Question

An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet $S$ having surface charge density $+\sigma$. The electron at $t = 0$ is at a distance of 1 m from $S$ and has a speed of 1 m/s. The maximum value of $\sigma$ if the electron strikes $S$ at $t = 1 \, \text{s}$ is $\alpha \left[ \frac{m \epsilon_0}{e} \right] \frac{\text{C}}{\text{m}^2}$. The value of $\alpha$ is ______.

Answer 1

Step 1. Given Information and Variables:

Initial velocity u = 1 m/s - Acceleration a = - $\frac{\sigma e}{2 \varepsilon_0 m}$ - Time t = 1 s - Displacement S = -1 m

Step 2. Use Kinematic Equation:

The kinematic equation for displacement S is:

S = ut + $\frac{1}{2}$at 2

Substitute values:

-1 = 1 × 1 + $\frac{1}{2}$ × $- \frac{\sigma e}{2 \varepsilon_0 m}$ × (1)2

Step 3. Solve for σ :

-1 = 1 - $\frac{\sigma e}{4 \varepsilon_0 m}$

$\frac{\sigma e}{4 \varepsilon_0 m}$ = 2

Therefore, σ = $8 \frac{\varepsilon_0 m}{e}$

Step 4. Determine α:

Given σ = α $\left[ \frac{m \varepsilon_0}{e} \right]$, we find α = 8.

Conclusion:

So, the correct answer is: α = 8