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

Correct Answer: 8

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 + \(\frac{1}{2}\)at²

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