A resistor is in the shape of a truncated right circular cone. The end radii are a and b, and the altitude is L. If the taper is small, we may assume that the current density is constant across any cross section. (a) We have to calculate the resistance of this object. (b) We have to show that our answer reduces to for the special case of zero taper .

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(a)

We consider the resistor as shown in the figure. It is in the shape of a truncated right circular cone of end radii a and b. We first determine the equation of the taper.

Assuming that it is linear, we describe it by the equation

We determine its constants using the boundary conditions

We find the constants and fix the line by the equation

We assume that the current density is uniform across any cross section of the resistor. Contribution to the resistance from the section of the cone of width at a distance x from the top end will be

Therefore, the resistance of the right-circular cone will be

(b)

If