Question

Two cylindrical resistors are made of the same material and have the same resistance. The resistors, R1 and R2, have different radii, r1 and r2, and different lengths, L1 and L2. Which of the following relative values for radii and lengths would result in equal resistances?
a. r1 = r2 and L1 = 2L2
b. 2r1 = r2 and L1 = 2L2
c. r1 = r2 and 4L1 = L2
d. 2r1 = r2 and 4L1 = L2

Answer 1

Answer:

Option d is correct.

Explanation:

We know , resistance of a body is directly proportional to its length and inversely proportional to its area.

$R=\dfrac{\rho\ L}{A}=\dfrac{\rho\ L}{\pi r^2}$      ( Here, $\rho$ is constant dependent on object material )

Writing $R_1 \ and\ R_2$ also :

$R_1=\dfrac{\rho\ L_1}{\pi r_1^2}\ , \ R_2=\dfrac{\rho\ L_2}{\pi r_2^2}$      ( since they are of same material therefore, $\rho$ is same.)

Now , if $r_2=2r_1 \ and \ 4L_1=L_2$.

Then $R_1=R_2$

Therefore, option d. is correct.

Hence, this is the required solution.