Physics Resistor

Question

There is a current of 232 pA when a certain potential is applied across a certain resistor. When that same potential is applied across a resistor made of the identical material but 24 times longer, the current is 0.034 pA.

Compare the effective diameters of the two resistors (i.e. find the ratio of the diameter of the second resistor to the diameter of the first resistor)

Answer 1

Using Ohm's law (V=IR), for the first resistor we have R1 = V/I = V/(232 pA). For the second resistor we have R2 = V/I = V/(0.034 pA). Because the second resistor is 24 times longer, its resistance is 24 times, we can divide the R2 by 24 so that we can compare each resistors resistance per unit length instead of their absolute resistance, so R1/L = V/(232 pA) and R2/L = V/(24 * 0.034 pA) = V/(0.816 pA). Because the resistance per unit length will be proportional to the cross sectional area so proportional to the diameter squared, we can divide the resistance per unit length of R2 by R1, so we get the ratio of cross sectional areas as 232 / .816 = 284.3. Taking the square root of this, we get 16.86 as the ratio of the diameter of resistor 2 to resistor 1. Hope this helps!