Set 52 Problem number 3

Problem

Under certain conditions of temperature and potential difference, a uniform wire with cross-sectional radius 6.6 cm carries a current of 1.3 amps.  Under the same conditions, what will be the current of a second wire whose length is identical to that of the first, but which has diameter 11.1 cm?

Solution

Since the potential differences and the lengths of the wires are identical the potential  gradients, or electric fields, will be identical. Thus the drift velocities will be equal.

• Since the  second wire has 11.1 / 6.6 = 1.681 the diameter of the  first, it will have ( 11.1 / 6.6)^2 = ( 1.681) ^2 = 2.825761 the cross-sectional area of the first.
• It will therefore  have 2.825761 times as many electrons per unit length available to carry the current. 
• Thus the drift of electrons will carry 2.825761 times as many electrons past  a given point every second.

The current in the second wire will therefore be 2.825761 * 1.3 amps = 3.673489 amps.

Generalized Solution

If wires having uniform circular cross-sections are of identical length and have the same potential difference from end to end, with one wire having diameter d1 and the other diameter d2, then the cross-sectional area of the second is (d2 / d1) ^ 2 times that of the first.

• The second wire will therefore have (d2 / d1) ^ 2 times as many available charge carriers per unit length.
• Since the drift velocity depends only on the potential gradient, the drift velocities are the same.
• It follows that in any given time interval, the number of electrons drifting past a point in wire 2 is (d2 / d1) ^ 2 times as great as in wire 1.

The current in wire 2 is therefore (d2 / d1) ^ 2 times as great as in wire 1.