Question

Two solid metal spheres one with 0.1m radius and the other with 0.01m radius are charged to 1.0 x 10-6 C. They are placed in two dry and identical environments. Almost immediately the smaller sphere send sparks out as the larger one stayed with the same charge for a longer period of time. Provided the electrical breakdown threshold electric field of dry air is 3 x 106 V/m. Explain through a numerical calculation why only the smaller sphere quickly discharged.

Answer 1

The smaller sphere has higher value of electric field than the threshold value of electric field.

Solution:

As per the question:

The radius of the first metal sphere, R = 0.1 m

The radius of the first metal sphere, R' = 0.01 m

Charge on the sphere, Q =
`1.0\tiems 10^(- 6)\ C`

The value of the critical electric field,
`\vec{E_(c)} = 3* 10^(6) V/m`

Now,

The electric field on the surface of a sphere is given by:

`\vec{E} = K(Q)/(R^(2))`

where

K = Electrostatic constant =
`(1)/(4\pi\epsilon_(o)) = 9* 10^(9)N.m^(2)C^(- 2)`

Now, for the first metal sphere:

`\vec{E} = K(Q)/(R^(2))`

`\vec{E} = (9* 10^(9))* ((1.0\tiems 10^(- 6)))/(0.1^(2)) = 9* 10^(5)\ V/m`

Now, for the second metal sphere:

`\vec{E'} = K(Q)/(R')`

`\vec{E'} = (9* 10^(9))* ((1.0\tiems 10^(- 6)))/(0.01^(2)) = 9* 10^(7)\ V/m`

The Electric field for second metal sphere,i.e., smaller sphere is higher than the threshold or critical value for the Electric field thus smaller sphere discharges quickly as above critical field breakdown occurs.