Modern Physics 7 Question 15

15. A plane electromagnetic wave travels in free space along the $x$-direction. The electric field component of the wave at a particular point of space and time is $E=6 Vm^{-1}$ along $y$-direction. Its corresponding magnetic field component, $B$ would be

(Main 2019, 8 April I)

(a) $2 \times 10^{-8} T$ along $z$-direction

(b) $6 \times 10^{-8} T$ along $x$-direction

(c) $6 \times 10^{-8} T$ along $z$-direction

(d) $2 \times 10^{-8} T$ along $y$-direction

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Solution:

  1. Key Idea For an electromagnetic wave, ratio of magnitudes of electric and magnetic field is

$$ \frac{E}{B}=c $$

where, $c$ is the speed of electromagnetic wave in vacuum.

Given, $E=6 V / m, c=3 \times 10^{8} ms^{-1}$

So, $\quad B=\frac{E}{c}=\frac{6}{3 \times 10^{8}}=2 \times 10^{-8} T$

Also, direction of propagation of electromagnetic wave is given by

$$ \hat{\mathbf{n}}=\mathbf{E} \times \mathbf{B} $$

Here, $\hat{\mathbf{n}}=\hat{\mathbf{i}}$ and $\mathbf{E}=$ Unit vector of electric field $(\hat{\mathbf{j}})$

$\mathbf{B}=$ unit vector of magnetic field.

$\Rightarrow \quad \hat{\mathbf{i}}=\hat{\mathbf{j}} \times \mathbf{B} \Rightarrow \mathbf{B}=\hat{\mathbf{k}}$

Hence, magnetic field components,

$$ \mathbf{B}=2 \times 10^{-8} \hat{\mathbf{k}} T=2 \times 10^{-8} T $$

(along $z$-direction)



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