An electron is released from rest near an infinite non-conducting sheet of uniform charge density '-σ'. The rate of change of de-Broglie wave length associated with the electron varies inversely as nᵗʰ power of time. The numerical value of n is _____.

Solution

Physics sir · 3 steps

A uniform sheet gives a constant force, so speed grows as t and wavelength falls as 1/t.

1. Constant force. An infinite sheet makes a uniform field of magnitude E = σ/2ε₀ at every distance. The electron and the sheet are both negative, so the electron is pushed away with a constant force eE and a constant acceleration a = eσ/2mε₀.
2. Speed and wavelength. The electron starts from rest, so v = at. Its de Broglie wavelength is λ = h/mv = h/mat, which goes as 1/t.

   dλ/dt = d/dt(h/ma t⁻¹) = -h/ma 1/t²

3. So. The rate of change of wavelength is proportional to 1/t², so n = 2.

The common slip: 1. Stopping at λ ∝ 1/t gives 1. The question asks for the rate dλ/dt, which goes as 1/t².

The idea to keep. Find how v depends on t first, then use λ = h/mv. With a constant force, λ ∝ 1/t, and differentiating adds one more power of t.

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