Physics M.C.Q (1 Marks)

4. Double the frequency of the wave.

Explanation:

The energy per unit volume of an electromagnetic wave,

$\text{u}=\frac{1}{2}\in_0\text{E}^2+\frac{\text{B}^2}{2\mu_0}$

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

$\text{U}=\text{u}\times\text{V}=\Big(\frac{1}{2}\in_0\text{E}^2+\frac{\text{B}^2}{2\mu_0}\Big)\times\text{V}\ ....(\text{i})$

Let the direction of propagation of the electromagnetic wave be along the z-axls. Then, the electric and magnetic fields at a particular point are given by,

$\text{E}_\text{x}=\text{E}_0\sin(\text{kz}-\omega\text{t})$

$\text{B}_\text{y}=\text{B}_0\sin(\text{kz}-\omega\text{t})$

Substituting the values of electric and magnetic fields in (1) we get,

$\text{U}=\Big(\frac{1}{2}\in_0\big(\text{E}_0^2\sin^2(\text{kz}-\omega\text{t})+\frac{\text{B}^2_0\sin^2(\text{kz}-\omega\text{t})}{2\mu_0}\Big)\times\text{V}$

$\text{U}=\Big(\in_0\text{E}^2_0\frac{(1-\cos(2\text{kz}-2\omega\text{t}))}{4}+\frac{\text{B}_0^2(1-\cos(2\text{kz}-2\omega\text{t}))}{4\mu_0}\Big)\times\text{V}$

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

3. $\text{p}\neq0,\text{ E}\neq0$

Explanation:

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 $\text{E}=\text{pc}$ Therefore, $\text{p}\neq0,\text{ E}\neq0$

1. Increases.
2. Decreases.

Explanation:

Displacement current inside a capacitor,

$\text{i}_\text{d}=\in_0\frac{\text{d}\phi_\text{E}}{\text{dt}},$ where

$\phi_\text{E}$ 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.

3. $\frac{\text{L}^2}{\text{T}^2}$

Explanation:

The speed of light, $\text{C}=\frac{1}{\sqrt{\mu_0\in_0}}$

The dimensions of $\frac{1}{\sqrt{\mu_0\in_0}}$ are of velocity, i.e., $\frac{\text{L}}{\text{T}}$

Therefore, $\frac{1}{\in_0\mu_0}$ will have dimensions $\frac{\text{L}^2}{\text{T}^2}$

3.  $\frac{\text{k}}{\omega}$

Explanation:

The given quantities can be expressed as,

k is given by,

$\omega$ is given by,

$\omega=2\pi\text{v}$

$\text{c}=\text{v}\lambda$

$\omega=2\pi\frac{\text{c}}{\lambda}$

$\frac{\text{k}}{\omega}$ is given by,

$\frac{\text{k}}{\omega}=\frac{\frac{2\pi}{\lambda}}{\frac{2\pi\text{c}}{\lambda}}=\frac{1}{\text{c}}$

$\text{k}\omega$ is given by,

$\text{k}\times\omega=\frac{2\pi}{\lambda}\times\frac{2\pi\text{c}}{\lambda}=\frac{4\pi^2\text{c}}{\lambda^2}$

Thus, $\frac{\text{k}}{\omega}$ is independent of the wavelength. 

1. $\text{E}_0\text{k}=\text{B}_0\omega$

Explanation:

The relation between E₀ and B₀ id given by $\frac{\text{E}_0}{\text{B}_0}=\text{c}\ ....(\text{i})$

Here, c = Speed of the electromagnetic wave,

The relation between $\omega$ (the angular frequency) and k(wave number),

$\frac{\omega}{\text{k}}=\text{c}\ ...(\text{ii})$

Therefore, from (i) and (ii), we get

$\frac{\text{E}_0}{\text{B}_0}=\frac{\omega}{\text{k}}=\text{c}$

$\text{E}_0\text{k}=\text{B}_0\omega$

4. Deflects and gradually comes to the original position in a time that is large compared to the time constant.

Explanation:

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 tis given by,

$\text{Q}=\text{CV}\Big(1-\text{e}^{-\tau/\text{RC}}\Big),$

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, $\tau=\text{RC}$

The capacitor keeps on charging up to the time $\tau$. The development of charge on the plates will be gradual after $​​\tau=\text{RC}$ 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.

4. Both of them.

Explanation:

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,

$\text{i}_\text{d}=\in_0\frac{\text{d}\phi_\text{E}}{\text{dt}}$ $\big(\because\phi_\text{E}$ is the electric flux$\big)$

Thus, the magnetic field is produced by the moving charge as well as the electric field.

3. For all intensities.

Explanation:

For any given medium, the speed (c) of an electromagnetic wave is given by,

$\text{C}=\text{v}\lambda$

Where,

V = Frequency of the electromagnetic wave.

$\lambda=$ wavelength of the electromagnetic wave.

As the frequency and wavelength are changed, the speed of the electromagnetic wave changes. So, the speed of an electromagnetic wave is not same for all wavelengths and all frequencies in any medium. The velocity of an electromagnetic wave changes with change in medium. Also, the speed of an electromagnetic wave is same for all the intensities in any medium.

1. Electric field.
2. Magnetic field.

Explanation:

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,

$\text{E}_\text{x}=\text{E}_0\sin(\text{kz}-\omega\text{t})$

$\text{B}_\text{y}=\text{B}_0\sin(\text{kz}-\omega\text{t})$

These are sinusoidal functions. Therefore, for a fixed value of z. the average value of the electric and magnetic fields are zero.

1. Both A and B are true.

Explanation:

For a linearly polarised, plane electromagnetic wave,

$\text{E}=\text{E}_0\sin\omega\Big(\text{t}-\frac{\text{x}}{\text{c}}\Big)$

$\text{B}=\text{B}_0\sin\omega\Big(\text{t}-\frac{\text{x}}{\text{c}}\Big)$

The average value of either E or Bover a cycle is zero (average of $\sin(\theta)$ over a cycle is zero).

Also the electric energy density (U_E) and magnetic energy density (U_B)are equal.

$\text{u}_\text{E}=\frac{1}{2}\in_0\text{E}^2=\frac{\text{B}^2}{2\mu_0}=\text{u}_\text{B}$

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.