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Question
Consider an electron in front of metallic surface at a distance d (treated as an infinite plane surface). Assume the force of attraction by the plate is given as `1/4 q^2/(4πε_0d^2)`. Calculate work in taking the charge to an infinite distance from the plate. Taking d = 0.1 nm, find the work done in electron volts. [Such a force law is not valid for d < 0.1nm].
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Solution
As per question F = `1/4 q^2/(4πε_0d^2)`
Let at any instant electron is at a distance x from the metal surface. Force of attraction between. metal surface and electron is F
F = `1/4 q^2/(4πε_0x^2)`
Work done by an external agency in taking the electron from distance d to ∞ is W.D. = `int_d^∞ F*dx`
W.D. = `int_d^∞ (q^2dx)/(4 xx 4πε_0x^2) = q^2/(4 xx 4πε_0) int_d^∞ x^-2 dx = q^2/(4 xx 4πε_0) x^-1/(-1)`
W.D. = `(-q^2)/(4.4πg e_0) [1/x]_d^∞ = (-q^2k)/4 [1/∞ - 1/d]`
W.D. = `(+ kq^2)/(4d)` [d = 0.1 nm = 10–10 m]
Work done is positive
So W.D. = `((1.6 xx 10^-19)^2 xx 9 xx 10^9)/(4 xx 10^-10)`
J = `(1.6 xx 9 xx 1.6 xx 10^(-33+9+10))/(4 xx 1.6 xx 10^-19)` eV
= `(9 xx 1.6 xx 10^(19+19))/4`
= 3.6 eV
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