~cm^2}\)

\(\Rightarrow R = 0.18~\Omega\)

Problem 2: What is the cell constant of the circuit when the conductivity is 20 Siemens/m having resistance 100 Ω?

Solution:

Resistance = 100 Ω

Conductivity = 20 Siemens/m

Cell constant =?

The Formula used,

\(R = \frac{Cell Constant}{Conductivity}\)

\(\Rightarrow Cell~Constant = Conductivity \times R\)

\(\Rightarrow Cell~Constant = 20~Siemens/m \times 100~Ω\)

\(Cell~Constant = 2000~m^{-1}\)

Frequently asked questions

1.What is the importance of resistance in electricity?

Ans: Resistance is a crucial component of electrical circuits; as resistance increases, current flow becomes more challenging, and as resistance decreases, current flow becomes easier. The resistance is a crucial component in conduction. Conduction greatly benefits from electron flow. The conductor turns into a semiconductor and an insulator as a result of an increase in resistance.

2. Do conductors have high or low resistance?

Ans: Insulators have a very high resistance to electrical current, compared to conductors’ extremely low resistance. Resistance turns become an insulator as it rises. Since there is no interruption to the high flow of electrons, the conductor’s resistance is very low. Conduction and resistance are inversely correlated.

3. Is it a light bulb resistor?

Ans: Despite not actually being resistors, light bulbs exhibit resistive behaviour. Electrons cannot pass through resistors, which also transform energy into another form. The process by which electricity passes through a light bulb to produce light and heat is the same. The light bulb’s filament serves as a resistor. The law of conservation of energy states that as energy cannot be created or destroyed, it can only be transformed from one form to another.

Introduction

Have you ever questioned why straw that has been dipped in water appears warped? When the water is added to the glass, the straw appears to bend, but when you remove it, you can see that it hasn’t actually bent at all. Refraction is to blame for this; rather than the straw itself, it bends the light around it. Likewise occurs when a pencil is inserted into a glass of only partially full water. If you look at the pencil, you can see that it looks normal above the water but twisted and a little bigger below. There are numerous such instances of refraction in daily life that will be covered in this chapter.

Refractive index of light

The refractive index of any material is given as the ratio of the speed of light in the vacuum divided (c) by the speed of light in a medium (v), and is presented with a symbol, n, such that,

$$
n=\frac{c}{v}
$$

We can infer from this relationship that optical density and light speed both affect the refractive index. With an increase in optical density, the refractive index often rises. Light may bend more when it enters a denser material than when it enters a rarer one. Additionally, as the medium’s medium’s light speed decreases, the refractive index rises.

Prism

A transparent substance that can reflect light and has at least two lateral surfaces that are obliquely inclined to one another is referred to as a prism. It contains five surfaces, including three rectangle lateral surfaces and two triangle bases. The angle of the prism refers to the angle created by two lateral surfaces. For a standard prism, the prism’s angle is always 60°.

Refraction of light through a prism

A light ray NP is seen entering glass at the initial surface OB in the diagram as it travels from air. Because glass is denser than air, incoming light is bent toward the normal after refraction. When light enters from glass into air at the second surface BC, it bends away from the usual. A line drawn perpendicular to the surface at the incident ray entry point is called a normal. In contrast to angle of emergence, which is the angle produced between the emergent ray and the normal, angle of incidence is the angle formed between the incident ray and the normal. The angle that the emergent ray (stretched rearward) creates with the incident light is known as the angle of deviation ($\angle D$) (extended forward). The equation can be written from angle of prism (∠A), angle of incidence (∠i) and angle of emergence (∠e). Therefore, the expression is given as,

$$
\angle D=\angle i+\angle e-\angle A
$$

Refractive index of a prism

Refractive index of a prism made up of glass is given by the formula,

$$
n=\frac{\sin \frac{(A+D)}{2}}{\sin \left(\frac{A}{2}\right)}
$$

Where n is the refractive index of the prism, A is the angle of the prism and D is the deviation. The deviation is minimum at one point and is called minimum deviation. Using this formula, we can easily calculate the refractive index of the prism.

Angle of deviation

It refers to the angle at which the emergent ray and incident ray make contact. Angle of deviation is affected by a variety of factors.

Refractive index

The refractive index is directly proportional to the angle of deviation. 

Angle of prism 

The magnitude of the angle of deviation increases with an increase in the angle of the prism.

Wavelength of light

As the wavelength increases, the angle of deviation decreases. Therefore, violet deviates the most, because it has a shorter wavelength.

Temperature

As temperature increases, intermolecular space also increases, density decreases, refractive index decreases and angle of deviation decreases. 

Dispersion of light through prism

The white light, composed of the whole spectrum) is divided further into its components called the spectrum when it passes through a prism. This phenomenon is called  The white light divides into its seven individual colours. These hues are a part of VIBGYOR (V-violet, I- indigo, B-blue, G-green, Y-yellow, O-orange, R-red). The wavelengths of these hues determine their deviation. Red has the longest wavelength among these colours, whereas violet has the shortest. As we previously established, the angle of departure increases with decreasing wavelength. Due to the fact that light propagates at different speeds, it “bends” or is “refracted” when it travels through a medium. At this point, light passing through a prism is deflected in the direction of the triangle’s base. Each of the many colours that make up light has a unique wavelength. Because of this, each of them bends at a different rate depending on its wavelength, with violet bending at the fastest rate because it has the shortest wavelength and red bending at the slowest rate because it has the longest. As a result, the spectrum of colours in white light are separated into their individual colours when it is refracted via a prism.

Summary

Refraction, which is the bending of light as it travels through two distinct media, has a variety of uses. Light slows down and bends when it passes through a prism. The pool appears to be shallower than it actually is. This results from the way light beams from the water’s bottom curve when they exit the water and enter the atmosphere. Have you ever noticed the water layer forming over a short distance in a desert or on a road on a sunny day? “Mirage” is the name given to this occurrence.

Frequently asked questions

1. Why do stars twinkle in the night sky?

Ans: A significant factor in this phenomenon is atmospheric refraction. The refraction of light caused by the earth’s atmosphere, which is made up of air layers with various optical densities, is referred to as “atmospheric refraction.” Light beams from stars are constantly changing their direction as they pass through the earth’s atmosphere due to the changing optical density of the atmosphere. It might affect the amount of starlight that reaches our eyes. The stars in the night sky appear to twinkle as a result.

2. How many refraction patterns are possible for a light beam when it passes through a prism? Explain.

Ans: The speed of the beam may decrease as it passes through air toward a prism, finally slowing down and bending. It also experiences additional refraction as it passes through the prism. Snell’s law of refraction allows us to draw this conclusion. This law states that a light beam moving from a rarer to a denser material may slant in the direction of normal. In a similar way, light beams can stray from the usual when they move from a denser to a rarer medium. As a result, there are two possible refractions.

3. Identify the graph and predict which colour of VIBGYOR has the minimum deviation.

Ans: This is a plot along the X and Y axes against the angle of incidence and the angle of deviation (a). This graph shows how a light beam deviates when it passes through a glass prism. White light splits up into its individual colours when it enters the prism. The wavelength of the light coming in determines how far the deviation extends. Light deviates the least and has the highest wavelength. Red has the highest wavelength among VIBGYOR. Red therefore deviates the least from the norm.

 

Introduction

We know that a capacitor consists of two plates of conductors separated by an isolated distance and is also known as a dielectric. The capacitor limits or regulates the current when connected to an alternating current source, but it does not completely prevent charge drift. The capacitor gradually charges and discharges as the current reverses throughout each half-cycle. The highest charging current occurs while the capacitor’s plates are not charged, hence the charging process is not linear or instantaneous. Similar to the capacitor, once it is completely charged, its charge starts to drop dramatically. The capacity of a capacitor to hold a charge on its plates is known as capacitance. When a capacitor is connected to a voltage source in a DC circuit, current flows for the brief period of time required to charge the capacitor. The voltage across the conductive plates increases as charge accumulates on them, reducing the current. The circuit current zeroes out after the capacitor is fully charged.

Capacitance in AC circuits and capacitive reactance

A capacitor’s estimated capacity to store energy in an AC circuit is known as capacitance. The ratio of an electric charge to the corresponding difference in its electric potential is known as capacitance.

$$C=\frac{d Q}{d V}$$

Where dQ and dV are the charge and potential difference across capacitors, respectively. The capacitance may also be defined as the property of a capacitor to store the charge. The correlation between charging current (I) and the capacitors at which the capacitors supply voltage changes is given by 

$$I=C \frac{d Q}{d V}$$

Capacitive reactance

Capacitive reactance is the resistance to the flow of electricity through the AC capacitor. It is calculated in ohm and denoted by \(X_C\) and measured in the units of Ω. It is calculated mathematically using the provided formula.

$$X_C=\frac{1}{2 \pi f C}=\frac{1}{\omega C}$$

Where f is the frequency, C is the capacitance and ⍵=2πf.

The ratio of the effective current to the voltage across the capacitor is another way to describe the capacitive reactance. We get the conclusion that capacitive reactance is inversely linked to frequency from the aforementioned connection. This implies that a drop in frequency across the capacitor will result in a decrease in capacitive reactance, and vice versa.

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How does a capacitor work in AC?

The capacitor is directly linked to the AC supply in an AC circuit. The capacitor goes through a process of charging or discharging and blocks DC when an AC source is applied. The capacitor also partially obstructs the AC signal. Reactance is the term used to describe a capacitor’s properties in reaction to an AC signal. The capacitor has a short circuit in AC.

AC Capacitor Circuits?

An AC capacitor circuit directly connects the AC supply to the capacitor to allow current to flow through the circuit. The capacitor’s plates are constantly being charged and discharged as a result of the AC supply.

Role of capacitor in AC circuit

As long as there is a source, the capacitor will constantly charge and discharge. The time constant, however, governs whether it fully charges (transforms electrical energy into charge to store between two plates) or fully discharges (charges into electrical energy). We must use a load to charge a capacitor. The time constant is RC, where C is the capacitance and R is the load resistance of the circuit. The capacitor starts to charge when a power source is placed in its path. When fully charged, it will wait for the appropriate time to release the energy it has accumulated.

Role of capacitor in DC circuit

The capacitor starts to charge as soon as a DC supply is connected since DC sources have continuous voltage. Once fully charged, it will wait for the right time to release the charge it has saved. The outcome is that it is an open circuit after being fully charged. As a result, the capacitor acts as a component of an open circuit. The charge is continually charged and discharged with an alternating current, though, due to the variable voltage. The capacitor, therefore, performs the role of a resistor. In this instance, reactance is used in place of resistance, and a capacitor’s reactance is equal to

$$
\frac{1}{2 \pi f C} .
$$

The function of a capacitor in an AC circuit

Electrical circuits contain capacitors, which store electrical energy and raise the circuit’s power factor.

$$
\text { Power factor }=\frac{\text { Real Power }}{\text { Apparent Power }}
$$

AC through the capacitor (Derivation)

Suppose Q is the charge on the capacitor at a given time t, and the instantaneous voltage is V across the capacitor, then we can write,

$$
V=\frac{Q}{C}
$$

The voltage across the source and the capacitor is uniform. Then, according to Kirchhoff’s loop rule

$$
V=V_m \sin \omega t
$$

From the above two equations, we can write that,

$$
V_m \sin \omega t=\frac{Q}{C}
$$

Again,

$$
I=\frac{d Q}{d t}
$$

$$
I=\frac{d}{d t}\left(C V_m \sin (\omega t)\right)=\omega C V_m \cos (\omega t)
$$

Now, as we know,

$$
\begin{gathered}
\cos (\omega t)=\sin \left(\omega t+\frac{\pi}{2}\right) \\
I=I_m \sin \left(\omega t+\frac{\pi}{2}\right) \\
I_m=\frac{V_m}{\left(\frac{1}{\omega C}\right)}
\end{gathered}
$$

\(\frac{1}{2 \pi f C} \) is the capacitive reactance and is denoted by \(X_C\).

So,

$$
I_m=\frac{V_m}{X_C}
$$

Summary

The capacitor is an electrical part that creates a direct connection with the voltage of the source of alternating current. The capacitor alters its charge or discharge in response to a change in the supply voltage. With no real current travelling through the capacitor, the circuit’s current will first flow in one direction before switching to the other. In a circuit with direct current, things are different. The capacitor plate contains both positive and negative charges when current passes through it when it is linked to a direct current circuit. In many diverse sectors, including energy storage, filters, rectifiers, and other things, capacitors are used. Additionally, it is utilised in circuits to increase voltage and smooth out current swings.

Frequently Asked Questions

1. What is capacitive reactance?

Ans: The capacitive reactance in an electric circuit is the resistance that a capacitor presents to the flow of alternating current

2. State Kirchhoff’s voltage law.

Ans: The algebraic sum of potential differences and electromotive forces is zero in a closed loop.

3. State the role of the capacitor in the AC circuit.

Ans: The charge is continually charged and discharged in an AC circuit due to variable voltage. The capacitor, therefore, performs the role of a resistor. In this case, reactance is used in place of resistance, and a capacitor’s reactance is equal to \(\frac{1}{2 \pi f C} \).

4. State the role of the capacitor in the DC circuit.

Ans: The capacitor starts to charge as soon as a DC supply is connected because a DC source’s voltage is constant. Once fully charged, it will wait for the right time to release the charge it has saved. The outcome is that it is an open circuit after being fully charged. As a result, the capacitor acts as a component of an open circuit.

5. What is an electrolytic capacitor?

Ans: An electrolytic capacitor is a capacitor in which ion mobility makes conduction feasible. A liquid or gel with a high ion concentration is called an electrolyte.