JEE Mains Physics · Electromagnetic Waves

E and B Fields of a Plane Wave

In a plane wave E and B are perpendicular to each other and to the direction of travel, oscillate in phase with E = cB, and B = (k × E)/ω fixes which way B points.

All Electromagnetic Waves notes JEE Mains Physics strategy

Why this matters

Thirty-four PYQs, one of them asking for a number, and six from 2026. Fourteen give one field as a full wave equation and ask for the other. Sixteen give E or B at one point and instant, or ask which components and which direction of travel go together. Four ask for the electric and magnetic forces on a moving charge. All of them use the same two facts: E = cB, and E × B points along the direction of travel.

Concept 1 of 3: Writing the magnetic field of a wave from its electric field

E and B rise and fall together, at right angles to each other and to the direction of travel. So once one field is known, the other has the same phase, an amplitude divided (or multiplied) by c, and a direction fixed by a cross product. Only the direction needs thought.

Definition

The wave travels along $\vec E \times \vec B$ . Equivalently $\hat B = \hat n \times \hat E$ and $\hat E = \hat B \times \hat n$ , where $\hat n$ is the unit vector along the direction of travel.
Vector form: $\vec B = \dfrac{\vec k \times \vec E}{\omega}$ . Its size is $B_0 = kE_0/\omega = E_0/c$ in vacuum.
Same phase: B carries exactly the same argument as E. A sine stays a sine and $kx - \omega t$ stays $kx - \omega t$ .
Direction of travel from the phase: $kx - \omega t$ , $\omega t - kx$ and $\omega(t - x/c)$ all travel along +x; $kx + \omega t$ travels along −x.
Oblique travel: a phase in $ax + by$ travels along $(a\hat i + b\hat j)/\sqrt{a^{2} + b^{2}}$ . The size of the vector that multiplies E is part of the amplitude.
Cross products: $\hat i \times \hat j = \hat k$ , $\hat j \times \hat k = \hat i$ , $\hat k \times \hat i = \hat j$ ; reversing the order changes the sign.

One field from the other

\vec B = \frac{\vec k \times \vec E}{\omega}, \qquad B_0 = \frac{E_0}{c}, \qquad \hat n = \hat E \times \hat B

Worked example

A wave in vacuum has

\vec E = 45\sin(2 \times 10^{6}z - 6 \times 10^{14}t)\,\hat i

V/m. Write its magnetic field. (

c = 3 \times 10^{8}

m/s)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 21 Jan 2026 Shift 1 · Q10Moderate

Example 1 · Electromagnetic Waves · E and B Fields of a Plane Wave

The electric field a plane electromagnetic wave is given by :

E_{y}= 69\sin\left\lbrack 0.6 \times10^{3}x - 1.8 \times10^{11}t \right\rbrack V/m

. The expression for magnetic field associated with this electromagnetic wave is

\_\_\_\_

The order of the cross product matters

The magnetic field is along k × E, not E × k. Reversing the order gives the right axis with the wrong sign, and the options usually offer both.

B keeps the phase of E

The two fields oscillate together, so B has the same argument as E. An option that changes kx − ωt to kx + ωt describes a wave going the other way.

Divide by c, do not multiply

In SI units B₀ = E₀/c is tiny, of order 10⁻⁷ T for fields of tens of volts per metre. An option that gives B the same number as E has skipped the division.

Travel along −x flips a sign

When the phase is kx + ωt, put −î in the cross product. Using +î out of habit reverses the direction of the field you are finding.

Drill 13 more on writing the magnetic field of a wave from its electric field

Concept 2 of 3: Electric and magnetic fields at one point of a wave

At one point and one instant, E and B are two perpendicular arrows whose sizes are in the ratio c, arranged so that E × B points along the direction of travel. Nothing else about the wave matters for this: the frequency in the question is a distractor.

Definition

At every point and instant $|\vec B| = |\vec E|/c$ , not only for the amplitudes.
With $\hat n$ along the direction of travel: $\hat B = \hat n \times \hat E$ , $\hat E = \hat B \times \hat n$ , and $\hat n$ is $\hat E \times \hat B$ .
E and B lie along the two axes other than the travel axis, one each. Neither field is ever along the direction of travel.
The amplitudes satisfy $kE_0 = \omega B_0$ , which is $E_0 = cB_0$ in vacuum.
With the magnetic intensity $H = B/\mu_0$ : $E_0/H_0 = \sqrt{\mu_0/\varepsilon_0} \approx 377\ \Omega$ .
E, B and the direction of travel are mutually perpendicular; the energy is shared equally between the two fields; and the wave carries no charge, so electric and magnetic fields do not deflect it.

Fields at a point

|\vec B| = \frac{|\vec E|}{c}, \qquad \hat B = \hat n \times \hat E, \qquad \frac{E_0}{H_0} = \sqrt{\frac{\mu_0}{\varepsilon_0}} \approx 377\ \Omega

Worked example

A 50 MHz wave travels along +z in vacuum. At one point and instant,

\vec E = 7.5\,\hat i

V/m. Find

\vec B

there. (

c = 3 \times 10^{8}

m/s)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 11 April 2023 · Q93Moderate

Example 2 · Electromagnetic Waves · E and B Fields of a Plane Wave

A plane electromagnetic wave of frequency 20

MHz

propagates in free space along

x

-direction. At a particular space and time,

\overrightarrow{E}= 6.6\widehat{j}V/m

. What is

\overrightarrow{B}

at this point?

The frequency is a distractor

At a point and an instant, B = E/c. The frequency given in the question changes nothing in this calculation.

B over E is 1/c, not c

The magnetic amplitude is the electric amplitude divided by c. A statement that B₀/E₀ equals the speed of light has the ratio upside down.

E₀/B₀ is c, but E₀/H₀ is 377 Ω

The factor √(μ₀/ε₀) links E with the magnetic intensity H = B/μ₀, not with B. Writing E₀ = √(μ₀/ε₀)B₀ mixes the two.

Neither field lies along the travel direction

A wave along y can have E and B only along x and z, one each. Any pair that puts a field along y, or both fields on the same axis, is not a plane electromagnetic wave.

Drill 15 more on electric and magnetic fields at one point of a wave

Concept 3 of 3: Electric and magnetic forces on a charge in a wave

A charge in the wave feels qE from the electric field and qvB from the magnetic field. Since B is E/c, the magnetic force is only v/c of the electric force. It matters only when the charge moves fast.

Definition

Electric force $F_e = qE$ ; magnetic force $F_m = qvB\sin\theta$ .
A charge moving along E moves at right angles to B, so $F_m = qvB = qvE/c$ .
The ratio is $F_e/F_m = c/v$ , which is always more than 1.
The largest forces use the amplitudes: $F_{e,\max} = qE_0$ and $F_{m,\max} = qvB_0 = qvE_0/c$ .
At a given point the total force is the Lorentz force $\vec F = q(\vec E + \vec v \times \vec B)$ . A charge at rest feels no magnetic force.

Forces on a moving charge

F_e = qE_0, \qquad F_m = qvB_0 = \frac{qvE_0}{c}, \qquad \frac{F_e}{F_m} = \frac{c}{v}

Worked example

A wave in vacuum has

E_z = 600\sin\omega(t - y/c)

V/m. A proton moves along z at

6 \times 10^{6}

m/s. Find the largest electric force, the largest magnetic force and their ratio. (

e = 1.6 \times 10^{-19}

c = 3 \times 10^{8}

m/s)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 27 July 2022 · Q15Moderate

Example 3 · Electromagnetic Waves · E and B Fields of a Plane Wave

A beam of light travelling along

X

-axis is described by the electric field

E_{y}= 900\sin\omega(t - x/c)

. The ratio of electric force to magnetic force on a charge

q

moving along

Y

-axis with a speed of

3 \times10^{7}{\text{ }ms}^{- 1}

will be: [Given speed of light

= 3 \times10^{8}{\text{ }ms}^{- 1}

]

Use B₀ = E₀/c in the magnetic force

The magnetic force is qvB₀, and B₀ is the electric amplitude divided by c. Putting E₀ in place of B₀ makes the magnetic force c times too large.

The electric force is the larger one

The ratio of electric to magnetic force is c/v. Inverting it gives a ratio less than 1, which would mean the magnetic force wins; it never does for a charge slower than light.

Drill 3 more on electric and magnetic forces on a charge in a wave

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Writing the magnetic field of a wave from its electric field
One field from the other
$\vec B = \frac{\vec k \times \vec E}{\omega}, \qquad B_0 = \frac{E_0}{c}, \qquad \hat n = \hat E \times \hat B$
Electric and magnetic fields at one point of a wave
Fields at a point
$|\vec B| = \frac{|\vec E|}{c}, \qquad \hat B = \hat n \times \hat E, \qquad \frac{E_0}{H_0} = \sqrt{\frac{\mu_0}{\varepsilon_0}} \approx 377\ \Omega$
Electric and magnetic forces on a charge in a wave
Forces on a moving charge
$F_e = qE_0, \qquad F_m = qvB_0 = \frac{qvE_0}{c}, \qquad \frac{F_e}{F_m} = \frac{c}{v}$

Watch out for (10)

The order of the cross product matters
→ Writing the magnetic field of a wave from its electric field
B keeps the phase of E
→ Writing the magnetic field of a wave from its electric field
Divide by c, do not multiply
→ Writing the magnetic field of a wave from its electric field
Travel along −x flips a sign
→ Writing the magnetic field of a wave from its electric field
The frequency is a distractor
→ Electric and magnetic fields at one point of a wave
B over E is 1/c, not c
→ Electric and magnetic fields at one point of a wave
E₀/B₀ is c, but E₀/H₀ is 377 Ω
→ Electric and magnetic fields at one point of a wave
Neither field lies along the travel direction
→ Electric and magnetic fields at one point of a wave
Use B₀ = E₀/c in the magnetic force
→ Electric and magnetic forces on a charge in a wave
The electric force is the larger one
→ Electric and magnetic forces on a charge in a wave

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