A wire of length `L_0` is supplied heat to raise its temperature by T. if `gamma` is the coefficient of volume expansion of the wire and Y is Young's modulus of the wire then the energy density stored in the wire is

Options

A. `(1)/(2)gamma^2T^2Y`
B. `(1)/(3)gamma^2T^2Y^3`
C. `(1)/(18)(gamma^2T^2)/(T)`
D. `(1)/(18)gamma^2T^2Y`

Answer

D

Explanation

Due to heating the length of the wire increases.
`:.` Longitudinal strain is produced `implies(DeltaL)/(L)=alphaxxDeltaT`
Elastic potential energy per unit volume
`E=(1)/(2)xx` stress `xx` strain `=(1)/(2)xxYxx("strain")^2`
`impliesE=(1)/(2)xxYxx((DeltaL)/(L))^2=(1)/(2)xxYxxalpha^2xxDeltaT^2`
or `E=(1)/(2)xxYxx((gamma)/(3))^2xxT^2=(1)/(18)gamma^2YT^2`