Question

Lightning can be studied with a Van de Graaff generator, which consists of a spherical dome on which charge is continuously deposited by a moving belt. Charge can be added until the electric field at the surface of the dome becomes equal to the dielectric strength of air. Any more charge leaks off in sparks. Assume the dome has a diameter of 43.0 cm and is surrounded by dry air with a "breakdown" electric field of 3.00 10^6 V/m. (a) What is the maximum potential of the dome?
(b) What is the maximum charge on the dome?

Answer 1

(b) $1.54\cdot 10^{-5} C$

We can actually solve first part (b) the problem. In fact, we know that the electric field strength at the surface of the a sphere is given by

$E=\frac{kQ}{R^2}$

where

k is the Coulomb's constant

Q is the charge on the surface of the sphere

R is the radius

For this sphere, the radius is half the diameter, so

$R=\frac{43.0 cm}{2}=21.5 cm = 0.215 m$

We also know that the maximum charge is the value of charge deposited at which the electric field of the sphere becomes equal to the breakdown electric field, so

$E=3.00 \cdot 10^6 V/m$

Solving the formula for Q, we find the maximum charge:

$Q=\frac{ER^2}{k}=\frac{(3.00\cdot 10^6)(0.215)^2}{9\cdot 10^9}=1.54\cdot 10^{-5} C$

(a) $6.45\cdot 10^5 V$

The maximum potential of a charged sphere occurs at the surface of the sphere, and it is given by

$V=\frac{kQ}{R}$

where we already found at point b)

$Q=1.54\cdot 10^{-5} C$

and we know that

R = 0.215 m

Solving for V, we find:

$V=\frac{(9\cdot 10^9)(1.54\cdot 10^{-5})}{0.215}=6.45\cdot 10^5 V$