Question

The dielectric strength of air (that is, the maximum electric field air can withstand before it becomes a conductor due to ionization) is 3.0 times 10^6 V/m. Small van de Graaf generators are commonly used in hair-raising demonstrations that must achieve a high electric potential. A spherical conductor has a radius of 30 cm (about 1 ft). What is the maximum charge that can be placed on the sphere before dielectric breakdown of the surrounding air occurs? For the charge calculated in (a), what is the electric potential V at the surface of the sphere?

Answer 1

given electric strength is 3 *10^6

radius of the spherical conductor is 30 cms?

we have to find th max charge placed on the sphere before dielectric breakdown occur?

for this charge what is the electric potential?

E = 3.0×10^6 V/m
The permittivity of air is approximately:
εo ≈ 8.85*10^-12 C^2.N-1.m-2
r = 0.3m

Via Gauss's law, the the maximum charge is:
∫s E.dA = Q/εo
= E*4πr^2 = Q/εo
Q = E*4πr^2*εo
Q = 3*10^6 *4*3.14*0.3*0.3*8.85*10^-12

Q= 30.01*10-^6 on sphere surface:

chargeQ = 30 micro coumbs

now potential

V = Q/(4πεo*r)

V = 30.01*10-^6 /4*3.14*8.85*10^-12*0.3

V = 0.9 *10^6N/C