Savchenko Problem 6.1.20
Physics · electromagnetism · modern-physics
Problem
a. A uniformly charged sphere with charges fixed on its surface was compressed in one direction n times, turning it into an ellipsoid. Prove that the electric field inside such an ellipsoid is zero. To prove this, divide the surface of the ellipsoid into pairs of small areas as shown in the figure. 190 b. Will there still be no field inside a long circular tube with a uniformly charged surface if it is compressed along with the fixed surface charges in the transverse direction?Intuitive explanation
The key insight is that uniform surface charge density creates a special symmetry: when you pair up surface elements on opposite sides of any closed surface and account for their distances and angles correctly, their electric field contributions cancel everywhere inside. For the ellipsoid, compression preserves this pairing property—each small area element on one side still has a corresponding partner whose field contributions sum to zero at any interior point. This elegant cancellation works because the inverse-square law and geometry conspire so that the distance factor and the solid angle subtended by each pair always balance out, regardless of the surface shape.
Intuitive explanation
The key insight is that uniform surface charge density creates a special symmetry: when you pair up surface elements on opposite sides of any closed surface and account for their distances and angles correctly, their electric field contributions cancel everywhere inside. For the ellipsoid, compression preserves this pairing property—each small area element on one side still has a corresponding partner whose field contributions sum to zero at any interior point. This elegant cancellation works because the inverse-square law and geometry conspire so that the distance factor and the solid angle subtended by each pair always balance out, regardless of the surface shape.