First slide

Introduction-Electromagnetic waves

Question

The electric field of a plane electromagnetic wave propagating along the x direction in vacuum is $\vec{E} = E_{0} \hat{j} \cos (ω t - k x)$ . The magnetic field $\vec{B}$ , at the moment t = 0 is

Moderate

A

$\vec{B} = E_{0} \sqrt{μ_{0} \in_{0}} \cos (k x) \hat{k}$
B

$\vec{B} = \frac{E_{0}}{\sqrt{μ_{0} \in_{0}}} \cos (k x) \hat{j}$
C

$\vec{B} = E_{0} \sqrt{μ_{0} \in_{0}} \cos (k x) \hat{j}$
D

$\vec{B} = \frac{E_{0}}{\sqrt{μ_{0} \in_{0}}} \cos (k x) \hat{k}$

Solution

Amplitude of magnetic field    $B_{0} = \frac{E_{0}}{C}$
   $B_{0} = E_{0} \sqrt{μ_{0} ε_{0}}$
$\vec{E} \times \vec{B}$ is along direction of propagation
$∴ \hat{j} \times \hat{B}$ must be parallel to $\hat{i}$ $∴ \vec{B}$ should be along $\hat{k}$
$\vec{E}, \vec{B}$ are in same phase
$∴ \vec{B} = B_{0} \hat{k} \cos (ω t - k x)$
$\vec{B} = E_{0} \sqrt{μ_{0} ε_{0}} \cos (ω t - k x) \hat{k}$

At t = 0
$\vec{B} = E_{0} \sqrt{μ_{0} ε_{0}} \cos (k x) \hat{k}$

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