Question

A 15.7mW laser beam has a diameter of 4mm. If the amplitude of the associated magnetic field is expressed as \(\frac{𝐴}{\sqrt{ε_0𝑐^3}}\) , the value of A is: (𝜀₀ is the free space permittivity and 𝑐 is the speed of light)

• A. 50
• B. 35.4
• C. 100
• D. 70.8

Answer 1

The Correct Option is A

Step 1: Calculate the intensity of the laser beam

The power of the laser beam is \( P = 15.7 \, \text{mW} = 15.7 \times 10^{-3} \, \text{W} \), and the diameter of the beam is \( d = 4 \, \text{mm} = 4 \times 10^{-3} \, \text{m} \). The area of the beam cross-section is: \[ A_{\text{beam}} = \pi \left(\frac{d}{2}\right)^2 = \pi \left(\frac{4 \times 10^{-3}}{2}\right)^2 = \pi (2 \times 10^{-3})^2 = 4\pi \times 10^{-6} \, \text{m}^2. \] The intensity \( I \) is given by: \[ I = \frac{P}{A_{\text{beam}}}. \] Substituting the values: \[ I = \frac{15.7 \times 10^{-3}}{4\pi \times 10^{-6}} = \frac{15.7 \times 10^{-3}}{12.566 \times 10^{-6}} \approx 1250 \, \text{W/m}^2. \]

Step 2: Relate intensity to the magnetic field amplitude

The intensity \( I \) of an electromagnetic wave is related to the amplitude of the magnetic field \( B_0 \) by: \[ I = \frac{B_0^2 c}{2\mu_0}, \] where: - \( \mu_0 = \frac{1}{\epsilon_0 c^2} \) is the permeability of free space, - \( c \) is the speed of light.

Step 3: Solve for \( B_0 \)

Rearrange the formula to solve for \( B_0^2 \): \[ B_0^2 = \frac{2 I \mu_0}{c}. \] Using \( \mu_0 = \frac{1}{\epsilon_0 c^2} \), the expression becomes: \[ B_0^2 = \frac{2 I}{\epsilon_0 c^3}. \] Therefore: \[ B_0 = \sqrt{\frac{2 I}{\epsilon_0 c^3}}. \] Comparing with the given form \( B_0 = \frac{A}{\sqrt{\epsilon_0 c^3}} \), we identify: \[ A = \sqrt{2I}. \]

Step 4: Calculate \( A \)

Substituting \( I = 1250 \, \text{W/m}^2 \): \[ A = \sqrt{2 \times 1250} = \sqrt{2500} = 50. \]

Final Answer:

The value of \( A \) is: 50.