Q.Suppose that the electric field amplitude of an electromagnetic wave is $E_0 = 120\ \text{N/C}$ and that its frequency is $\nu = 50.0\ \text{MHz}$.

Concept: Electromagnetic Wave Relation — in a vacuum, $E$ and $B$ are linked by $c$, and the wave parameters $\omega$, $k$, $\lambda$ follow from frequency.

(a)

Speed of light $c = 3.00 \times 10^8\ \text{m/s}$.

Magnetic amplitude:

Angular frequency:

Wavelength:

Wave number:

(b)

Assuming propagation along $+x$ with $\mathbf{E}$ along $y$ and $\mathbf{B}$ along $z$:

$\mathbf{E}(x,t) = (120\ \text{N/C})\, \sin(1.05x - 3.14 \times 10^8 t)\,\hat{\mathbf{j}}$ …

The core idea is that in an electromagnetic wave, the electric and magnetic fields are intimately linked through Maxwell’s equations. In free space, the wave speed is $c = 3.00 \times 10^8\ \text{m/s}$, and the amplitudes satisfy $E_0 = c B_0$. The frequency $\nu$ tells us how fast the fields oscillate, and from it we get the angular frequency $\omega = 2\pi\nu$. The wavelength $\lambda$ follows from $\lambda = c / \nu$, and the wave number $k = 2\pi / \lambda$. Once we have these parameters, we can write the full expressions for $\mathbf{E}$ and $\mathbf{B}$ as sinusoidal waves, choosing a direction of propagation (say $+x$) and mutually perpendicular polarization directions.

Let’s work through each part.

1. Magnetic amplitude $B_0$ The relation between the peak electric and magnetic fields in vacuum is $E_0 = c B_0$. So

This is a very small field — typical for radio waves.

2. Angular frequency $\omega$ $\omega = 2\pi \nu$. With $\nu = 50.0\ \text{MHz} = 50.0 \times 10^6\ \text{Hz}$,

(Using $\pi \approx 3.14$.)

3. Wavelength $\lambda$ For any wave, $\lambda = c / \nu$.

This is in the radio band — about the length of a car.

4. Wave number $k$ $k = 2\pi / \lambda$, so

Alternatively, $k = \omega / c$ gives the same: $3.14 \times 10^8 / 3.00 \times 10^8 = 1.05\ \text{rad/m}$.

Now for part (b): we need expressions for $\mathbf{E}$ and $\mathbf{B}$. We must choose a direction of propagation and orientations for the fields. The standard choice: let the wave travel along the $+x$ axis. Then $\mathbf{E}$ and $\mathbf{B}$ are perpendicular to each other and to the direction of propagation. A common convention is to take $\mathbf{E}$ along the $y$-axis and $\mathbf{B}$ along the $z$-axis. The wave is sinusoidal, so we write:

$\mathbf{B}(x,t) = B_0 \sin(kx - \omega t)\,\hat{\mathbf{k}}.$ …

Method: Standard Wave Parameter Extraction from Field Amplitude and Frequency

This method uses the fundamental relations of electromagnetic waves in vacuum — the wave equation, Maxwell’s speed relation, and the angular frequency/wave number definitions.

Step 1: Identify given data

• $E_0 = 120\ \text{N/C}$
• $\nu = 50.0\ \text{MHz} = 50.0 \times 10^6\ \text{Hz}$
• Speed of light in vacuum: $c = 3.00 \times 10^8\ \text{m/s}$

Step 2: Find $B_0$ (magnetic field amplitude)

From Maxwell’s relation for EM waves in vacuum:

So:

Step 3: Find $\omega$ (angular frequency)

Angular frequency is related to ordinary frequency by:

Step 4: Find $k$ (wave number)

Wave number is related to angular frequency and speed:

Step 5: Find $\lambda$ (wavelength)

Wavelength from frequency:

$\lambda = \frac{3.00 \times 10^8}{50.0 \times 10^6}$ …

Common Mistakes & How to Avoid Them

Mistake 1: Using $c = 3 \times 10^8$ without checking units

The error: Students plug $c = 3 \times 10^8$ directly, forgetting that frequency is given in MHz, not Hz.

How to avoid: Always convert to SI units first.

• $\nu = 50.0\ \text{MHz} = 50.0 \times 10^6\ \text{Hz}$
• Then use $c = 3.00 \times 10^8\ \text{m/s}$

Mistake 2: Confusing $B_0$ formula sign or missing the factor

The error: Using $B_0 = E_0 \times c$ instead of $B_0 = \dfrac{E_0}{c}$.

How to avoid: Remember the fundamental relation:

Correct calculation:

Mistake 3: Mixing up $\omega$ and $\nu$ (angular vs ordinary frequency)

The error: Writing $\omega = \nu$ or $\omega = \frac{1}{\nu}$.

How to avoid: Memorise the exact relation:

Correct:

Mistake 4: Using wrong formula for $k$ (wave number)

The error: Using $k = \frac{2\pi}{\nu}$ or $k = \frac{\omega}{c}$ incorrectly.

How to avoid: Two equivalent correct formulas:

First find $\lambda$:

Then:

Mistake 5: Writing $\mathbf{E}$ and $\mathbf{B}$ expressions with wrong phase or direction

The error: Writing $\mathbf{E}$ and $\mathbf{B}$ as if they are in the same direction, or forgetting the $\hat{k}$ (direction of propagation).

How to avoid: Remember the right-hand rule:

• $\mathbf{E} \times \mathbf{B}$ gives the direction of propagation
• For a wave travelling along $+x$:
  • If $\mathbf{E}$ is along $\hat{j}$ (say $y$-direction) …