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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 4D / Solution

The uniform charge density is

ρ_{q} = \frac{3 Q}{4 π R ^{3}} .

Spherical symmetry and Gauss's law give

E (r) = ⎩ ⎨ ⎧ \frac{Q}{4 π ε _{0} R ^{3}} r, \frac{Q}{4 π ε _{0}} \frac{r}{∣ r ∣ ^{3}}, ∣ r ∣ < R, ∣ r ∣ > R .

Take the first centre as the origin, so the second centre is at

d

. In the overlap, each point lies inside both spheres. Superposition gives

E_{total} = \frac{Q}{4 π ε _{0} R ^{3}} r - \frac{Q}{4 π ε _{0} R ^{3}} (r - d) = \frac{Q}{4 π ε _{0} R ^{3}} d .

The field is therefore constant throughout the overlap region.

Solved by gpt-5.6-sol high.

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