Q.A plane electromagnetic wave travels in vacuum along $z$-direction. What can you say about the directions of its electric and magnetic field vectors? If the frequency of the wave is $30\ \text{MHz}$, what is its wavelength?

Concept: Electromagnetic Wave Relation — in vacuum, $\vec{E}$, $\vec{B}$, and the propagation direction $\vec{k}$ form a right-handed orthogonal triad.

Reasoning:

1. For a plane wave travelling along $z$, the wave vector is $\vec{k} = k\,\hat{z}$.
2. Maxwell’s equations demand $\vec{E} \perp \vec{k}$ and $\vec{B} \perp \vec{k}$, so both fields lie in the $xy$-plane.
3. Additionally, $\vec{E} \perp \vec{B}$ and the direction $\vec{E} \times \vec{B}$ gives $\vec{k}$. Thus if $\vec{E}$ is along $x$, $\vec{B}$ must be along $y$ (or vice versa). …

The key to this question is the electromagnetic wave relation — a set of three orthogonal directions that every plane EM wave obeys in vacuum. Let’s build the intuition first.

Why this approach works

An electromagnetic wave is a self-sustaining oscillation of electric and magnetic fields. In vacuum, Maxwell’s equations force a strict geometry: the electric field $\vec{E}$, the magnetic field $\vec{B}$, and the direction of propagation $\vec{k}$ (the wave vector) are always mutually perpendicular. This is not an assumption — it’s a consequence of Faraday’s law and Ampère’s law working together.

Think of it like this: if the wave travels along $z$, then $\vec{E}$ cannot have a $z$-component, because that would mean the wave is longitudinal (like sound), but EM waves are transverse. Similarly, $\vec{B}$ cannot have a $z$-component. So both fields lie entirely in the $xy$-plane. Moreover, $\vec{E}$ and $\vec{B}$ are perpendicular to each other — if $\vec{E}$ points along $x$, then $\vec{B}$ must point along $y$ (or vice versa), and the cross product $\vec{E} \times \vec{B}$ gives the direction of propagation.

Now, the second part is a straightforward application of the wave equation: $c = f \lambda$.

1. Direction of $\vec{E}$ and $\vec{B}$

   The wave travels along $z$. Therefore, both $\vec{E}$ and $\vec{B}$ have no $z$-component — they are transverse. They lie in the $xy$-plane.

   Additionally, $\vec{E}$ and $\vec{B}$ are perpendicular to each other. So if $\vec{E}$ is along $x$, then $\vec{B}$ is along $y$ (or the negative $y$, depending on phase). The exact orientation depends on the source, but the key point is: both are perpendicular to $z$, and to each other.

2. Wavelength from frequency …

Method: Right-Hand Rule for EM Wave Propagation

This problem uses the right-hand rule for electromagnetic waves and the wave equation relation $c = f \lambda$.

Step 1: Direction of Electric and Magnetic Fields

In a plane electromagnetic wave traveling in vacuum:

• The electric field $\vec{E}$ and magnetic field $\vec{B}$ are perpendicular to each other.
• Both are perpendicular to the direction of propagation.

Since the wave travels along the $z$-direction:

• $\vec{E}$ and $\vec{B}$ lie in the $xy$-plane.
• Using the right-hand rule: if your thumb points in the direction of propagation ($+z$), your fingers curl from $\vec{E}$ to $\vec{B}$.

Answer: The electric and magnetic field vectors are perpendicular to each other and both lie in the plane perpendicular to the $z$-axis (the $xy$-plane).

Step 2: Wavelength Calculation

Use the fundamental wave relation for electromagnetic waves in vacuum:

Where: …

Common Mistakes & How to Avoid Them

Mistake 1: Confusing the Direction of Propagation with Field Directions

The Mistake:

Students often think that if the wave travels along the $z$-axis, then both $\vec{E}$ and $\vec{B}$ also point along $z$.

Why it's wrong:

Electromagnetic waves are transverse — the electric and magnetic fields oscillate perpendicular to the direction of propagation. For a wave traveling along $z$, both $\vec{E}$ and $\vec{B}$ lie in the $xy$-plane.

How to avoid:

• Memorise the right-hand rule: thumb = direction of wave travel ($\vec{k}$), index finger = $\vec{E}$, middle finger = $\vec{B}$.
• Remember: $\vec{E} \times \vec{B}$ gives the direction of propagation. So if propagation is along $z$, then $\vec{E}$ and $\vec{B}$ must be perpendicular to $z$ and to each other.

Mistake 2: Forgetting That $\vec{E}$ and $\vec{B}$ Are Perpendicular to Each Other

The Mistake:

Some students say "both fields are perpendicular to $z$" but forget to mention that $\vec{E} \perp \vec{B}$ as well.

Why it's wrong:

Maxwell's equations demand that in a plane wave, $\vec{E}$ and $\vec{B}$ are mutually perpendicular. If you only say "perpendicular to $z$", you haven't fully described the relationship.

How to avoid:

• Always state: $\vec{E}$ and $\vec{B}$ are perpendicular to each other and both are perpendicular to the direction of propagation.
• Draw a quick sketch: $z$-axis as arrow, $\vec{E}$ along $x$, $\vec{B}$ along $y$.

Mistake 3: Using the Wrong Formula for Wavelength

The Mistake:

Students sometimes use $v = f \lambda$ but plug in the wrong speed (e.g., speed in a medium instead of vacuum).

Why it's wrong:

The problem explicitly says "in vacuum". The speed of light in vacuum is $c = 3 \times 10^8\ \text{m/s}$, not any other value.

How to avoid:

• Always check the medium first. If it says "vacuum" or "free space", use $c$.
• Write the formula clearly:

Mistake 4: Unit Conversion Errors

The Mistake:

Frequency is given as $30\ \text{MHz}$. Students forget to convert MHz to Hz before plugging in.

Why it's wrong:

$1\ \text{MHz} = 10^6\ \text{Hz}$. If you use $f = 30$ instead of $f = 30 \times 10^6$, your answer will be off by a factor of $10^6$.

How to avoid:

• Always convert to SI units before calculation.
• Write: …