Two wires of the same material and length but diameters in the ratio 1 : 2 are stretched by the same force. The potential energy per unit volume for the two wire when stretched will be in the ratio

Options

A. `1:2`
B. `4:1`
C. `1:1`
D. `16:1`

Answer

D

Explanation

Potential energy per unit volume of the wire is `u= (1)/(2) (("stress")^(2))/("Young.s modulus") = (1)/(2) (S^(2))/(Y)`
As stress, `S= ("Force")/("Area") :. (S_(1))/(S_(2)) = ((F_(1))/(F_(2))) ((A_(2))/(A_(1)))`
But `F_(1) = F_(2) :. (S_(1))/(S_(2)) = (A_(2))/(A_(1))`
As the two wires are of the same material, therefore their Young.s moduli are the same. i.e, `Y_(1) = Y_(2)`
`:. (u_(1))/(u_(2)) = ((S_(1))/(S_(2)))^(2) = ((A_(2))/(A_(1)))^(2) = ((d_(2))/(d_(1)))^(4) = ((2)/(1))^(4) = (16)/(1)`