Question

The electric field vector associated with plane electromagnetic wave is given as $\vec{E}(x, t) = 4 \cos(3 \times 10^8t - x)\hat{j} \mu V/m$. This wave strikes on a perfectly absorbing surface of area $1m^2$ at normal incident. The total momentum transferred to the surface in 12.5 second is $(\epsilon_0 \times 10^{-n}) kg.m/s$. Find n ($\epsilon_0$ is permittivity of medium)

Answer 1

10

Answer 2

1. Extract $E_0$, $\omega$, and $k$ from the given electric field equation:
   $\vec{E}(x, t) = 4 \cos(3 \times 10^8t - x)\hat{j} \mu V/m$ $E_0 = 4 \times 10^{-6} V/m$, $\omega = 3 \times 10^8 rad/s$, $k = 1 rad/m$

2. Calculate the speed of light $c = \omega/k$:
   $c = \frac{3 \times 10^8}{1} = 3 \times 10^8 m/s$

3. Compute the average intensity $I = \frac{1}{2} c \epsilon_0 E_0^2$:
   $I = \frac{1}{2} \times 3 \times 10^8 \times \epsilon_0 \times (4 \times 10^{-6})^2 = 24 \times 10^{-4} \epsilon_0 W/m^2$

4. Determine the radiation pressure $P_{rad} = I/c$ for a perfectly absorbing surface:
   $P_{rad} = \frac{24 \times 10^{-4} \epsilon_0}{3 \times 10^8} = 8 \times 10^{-12} \epsilon_0 N/m^2$

5. Calculate the force $F = P_{rad} \times A$:
   $F = 8 \times 10^{-12} \epsilon_0 \times 1 = 8 \times 10^{-12} \epsilon_0 N$

6. Find the total momentum transferred $p = F \times \Delta t$:
   $p = 8 \times 10^{-12} \epsilon_0 \times 12.5 = 100 \times 10^{-12} \epsilon_0 = 10^{-10} \epsilon_0 kg.m/s$

7. Compare the calculated momentum with the given format $(\epsilon_0 \times 10^{-n})$ to find $n$:
   $10^{-10} \epsilon_0 = \epsilon_0 \times 10^{-n}$
   $n = 10$