Consider a rod of uniform thermal conductivity whose one end $(x = 0)$ is insulated and the other end $(x =L)$ is exposed to flow of air at temperature $T_{\infty}$ with convective heat transfer coefficient $h$. The cylindrical surface of the rod is insulated so that the heat transfer is strictly along the axis of the rod. The rate of internal heat generation per unit volume inside the rod is given as $$\dot{q} = \cos \frac{2 \pi x }{L}$$ The steady state temperature at the mid-location of the rod is given as $T_{A}$. What will be the temperature at the same location, if the convective heat transfer coefficient increases to $2h$?

1. $T_{A} + \frac{\dot{q}L}{2h}$
2. $2T_{A}$
3. $T_{A}$
4. $T_{A}\left ( 1 - \frac{\dot{q}L}{4 \pi h} \right ) + \frac{\dot{q}L}{4 \pi h} T_{\infty }$