Question

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 $'b'$ filled in between with an electrolyte of resistivity $\rho$ (see figure). If the battery is connected to a resistance of value $R ,$ the maximum Joule heating in $R$ will take place for :

• A. $R =\frac{2 \rho}{\pi l} \ln \left(\frac{ b }{ a }\right)$
• B. $R =\frac{\rho}{\pi l} \ln \left(\frac{ b }{ a }\right)$
• C. $R =\frac{\rho}{2 \pi l}\left(\frac{ b }{ a }\right)$
• D. $R =\frac{\rho}{2 \pi l} \ln \left(\frac{ b }{ a }\right)$

Answer 1

The Correct Option is D

The problem involves calculating the resistance of an electrolytic cell that is shaped as two concentric cylinders, where the inner cylinder has radius $a$, the outer cylinder has radius $b$, and both have a length $l$. The space between these cylinders is filled with an electrolyte that has a resistivity $\rho$.

To solve this problem, we need to determine the resistance $R$ of the cylindrical shell of the electrolyte.

1. Consider an infinitesimally thin cylindrical shell of radius $r$ and thickness $dr$ filled with the electrolyte. The elemental resistance $dR$ of this shell can be calculated using the formula for resistance in a cylindrical shell:

\[ dR = \frac{\rho}{2\pi r l} \, dr \]

2. To find the total resistance $R$, integrate the elemental resistance from $a$ to $b$:

\[ R = \int_a^b \frac{\rho}{2\pi r l} \, dr \]

3. This integral evaluates to:

\[ R = \frac{\rho}{2\pi l} \ln \left(\frac{b}{a}\right) \]

Thus, the correct expression for the total resistance $R$ of the cylindrical shell filled with the electrolyte is:

\[ R = \frac{\rho}{2\pi l} \ln \left(\frac{b}{a}\right) \]

This matches the given correct answer option.