Electric Potential in Geometric Shapes · USAPhO 2018 #B1
Physics · electromagnetism
Problem
The electric potential at the center of a cube with uniform charge density ρ and side length a is Φ≈0.1894ρa2/ε0.
You do not need to derive this.
For the entirety of this problem, any computed numerical constants should be written to three significant figures.
a. What is the electric potential at a corner of the same cube? Write your answer in terms of ρ, a, ε0, and any necessary numerical constants.
b. What is the electric potential at the tip of a pyramid with a square base of side length a, height a/2, and uniform charge density ρ? Write your answer in terms of ρ, a, ε0, and any necessary numerical constants.
c. What is the electric potential due to a square plate with side length a of uniform charge density σ at a height a/2 above its center? Write your answer in terms of σ, a, ε0, and any necessary numerical constants.
d. Let E(z) be the electric field at a height z above the center of a square with charge density σ and side length a. If the electric potential at the center of the square is approximately 0.281aσ/ε0, estimate E(a/2) by assuming that E(z) is linear in z for 0<z<a/2. Write your answer in terms of σ, a, ε0, and any necessary numerical constants.