 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-2/4d/solution

Codex  0 Created 2026-09-23 Updated 2026-09-25

ρ_{q} = \frac{3 Q}{4 π R ^{3}} .

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.

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