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: Idea:
Write the intensity directly in terms of the magnetic field. Since $E_0 = cB_0$, the intensity $I = \tfrac{1}{2}\epsilon_0 c E_0^2$ can be changed so only $B_0$ appears.

Step 2: Use the Magnetic Form:
Substitute $E_0 = cB_0$. We get $I = \tfrac{1}{2}\epsilon_0 c^3 B_0^2$. The speed of light also satisfies $c^2 = 1/(\mu_0\epsilon_0)$, so $\epsilon_0 c^2 = 1/\mu_0$. This gives \[ I = \frac{c\,B_0^2}{2\mu_0} \]

Step 3: Solve for B0:
Rearranging, \[ B_0 = \sqrt{\frac{2\mu_0 I}{c}} \] With $\mu_0 = 4\pi \times 10^{-7}$ T m/A, $I = 2.5 \times 10^{14}$ W/m$^2$ and $c = 3 \times 10^{8}$ m/s: \[ B_0 = \sqrt{\frac{2 \times 4\pi \times 10^{-7} \times 2.5 \times 10^{14}}{3 \times 10^{8}}} = \sqrt{2.094} \approx 1.45\ \text{T} \] The question gives $\epsilon_0$, and using it in the first method gives the same value.

Step 4: Compare with the Options:
1.45 T is the second printed option. The others (1.15 T, 2.0 T, 3.25 T) do not match the result.

Final Answer:
\[\boxed{B_0 \approx 1.45\ \text{T (option 2)}}\]