Question

Two cylindrical resistors are made from the same material. The shorter one has a length LL and diameter DD . The longer one has a length 16L16L and diameter 4D4D . How do their resistances compare? The resistance of the longer resistor is four times the resistance of the shorter resistor. The resistance of the longer resistor is twice the resistance of the shorter resistor. The resistance of the longer resistor is the same as the resistance of the shorter resistor. The resistance of the longer resistor is half the resistance of the shorter resistor. The resistance of the longer resistor is a quarter of the resistance of the shorter resistor.

Answer 1

Answer:

The resistance of the longer resistor is a quarter of the resistance of the shorter resistor.

Explanation:

If Two cylindrical resistors are made from the same material, then their resistivity will be the same.  Formula for calculating resistivity of a material is expressed as;

$\rho = \frac{RA}{L} \ where \ A = \frac{\pi d^2}{4}$ where;

R is the resistance

A is the cross sectional area of the material

L is the length of the material

For the shorter cylinder:

Length = L

diameter = D

$\rho = \dfrac{R_s(\frac{\pi D^2}{4})}{L} \\\\\rho = \dfrac{R_s{\pi D^2}}{4L}$

For the longer cylinder:

Length = 16L

diameter = 4D

$\rho = \dfrac{R_l(\frac{\pi (4D)^2}{4})}{16L} \\\\\\\rho = \dfrac{R_l(\frac{\pi (16D^2)}{4})}{16L} \\\\\rho = \dfrac{R_l{16\pi D^2}}{16L}\\\\\rho = \dfrac{R_l{\pi D^2}}{L}$

Since their resistivity are the same then;

$\dfrac{R_s{\pi D^2}}{4L} = \dfrac{R_l{\pi D^2}}{L} \\\\ \dfrac{R_s}{4} = {R_l} \\\\R_s = 4R_l\\\\R_l = \frac{R_s}{4}$

Hence the resistance of the longer resistor is a quarter of the shorter resistor.