IIT JEE and NEET Physics @ Venkats Academy

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Representation of Light as Wave

Representation of the light as a wave

We can consider the light travelling like a wave. The particles of the medium through which the light is passing experience a displacement and it can be represented as shown below. Even before the oscillation starts the particles can have some displacement and it is called as initial phase. The position of the particle with respect to the mean position is called phase. The wave moving along the positive direction and the wave moving along the negative direction are shown with a different signs as shown below. We have also derived a small relation between phase difference and path difference.

We also know that velocity of the wave is the product of frequency of the wave with its wavelength. We shall understand that the velocity of the particle is different from the velocity of the wave.

The intensity of the light at any point is directly proportional to Squire of the amplitude.

Principle of superposition

When two waves are superimposed one over each other, the resultant of displacement is going to be different from individual displacements.

Treating the displacement as a vector, we can calculate the resultant amplitude as shown below. The derivation is made basing on the parallelogram law of vectors. It is clear from the derivation that the maximum possible amplitude of the two waves is equal to sum of the individual waves amplitudes. The minimum possible amplitude of the resultant wave is equal to the difference between the amplitudes of the two waves.

We can calculate the ratio of maximum amplitude to minimum amplitude as shown below. As we know that the intensity is directly proportional to Squire of amplitude we can also calculate the ratio of maximum intensity to minimum intensity.

Doppler effect of light

The apparent change in the frequency due to the relative motion is called Doppler effect. The change in the apparent frequency is not dependent of change in the velocity of the observer. It is simply because when compared with the velocity of the light, the velocity of the observer is significantly small. Therefore the impact of motion of the observer is less on the apparent change in the frequency of light.

We can explain the concept of blue shift and the red shift basing on Doppler Effect of light. When an astronomical body is approaching the earth, its apparent frequency increases. We know that the wavelength is reciprocal of frequency. As the frequency of the approaching body is increasing, its wavelength decreases. Among all the visible colors, violet is having the least possible wavelength but it is not a primary color. As the closest color with the dominating wavelength is blue, the body approaching the earth appears in blue color.

If an astronomical body is going away from the earth, its wavelength increases and it appears like red in color. This is called the red shift of the star.

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Angular Dispersion and Dispersive Power

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Working of Human Eye and Simple Microscope

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Working of a Galilean Telescope

Different Theories of Light and Their Analysis

Doppler Effect and Its Applications

The apparent change in the frequency due to the relative motion of source and observer is called Doppler Effect. We can experience the change in the frequency only when there is a relative motion. The original frequency of the source is not actually changing. Due to the relative motion it is appearing like changing and that’s why it is called as apparent change in frequency.

We can derive the equation for the apparent frequency in different possible cases. When the observer is in the motion he will receive more number of the waves than when he is in the state of rest. It is simply because waves are not only crossing him and he is also crossing the waves.

When the observer is crossing the stationary source there will be difference in the frequencies. If the observer is approaching the apparent frequency increases and when the observer is receding the apparent frequency decreases. The difference between these frequencies can be heard like beats to the observer. We can calculate the number of the beats as shown below.

There will be apparent change in the frequency even when the observer is the state of rest and the source is moving towards the Observer. Here is the source is approaching the observer, its wavelength towards the observer decreases and hence frequency increases. We can derive the equation for the apparent frequency in this case as shown below.

When the observer is in the state of the rest and source is approaching him, apparent frequency increases. When the source is moving away from the stationary observer, apparent frequency decreases. The difference between these two frequencies can be heard like the beats to the observer. We can derive the equation for the number of the beats as shown below.

When a source is revolving around the stationary observer, he is not going to have any Doppler effect experience. It is simply because there is no relative motion between the source and observer. No component of the velocity of the source is acting towards the observer and hence we cannot find any change in the frequency.

If the source is moving in the circular path and observer is far away from the Centre of the circular path, he can hear apparent frequency with different possible frequencies. When the source is moving away from the observer, apparent frequency decreases and vice versa. We can write the equation is as shown below.

When a source is moving by making an angle to the direction of the observer, still there will be apparent change in the frequency due to the component of the velocity of the source towards Observer. We can write the equation for it as shown below.

Problem and solution

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Frequency of Stationary wave in a strecthed String

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Beats and Its Applications

Frequency of Stationary wave in a streched String

Speed of the transverse wave in a string

When a string is attached tightly between the two points there will be tension generated in the string. Linear density of the string can be defined as the mass per unit length of the string. It can be proved that velocity of the string is directly proportional to Squire root of the tension and inversely proportional to Squire root of linear density.We can express the equation different formats as per the requirement as shown.

If Young's modulus of the wire is given with can express the tension in terms of Young’s modulus as shown below.

Problem and solution

We need the find the velocity of the wave in a stretched string using the regular formula as shown below.

Standing waves

Two waves of same amplitude, frequency and velocity moving in opposite directions are superimposed then stationary waves are formed.

The superposition of the waves can be done basing on the vector laws of addition. In the stationary waves there are some points that the displacement is minimum and the points are called nodes. There are some other points where the displacement is maximum and that points are called anti-nodes. The interval between two successive who nodes as well as the anti-nodes is always fixed as shown below.

Depending on the point of disturbance stationary waves can be formed under different modes of vibration. At the point of disturbance the displacement is going to be maximum and there is a formation of anti-node. Depending on the point of disturbance, a string can vibrate under different modes of vibration.

Laws of stretches strings

The frequency of a stretched string is inversely proportional to its length when it’s tension and linear density are kept constant. This is called law of lengths.

The frequency of stretches string is directly proportional to Squire root of the tension when its length and linear densities are constants. This is called law of tensions.

The frequency of a stretched string is inversely proportional to Squire root of the linear density when the length and tension are kept constant. This law is called as law of linear densities.

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Progressive Wave Representation and Problems

Equation of a progressive wave

To represent the displacement of a particle in wave motion we have a mathematical equation.

When the particle is advancing from the origin towards the positive x-axis, any particle who is at a certain distance from the origin will receive the wave lately than the origin by a specified time.

Taking that into consideration we can write a mathematical equation as shown below.

When the wave is moving along the negative x-axis, the particles will receive the vibration not with the time lag but with the time addition. The corresponding representation of the wave in the different possible formats is as shown below.

Relation between phase difference and path difference

This can be obtained basing on the definitions of the basic terms itself. We know that when the particles are separated by a distance equal to wavelength they are going to have a phase difference of 360°. By calculating the phase difference for unit separation we can get the relation as shown.

Intensity of the sound energy is defined as the energy emitted by a body per unit surface area per unit time. When the number of the waves are acting simultaneously on the same point, we can get the resultant of them using the vector laws of addition.

Reflection of waves

When waves travel from one medium to other, a part of it returns to the other medium. When the wave strikes the obstacle, it reflects back in the opposite direction. The string through which the wave is travelling applies a force on a wall as an action. The wall applies the reaction the string in opposite direction which is in satisfaction with the Newton third law.

When the wave travels from wherever medium to denser medium velocity decreases and the pulses inverted upon reflection.

Problem and solution

The audible range of frequencies for a human being leaves from 20 to 20,000 Hz. Express them in terms of the wavelength?

We can solve the problem using the simple relation between where velocity and wavelength. We shall also remind ourselves that when the wave changes its medium its frequency is going to remain constant. Frequency is a characteristic property of the source and it is independent of the medium.

Problem and solution

We are going to solve the following problem basing on the very definition of frequency.It is the number of vibrations made per one second.Then time taken for one vibration is called time period.