Set 52 Problem number 3

Problem

Two uniform wires having circular cross-sections, both with the same length and made of  identical material, are both maintained at the same potential difference and the same  temperature. The first wire has diameter 4.2 mm, while the second has  diameter 8.2 mm. If the current in the first wire is .5 amps, then what is the current in the second wire?

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 8.2 / 4.2 = 1.952 the diameter of the  first, it will have ( 8.2 / 4.2)^2 = ( 1.952) ^2 = 3.810304 the cross-sectional area of the first.
• It will therefore  have 3.810304 times as many electrons per unit length available to carry the current. 
• Thus the drift of electrons will carry 3.810304 times as many electrons past  a given point every second.

The current in the second wire will therefore be 3.810304 * .5 amps = 1.905152 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.