$A$ compressive force, $F$ is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by $\Delta T$. The net change in its length is zero. Let $l$ be the length of the rod, $A$ its area of cross-section, $Y$ its Young's modulus, and $\alpha$ its coefficient of linear expansion. Then, $F$ is equal to

• A
  $l^2 Y \alpha \Delta T$
• B
  $l A Y \alpha \Delta T$
• C
  $A Y \alpha \Delta T$
• D
  $\frac{A Y}{\alpha \Delta T}$