Conduction in air

Question: In a particular experiment a high voltage is created by charging an isolated metal sphere. The sphere has a diameter 42 cm and any charge on its surface may be considered at its centre. The air surrounding the sphere loses its insulating properties, causing a spark, when the electric field exceeds 20 kV/cm.

(a) By reference to an atom in air, suggest the mechanism by which the electric field causes the air to become conducting.

(b) Calculate, for the charged sphere when a spark is about to occur,
(i) the charge on the sphere
(ii) it's potential.

(c) Under certain conditions, a spark sometimes occurs before the potential reaches that calculated in (b)(ii). Suggest a reason for this.

Answer:

(a) the electric field is great enough to tear the outer electrons off the atoms of the gases in the air and produce ions. These charged ions and free electrons then move within the electric field.

(b) Using the following equation and considering the charge as a point :

Electric field intensity (E_E) = [(1/4pe_o)]Q/r²         r = radius of sphere = 0.21 m
E_E = 20 kV/cm = 20x10⁵ Vm^-1 = [(1/4pe_o)]Q/0.21²

Therefore:

(i) Charge on the sphere (Q) = [20x10⁵x0.21²]/9x10⁹ = 9.8x10^-6 C
(ii) Potential (V) = (1/4pe_o)Q/r = E_Ex r = 9.8x10^-6x0.21 = 2x10^-6 V

(c) if the air is damp the breakdown potential is reduced. This means that ions are more easily conducted from the region outside the charged sphere