Question

Two wires P and Q are made of the same material. The wire Q has twice the diameter and half the length as that of wire P. If the resistance of wire P is R, the resistance of the wire Q will be:

Options

• R

• `R/2`

• `R/8`

• 2R

Answer 1

`bb(R/8)`

Explanation:

Given that wires P and Q are made of the same material, they have the same resistivity (ρ).

The resistance of a wire is given by the formula:

R = `rho L/A`

Here, all alphabets are used in their usual meanings.

For wire P: 

Its length be L,

diameter be d,

So its radius (r) = `d/2`

The cross-sectional area (A_p) = πr²

= `pi(d/2)^2`

= `(pi d^2)/4`

The resistance of wire P is given as R, so:

R = `rho L/A_p`

= `rho L/((pi d^2)/4)`

= `rho (4 L)/(pi d^2)`

For wire Q:

Its length is ha;f of wire P, so L_Q = `L/2`, diameter is twice that of wire P, so d_Q = 2d, and its radius is:

r_Q = `(2d)/2` = d

The cross-sectional area:

A_Q = `pi r_Q^2` = πd²

The resistance of wire Q:

R_Q = `rho L_Q/A_Q`

= `rho (L/2)/(pi d^2)`

= `rho (L/2)/(pi d^2)`

Now, dividing R_Q by R:

`(R_Q)/R = ((rho L)/(2 pi d^2))/((4 rho L)/(pi d^2))`

= `1/2 xx 1/4`

= `1/8`

R_Q = `R/8`