An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet S having surface charge density +σ. 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 σ if the electron strikes S at t=1 s is α[(m ε₀)/e] C/m², the value of α is ___________.

Solution

Physics sir · 4 steps

Bigger σ means a stronger pull, so for the maximum σ the electron must start off moving away from the sheet.

1. Field and pull. An infinite sheet gives a uniform field E = σ/2ε₀. The electron is pulled toward the positive sheet with constant acceleration a = eE/m = eσ/2mε₀.
2. Pick the direction. A larger σ gives a larger a, so the electron reaches the sheet sooner. It can still take a full 1 s only if it first moves away from S at 1 m s⁻¹, slows down and then comes back. Starting away from the sheet allows the largest a.

   1 = -(1)(1) + 1/2 a (1)² ⇒ a = 4 m s⁻²

3. Solve for σ. Take displacement toward the sheet as positive, as in the equation above. Then eσ/2mε₀ = 4, so σ = 8[mε₀/e].
4. So. α = 8.

The common slip: 4. Using E = σ/ε₀, the field near a conductor, instead of σ/2ε₀ for a charged sheet gives 4, not 8.

The idea to keep. Field of an infinite sheet is σ/2ε₀, uniform at every distance. For a maximum or minimum, let the initial velocity point whichever way stretches the motion to the given time.

Answer from the NTA final answer key. Solution by PrepWiser.