Q 12-03-234JEE MainJEE Main 2020 (3 Sep, Shift 1)Medium
Model a torch battery of length $l$ to be made up of a thin cylindrical bar of radius $a$ and a concentric thin cylindrical shell of radius $b$, with the space in between filled with an electrolyte of resistivity $\rho$. If the battery is connected to a resistance of value $R$, the maximum Joule heating in $R$ will take place for:
Answer: (B) $R = \dfrac{\rho}{2\pi l}\ln\left(\dfrac ba\right)$
Current flows radially through the electrolyte. A thin shell of radius $r$ and thickness $dr$ has resistance $\dfrac{\rho\,dr}{2\pi rl}$, so the internal resistance is
$$r_{int} = \int_a^b \frac{\rho\,dr}{2\pi rl} = \frac{\rho}{2\pi l}\ln\frac ba$$
Power in $R$ is maximum when $R = r_{int}$ (maximum power transfer).
Solution by Sreeraj P, M.Sc Physics