We have to find (a) the total charge q that a disk of radius 2.50 cm must carry in order that the electric field on the surface of the disk at its centre equals the value at which air breaks down electrically, producing sparks. (b) Suppose that each atom at the surface has an effective cross-sectional area of . We have to estimate the number of atoms at the disk’s surface. (c) If the charge in (a) results from one excess electron carried by some of the surface atoms, we have to find the fraction of the surface atoms that must be so charged.

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(a)

Electric breakdown in air takes place if the electric field strength exceeds .

Let us consider a uniformly charged disk of radius R and total charge q. The surface charge density will be

.

The electric field strength at the centre of a uniformly charged disk is equal to the electric field of an infinite plate having uniform charge density . That is

.

The charge q on a disk of radius R that will produce breakdown of air will be

.

It is given that the radius of the disk is 2.50 cm. Therefore, the total charge q that it must have for air to breakdown at its centre will be

(b)

It is given that the effective cross-sectional area of each atom on the surface of the disk is . Therefore, the total number of atoms on the disk is

(c)

It is given that the charge on the disk is due to those atoms which carry one excess electron, that is have a charge . Therefore, the total number of charged atoms on the disk is

Fraction of the surface atoms that are charged will be