Question

The intensity of a laser beam is \(2.5 \times 10^{14}\ \text{W/m}^2\). The amplitude of the magnetic field in the beam is
(Given, \(\epsilon_0 = 8.85 \times 10^{-12}\ \text{C}^2\text{N}^{-1}\text{m}^{-2}\))

• A. ~1.15 T
• B. ~1.45 T
• C. ~2.0 T
• D. ~3.25 T

Hint:

Use \(I = \tfrac{1}{2}\epsilon_0 c E_0^2\) to get \(E_0\), then \(B_0 = E_0/c\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
A laser beam is an electromagnetic wave. We know its intensity and need the amplitude of the magnetic field \(B_0\). The speed of light is \(c = 3 \times 10^{8}\ \text{m/s}\).

Step 2: Key Formula or Approach:
The average intensity of an electromagnetic wave is the energy carried per second per unit area. It is given by \[ I = \frac{1}{2}\,\epsilon_0 E_0^2\, c \] The two field amplitudes are linked by \(E_0 = cB_0\).

Step 3: Find the Electric Field Amplitude:
Rearrange the intensity formula. \[ E_0 = \sqrt{\frac{2I}{\epsilon_0 c}} = \sqrt{\frac{2 \times 2.5 \times 10^{14}}{8.85 \times 10^{-12} \times 3 \times 10^{8}}} \] The denominator is \(2.655 \times 10^{-3}\). So \[ E_0 = \sqrt{1.883 \times 10^{17}} \approx 4.34 \times 10^{8}\ \text{V/m} \]

Step 4: Find the Magnetic Field Amplitude:
Use \(B_0 = E_0/c\). \[ B_0 = \frac{4.34 \times 10^{8}}{3 \times 10^{8}} \approx 1.45\ \text{T} \]

Step 5: Checking Each Option:
Option 1 (1.15 T) and option 3 (2.0 T) do not match the calculated 1.45 T. Option 4 (3.25 T) is far too large. Only option 2 matches.

Final Answer:
The amplitude of the magnetic field is about 1.45 T, so option 2 is correct. \[ \boxed{B_0 \approx 1.45\ \text{T}} \]