Question

An electron and a proton are a distance r = 7 × 10^-9 m apart. How much energy is required to increase their separation by a factor of 3?

Answer 1

Coulumb's Law: The force felt by the electron is F = kQ[p]Q[e] / r^2
where k = 1/(4piε0)
and ε0 is the permittivity of free space = 8.85418782 × 10-12 C²/N.m²

Substituting in values for

Q[p] = charge of proton=1.60217646 × 10^-19 C,

Q[e] = charge of electron = 1.60217646 × 10^-19 C

we get that F = e2/4piε0 * 1/r^2 = 2.307077128 * 10^-28 / r^2

F = 2.307077128 * 10^-28 / r^2
Positive values taken for convenience.

Notice that if you move the electron a distance dr from its current position you would increase r by dr. Also notice that as you move it the force acting on it would continuously decrease because it gets further and further away from the proton.

Work done in moving the electron = Force x Distance for a constant force but in this case force is changing so;
Work done is therefore ∫ F dr where F is the expression for Force and dr is the change in distance from the other particles.

Since r = 7x 10^-9 m, triple that and,
r + dr = 3.43 x 10^-7m

Then Work done in moving dr from r is the definite integral W = ∫ (2.307077128 * 10^-28 / r^2) dr --- for limits 7.5 x 10^-9 to 3.43 x 10^-7

This works out to 3.0084 * 10^-20 Joules needed to increase their separation in this way.