HC Verma Questions and Solutions: Chapter 40: Electromagnetic Waves- 1 | HC Verma Solutions - JEE PDF Download

According to Ampere-Maxwell's Law, a magnetic field is produced due to the conduction current in a conductor and the displacement current. The conduction current is actually the motion of the charge. The displacement current is due to the changing electric field. The displacement current is given by 
Thus, the magnetic field is produced by the  moving charge as well as the changing electric field.

The compass needle deflects due to the presence of the magnetic field. Inside the capacitor, a magnetic field is produced when there is a changing electric field inside it. As the capacitor is connected across the battery, the charge on its plates at a certain time t is given by :

where
Q =  charge developed on the plates of the capacitor
R =  resistance of the resistor connected in series with the capacitor 
​C =  capacitance of the capacitor
V =  potential difference of the battery
The time constant of the capacitor is given, τ = RC
The capacitor keeps on charging up to the time τ. The development of charge on the plates will be gradual after t = RC. The change in  electric field will be up to the time the charge is developing on the plates of the capacitor. Thus, the compass needle ​deflects and gradually comes to the original position in a time that is large compared to the time constant.

The speed of light, 
The dimensions of  are of velocity, i.e L/T .
Therefore,  will have dimension L²/T²​.

A static charge produces an electrostatic field. A moving charge produces a magnetic field. Electromagnetic waves are produced by an accelerating charge.

The relation between E₀ and B₀ is given by 
Here, c = speed of the electromagnetic wave
The relation between ​ω (the angular frequency) and k (wave number):

Therefore, from (1) and (2), we get :

For an electromagnetic wave,electric field ,magnetic field and direction of propagation are mutually perpendicular to each other.We can have a region in which electric and magnetic fields are applied at an angle with each other.In transmission lines Different modes exist. In transverse electric (TE) mode-no electric field exist in the direction of propagation. These are sometimes called H modes because there is only a magnetic field along the direction of propagation (H is the conventional symbol for magnetic field).

For a linearly polarised, plane electromagnetic wave

The average value of either E or B over a cycle is zero ( average of sin(θ) over a cycle is zero).
Also the electric energy density (u_E) and magnetic energy density (u_B) are equal .

Energy can be found out by integrating energy density over the entire volume of full space.
As the energy of the electromagnetic wave is equally shared between electric and magnetic field so their average values will also be equal.

As the electron is at rest initially, only the electric field will exert force on it. There will be no magnetic force on the electron in the stating. Hence, the electron will start moving along the electric field.

When an electromagnetic wave strikes a material surface, it transports the momentum, as well as the energy, to the surface. The striking electromagnetic wave exerts pressure on the surface. The total energy transferred to the surface by the electromagnetic wave is given by E = pc. Therefore, ​p ≠ 0, E ≠ 0.

The given quantities can be expressed as :
k is given by

ω is given by 

k/ω is given by 

kω is given by 

Thus, ​k/ω is independent of the wavelength.

Displacement current inside a capacitor, Where   is the electric flux inside the capacitor.
Up to the time the electric flux changes, there will be a displacement current. This is possible when the charge on a capacitor changes. Therefore, the displacement current goes through the gap between the plates of a capacitor when the charge of the capacitor or electric field increases or decreases inside the capacitor.

In a vacuum, the speed of electromagnetic waves is constant for all intensities.

In a plane electromagnetic wave, the electric and the magnetic fields oscillate sinusoidally. For an electromagnetic wave propagating in the z-direction, the electric and magnetic fields are given by:
E_x = E₀ sin (kz – ωt)
B_y = B₀ sin (kz – ωt)
These are sinusoidal functions. Therefore, for a fixed value of z, the average value of the electric and magnetic fields are zero.

The energy per unit volume of an electromagnetic wave,

The energy of the given volume can be calculated by multiplying the volume with the above expression.

Let the direction of propagation of the electromagnetic wave be along the z-axis. Then, the electric and magnetic fields at a particular point are given by
E_x= E₀ sin (kz – ωt)
B_y= B₀ sin (kz – ωt)
Substituting the values of electric and magnetic fields in (1), we get :

From the above expression, it can be easily understood that the energy of the electric and magnetic fields have angular frequency 2ω. Thus, the frequency of the energy of the electromagnetic wave will also be double.