Electromagnetic Waves Test

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Electromagnetic Waves Test - 18

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Weekly Quiz Competition

Question 1
1 / -0

The charge on a parallel plate capacitor is varying as $$q={ q }_{ 0 }\sin { \left( 2\pi nt \right) } $$. The plates are very large and close together. neglecting the edge effects, the displacement current through the capacitor is:

A
$$\cfrac { q }{ { \varepsilon }_{ 0 }A } $$

B
$$\cfrac { { q }_{ 0 } }{ { \varepsilon }_{ 0 } } \sin { 2\pi } nt$$

C
$$2\pi n{ q }_{ 0 }\cos { 2\pi } nt $$

D
$$\cfrac { 2\pi n{ q }_{ 0 } }{ { \varepsilon }_{ 0 } } \cos { 2\pi } nt $$

Solution

Displacement current through a surface is $$\epsilon_0$$ times the rate of change of flux through the surface.
Since $$E=\dfrac{q}{A\epsilon_0}$$.
Therefore electric flux through the surface is $$\phi = EA=\dfrac{q}{\epsilon_0}$$
So real current though the capacitor $$i = \dfrac{d\phi}{dt} = \epsilon_0 \dfrac{d}{dt}(\dfrac{q}{\epsilon_0})=\dfrac{dq}{dt}=i$$
Thus displacement current is $$\dfrac{dq}{dt}=2\pi n q_0\cos2\pi ft$$
Question 2
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Directions For Questions

A plane electromagnetic wave frequency $$40MHz$$ travels in free space in the X-direction. At some point and at some distant, the electric field has its maximum value of $$750N/C$$ in Y-direction.

...view full instructions

The period of the wave will be

A
$$2.5\mu s$$

B
$$0.25\mu s$$

C
$$0.025\mu s$$

D
none of these

Solution

Time period, $$T= \dfrac{1}{\nu}$$
$$T= \dfrac{1}{40 \times 10^6}$$
$$\implies T= .025 \mu s$$
Question 3
1 / -0

Directions For Questions

A parallel plate capacitor made of circular plates each of radius $$R=6\ cm$$ has capacitance $$C=100\ pF$$. The capacitor is connected to a $$230\ V$$ AC supply with an angular frequency of $$300\ rad/s$$.

...view full instructions

The $$rms$$ value of conduction current will be:

A
$$5.7\mu A$$

B
$$6.3\mu A$$

C
$$9.6\mu A$$

D
$$6.9\mu A$$

Solution

Rms value of conduction current, $$I=V/X_c$$
Where,
$$X_c$$ = Capacitive reactance
$$=1/ωC$$
∴ $$ I = V × ωC $$
= $$230\times 300\times 100\times 10^{-12}$$
= $$6.9\times 10^{-6} $$ A
= $$6.9$$ μA
Question 4
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Maxwell's four equations are written as
(i) $$\oint { \overrightarrow { E } .\overrightarrow { d } s } =\cfrac { { q }_{ 0 } }{ { \varepsilon }_{ 0 } } \quad $$
(ii)$$\oint { \overrightarrow { B } .\overrightarrow { d } s } =0$$
(iii)$$\oint { \overrightarrow { E } .\overrightarrow { d } l } =\cfrac { d }{ dt } \quad \oint { \overrightarrow { B } .\overrightarrow { d } s } $$
(iv)$$\oint { \overrightarrow { B } .\overrightarrow { d } s } ={ \mu }_{ 0 }{ \varepsilon }_{ 0 }\cfrac { d }{ dt } \quad \oint { \overrightarrow { E } .\overrightarrow { d } s } $$
Which of the above Maxwell's equations shows that electric field lines do not form closed loops?

A
(i) only

B
(ii) only

C
(iii) only

D
(iv) only

Solution

$$\oint \vec{E}.\vec{d}s $$ represents electric flux which is equal to the number of electric field lines crossing through an area.
Now for $$q_o \neq 0$$, implies $$\oint \vec{E}.\vec{d}s \neq 0 $$ which indicates Electric field lines do not form closed loops.
Question 5
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Directions For Questions

A plane electromagnetic wave frequency $$40MHz$$ travels in free space in the X-direction. At some point and at some distant, the electric field has its maximum value of $$750N/C$$ in Y-direction.

...view full instructions

The wavelength of the wave is

A
$$3.5m$$

B
$$5.5m$$

C
$$7.5m$$

D
$$9.5m$$

Solution

Using $$v=\nu \lambda$$ where $$v$$ is the speed of EM wave.
As $$v= c= 3 \times 10^8 m/s $$
$$\implies 3 \times 10^8 = 40 \times 10^6 \lambda$$
$$\implies \lambda = 7.5 m$$
Question 6
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Directions For Questions

A plane electromagnetic wave frequency $$40MHz$$ travels in free space in the X-direction. At some point and at some distant, the electric field has its maximum value of $$750N/C$$ in Y-direction.

...view full instructions

The angular frequency of emf wave will be (in rad/s)

A
$$8p\times { 10 }^{ 7 }$$

B
$$4p\times { 10 }^{ 6 }$$

C
$$2p\times { 10 }^{ 5 }$$

D
$$\pi \times { 10 }^{ 4 }$$

Solution

Frequency of wave $$f = 40\times 10^6 \ Hz$$
Angular frequency, $$\omega = 2\pi \nu$$
$$\omega = 2\pi \times 40 \times 10^6$$
$$\implies \omega = 8 \pi \times 10^7 \ rad/s$$
Question 7
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Maxwell's four equations are written as
(i) $$\oint { \overrightarrow { E } .\overrightarrow { d } s } =\cfrac { { q }_{ 0 } }{ { \varepsilon }_{ 0 } } \quad $$
(ii)$$\oint { \overrightarrow { B } .\overrightarrow { d } s } =0$$
(iii)$$\oint { \overrightarrow { E } .\overrightarrow { d } l } =\cfrac { d }{ dt } \quad \oint { \overrightarrow { B } .\overrightarrow { d } s } $$
(iv)$$\oint { \overrightarrow { B } .\overrightarrow { d } s } ={ \mu }_{ 0 }{ \varepsilon }_{ 0 }\cfrac { d }{ dt } \quad \oint { \overrightarrow { E } .\overrightarrow { d } s } $$
Out of the four Maxwell's equation above, which one shows the non-existence of monopoles?

A
(i) and (iv)

B
(ii) only

C
(iii) only

D
none of these

Solution

Had there been a monopole, the integral of $$\vec{B}.\vec{ds}$$ on a closed surface enclosing the monopole would have been non-zero.
Thus the second equation shows the non-existence of monopole.
Question 8
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The Sun delivers $${{10}^{3}}W/{{m}^{2}}$$ of electromagnetic flux to the Earth's surface. The radiation force on the roof of dimensions $$8m\times 20m$$ will be:

A
$$3.33\times { 10 }^{ -5 }N$$

B
$$5.33\times { 10 }^{ -4 }N$$

C
$$7.33\times { 10 }^{ -3 }N$$

D
$$9.33\times { 10 }^{ -2 }N$$

Solution

Poynting vector, $$S={10}^3 W/m^2$$
Pressure, $$P=\dfrac{S}{c}=$$$$\dfrac{1000}{3\times 10^8}$$
$$=3.33\times {10}^{-6} N/m^2$$
Radiation force on the roof is given by:
$$F=P\times 8\times 20 = 5.33 \times {10}^{-4}N$$
Question 9
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Directions For Questions

A plane electromagnetic wave frequency $$40MHz$$ travels in free space in the X-direction. At some point and at some distant, the electric field has its maximum value of $$750N/C$$ in Y-direction.

...view full instructions

The propagation constant of the wave will be

A
$$8.38{{m}^{-1}}$$

B
$$0.838{{m}^{-1}}$$

C
$$4.19{{m}^{-1}}$$

D
$$0.419{{m}^{-1}}$$

Solution

Using $$v=\nu \lambda$$ where $$v$$ is the speed of EM wave.
As $$v= c= 3 \times 10^8 m/s $$
$$\implies 3 \times 10^8 = 40 \times 10^6 \lambda$$
$$\implies \lambda = 7.5 m$$
Propagation constant, $$K= \dfrac{2\pi}{\lambda}= \dfrac{2\pi}{7.5}$$
$$\implies K= 0.838 m^{-1}$$
Question 10
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An electromagnetic wave, going through vacuum is described by $$E=E_0\sin{(kx-\omega t)}$$. Which of the following is independent of wavelength?

A
$$k$$

B
$$\omega$$

C
$$\displaystyle\dfrac{k}{\omega}$$

D
$$k\omega$$

Solution

$$k=\dfrac{2\pi}{\lambda}$$

$$\omega = \dfrac{2\pi c}{\lambda}, $$where $$c$$ is the velocity of light

Hence, $$\dfrac{k}{2\pi} = \dfrac{\omega}{2\pi c} \implies \dfrac{k}{\omega}$$ is independent of the wavelength.

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Re-Attempt Weekly Quiz Competition

Electromagnetic Waves Test - 18

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Electromagnetic Waves Test - 18

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Question 1

$$\cfrac { q }{ { \varepsilon }_{ 0 }A } $$

$$\cfrac { { q }_{ 0 } }{ { \varepsilon }_{ 0 } } \sin { 2\pi } nt$$

$$2\pi n{ q }_{ 0 }\cos { 2\pi } nt $$

$$\cfrac { 2\pi n{ q }_{ 0 } }{ { \varepsilon }_{ 0 } } \cos { 2\pi } nt $$

Solution

Question 2

$$2.5\mu s$$

$$0.25\mu s$$

$$0.025\mu s$$

none of these

Solution

Question 3

$$5.7\mu A$$

$$6.3\mu A$$

$$9.6\mu A$$

$$6.9\mu A$$

Solution

Question 4

(i) only

(ii) only

(iii) only

(iv) only

Solution

Question 5

$$3.5m$$

$$5.5m$$

$$7.5m$$

$$9.5m$$

Solution

Question 6

$$8p\times { 10 }^{ 7 }$$

$$4p\times { 10 }^{ 6 }$$

$$2p\times { 10 }^{ 5 }$$

$$\pi \times { 10 }^{ 4 }$$

Solution

Question 7

(i) and (iv)

(ii) only

(iii) only

none of these

Solution

Question 8

$$3.33\times { 10 }^{ -5 }N$$

$$5.33\times { 10 }^{ -4 }N$$

$$7.33\times { 10 }^{ -3 }N$$

$$9.33\times { 10 }^{ -2 }N$$

Solution

Question 9

$$8.38{{m}^{-1}}$$

$$0.838{{m}^{-1}}$$

$$4.19{{m}^{-1}}$$

$$0.419{{m}^{-1}}$$

Solution

Question 10

$$k$$

$$\omega$$

$$\displaystyle\dfrac{k}{\omega}$$

$$k\omega$$

Solution

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