Q.A plane electromagnetic wave travelling in vacuum along the $z$-direction is given by $\mathbf{E} = E_0\sin(kz - \omega t)\,\hat{i}$ and $\mathbf{B} = B_0\sin(kz - \omega t)\,\hat{j}$. Consider a rectangular loop $1234$ lying in the $x$-$z$ plane. Its two vertical sides run parallel to the $x$-axis and have length $h$: corner $1$ (bottom) and corner $4$ (top) lie on the left, at position $z = z_1$; corner $2$ (bottom) and corner $3$ (top) lie on the right, at position $z = z_2$. The two horizontal sides ($1\to 2$ along the bottom and $4\to 3$ along the top) run parallel to the $z$-axis. The loop is traversed in the order $1\to 2\to 3\to 4\to 1$.

1. $\oint\mathbf{E}\cdot d\mathbf{l}=E_0 h\,[\sin(kz_2-\omega t)-\sin(kz_1-\omega t)]$.
2. $\Phi_B=\dfrac{B_0 h}{k}[\cos(kz_1-\omega t)-\cos(kz_2-\omega t)]$.
3. Faraday's law $\Rightarrow E_0=\dfrac{\omega}{k}B_0=cB_0$. …

(i) Line integral of $\mathbf{E}$ around loop 1234

Go round $1\to 2\to 3\to 4\to 1$. Since $\mathbf{E}=E_0\sin(kz-\omega t)\hat{i}$ is along $\hat{x}$:

• Sides $1\to 2$ (bottom) and $3\to 4$ (top) run along $\pm\hat{z}$, so $\mathbf{E}\cdot d\mathbf{l}=0$ there.
• Side $2\to 3$ runs along $+\hat{x}$ at fixed $z=z_2$: contributes $\displaystyle\int_0^h E_0\sin(kz_2-\omega t)\,dx=E_0 h\sin(kz_2-\omega t)$.
• Side $4\to 1$ runs along $-\hat{x}$ at fixed $z=z_1$: contributes $-E_0 h\sin(kz_1-\omega t)$.

(ii) Magnetic flux through the loop

The loop lies in the $x$-$z$ plane, so its area element is $d\mathbf{s}=dx\,dz\,\hat{j}$, parallel to $\mathbf{B}=B_0\sin(kz-\omega t)\hat{j}$:

(iii) Faraday's law $\Rightarrow E_0/B_0=c$

Differentiate the flux (using $\frac{d}{dt}\cos(kz-\omega t)=\omega\sin(kz-\omega t)$):

Setting $\oint\mathbf{E}\cdot d\mathbf{l}=-\dfrac{d\Phi_B}{dt}$ and cancelling the common factor $h\,[\sin(kz_2-\omega t)-\sin(kz_1-\omega t)]$:

since the phase speed of the wave is $\omega/k=c$.

(iv) Ampère-Maxwell law $\Rightarrow c=1/\sqrt{\mu_0\varepsilon_0}$

Repeat the argument with a rectangular loop of the same shape but lying in the $y$-$z$ plane, so that $\mathbf{B}$ (along $\hat{y}$) does the line integral and $\mathbf{E}$ (along $\hat{x}$, normal to that loop) supplies the flux. In vacuum the conduction current $I=0$, so …

Method: Deriving the Wave Relations $E_0/B_0=c$ and $c=1/\sqrt{\mu_0\varepsilon_0}$ from Maxwell's Loop Laws

This method applies to any question that gives explicit sinusoidal $\mathbf{E}$ and $\mathbf{B}$ expressions for a plane wave and asks you to derive a relation between them using a rectangular Amperian/Faraday loop.

Steps

Step 1: Choose a rectangular loop aligned with the fields, and identify which sides contribute to the line integral

A field that points along one axis (say $\hat{x}$) contributes to $\oint \vec{F}\cdot d\vec{l}$ only along the loop sides that run parallel to that axis; sides running perpendicular to the field give zero contribution. Evaluate the field at each contributing side's fixed coordinate and multiply by the side's length.

Step 2: Compute the flux of the perpendicular field through the same loop

The loop's area vector (normal) is fixed by its orientation. Only the field component parallel to that normal contributes to flux. Integrate that field over the loop's area (a double integral over the two in-plane coordinates) to get $\Phi = \int \vec{F}\cdot d\vec{s}$.

Step 3: Apply the appropriate Maxwell equation and cancel the common spatial factor …