Q.The magnetic field in a plane electromagnetic wave is given by $B_y = (2 \times 10^{-7})\ \text{T} \sin (0.5\times10^{3}x + 1.5\times10^{11}t)$.

The wave is $B_y=(2\times10^{-7})\ \text{T}\,\sin(0.5\times10^{3}x+1.5\times10^{11}t)$. Compare with $B=B_0\sin(kx+\omega t)$.

(a) Wavelength and frequency.

• $k=0.5\times10^{3}=500\ \text{rad/m}\Rightarrow \lambda=\dfrac{2\pi}{k}=\dfrac{2\pi}{500}\approx1.26\times10^{-2}\ \text{m}\ (=1.26\ \text{cm}).$
• $\omega=1.5\times10^{11}\ \text{rad/s}\Rightarrow f=\dfrac{\omega}{2\pi}=\dfrac{1.5\times10^{11}}{2\pi}\approx2.39\times10^{10}\ \text{Hz}.$ …

Compare with the standard form. Writing $B_y=B_0\sin(kx+\omega t)$ with $B_0=2\times10^{-7}\ \text{T}$, we read off

Step 1 — Wavelength.

Step 2 — Frequency.

Step 3 — Direction of propagation. In $\sin(kx+\omega t)$ the $x$ and $t$ terms carry the same sign, so a point of constant phase satisfies $kx+\omega t=\text{const}$; as $t$ increases, $x$ must decrease. Hence the wave travels along the $\mathbf{-x}$ direction.

Step 4 — Amplitude of the electric field. For a plane EM wave in vacuum $E_0=cB_0$:

$E_0=(3\times10^{8})(2\times10^{-7})=60\ \text{V/m}.$ …

Method: Standard Wave Analysis for EM Waves

We use the general wave equation comparison method — matching the given wave to the standard form to extract parameters, then applying the EM wave relation $E = Bc$.

Step 1: Identify the wave form

The given magnetic field:

Standard form for a wave traveling along $x$:

Here the + sign means the wave travels in the negative x-direction.

Step 2: Extract $k$ and $\omega$

From comparison:

• Wave number: $k = 0.5 \times 10^{3} = 500\ \text{rad/m}$
• Angular frequency: $\omega = 1.5 \times 10^{11}\ \text{rad/s}$

Step 3: Find wavelength $\lambda$ and frequency $f$

Wavelength:

Frequency:

Step 4: Write the electric field expression

For an EM wave in vacuum:

Given $B_0 = 2 \times 10^{-7}\ \text{T}$ and $c = 3 \times 10^{8}\ \text{m/s}$: …

Common Mistakes & How to Avoid Them

Mistake 1: Confusing the sign in the wave equation

The error:

Students see $B_y = (2 \times 10^{-7}) \sin (0.5\times10^{3}x + 1.5\times10^{11}t)$ and assume the wave travels along $+x$ because the coefficient of $t$ is positive.

Why it’s wrong:

The general form is $\sin(kx - \omega t)$ for a wave traveling in $+x$ direction. Here we have $\sin(kx + \omega t)$, which means the wave travels in $-x$ direction.

How to avoid:

• Always compare with the standard form:
  • $\sin(kx - \omega t)$ → wave moves along $+x$
  • $\sin(kx + \omega t)$ → wave moves along $-x$
• Memorise: The sign of the $t$ term tells you the direction — opposite to what intuition might suggest.

Mistake 2: Using the wrong formula for wavelength

The error:

Students write $k = \frac{2\pi}{\lambda}$ but then plug $k = 0.5 \times 10^3$ without checking units.

Why it’s wrong:

The given $k = 0.5 \times 10^3 \, \text{m}^{-1}$ is correct, but students sometimes forget to convert or misplace the decimal.

How to avoid:

• Always write:

• Substitute carefully:

• Double-check: $\lambda$ should be in metres — if you get a weird number, re-check $k$.

Mistake 3: Confusing angular frequency $\omega$ with frequency $f$

The error:

Students write $f = \omega$ or use $f = \frac{\omega}{2\pi}$ incorrectly.

Why it’s wrong:

Here $\omega = 1.5 \times 10^{11} \, \text{rad/s}$. The frequency is:

How to avoid:

• Remember:
  • $\omega$ = angular frequency (rad/s)
  • $f$ = ordinary frequency (Hz or s$^{-1}$)
  • Relation: $\omega = 2\pi f$
• Always write the unit — if you get rad/s, you know it’s $\omega$, not $f$.

Mistake 4: Forgetting the direction of the electric field

The error:

Students write $E_x$ or $E_z$ without checking the cross-product relation.

Why it’s wrong:

For an EM wave, $\vec{E}$, $\vec{B}$, and direction of propagation $\hat{k}$ are related by:

Here:

• $\vec{B}$ is along $+y$
• Wave travels along $-x$
• So $\vec{E}$ must be along $+z$ (use right-hand rule)

How to avoid:

• Use the right-hand rule:
  • Point fingers along $\vec{E}$
  • Curl them toward $\vec{B}$
  • Thumb points in direction of wave travel
• Check: If wave goes in $-x$, $\vec{B}$ along $+y$, then $\vec{E}$ must be along $+z$.

Mistake 5: Using the wrong relation between $E_0$ and $B_0$

The error:

Students write $E_0 = c B_0$ but forget that $c = 3 \times 10^8 \, \text{m/s}$.

Why it’s wrong:

The formula is correct, but students sometimes use $c = 3 \times 10^8$ incorrectly or forget to multiply.

How to avoid:

• Always write:

• Substitute: …