$A$ steel rod of length $\ell$,cross-sectional area $A$,Young's modulus of elasticity $Y$,and thermal coefficient of linear expansion $\alpha$ is heated so that its temperature increases by $t\,^\circ C$. The work that can be done by the rod on heating is:

• A
  $(YA\alpha t) \times (\ell\alpha t)$
• B
  $\frac{1}{2}(YA\alpha t) \times (\ell\alpha t)$
• C
  $\frac{1}{2}(YA\alpha t) \times (\frac{1}{2})(\ell\alpha t)$
• D
  $2(YA\alpha t) \times (\ell\alpha t)$