JEE Main · 2020 · 03 Sep 2020 · Shift 1 · Current Electricity

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

ρ

(see figure). If the battery is connected to a resistance of value R, the maximum joule heating in R will take place for:

(A) $R = \frac{ρ}{π ℓ} ln (\frac{b}{a})$
(B) $R = \frac{ρ}{2 π ℓ} (\frac{b}{a})$
(C) $R = \frac{2 ρ}{π ℓ} ℓ (\frac{b}{a})$
(D) $R = \frac{ρ}{2 π ℓ} ln (\frac{b}{a})$ ✓ Correct

Answer: (D)

Solution

The battery delivers maximum power to

R

when

R

equals the internal resistance of the cell. For the electrolyte filling the gap between the coaxial cylinders, a shell of radius

x

and thickness

d x

contributes

d R = \frac{ρ d x}{2 π x ℓ}

to the radial resistance, and integrating from

a

b

gives

\frac{ρ}{2 π ℓ} ln \frac{b}{a}

. Hence maximum heating in

R

occurs for

R = \frac{ρ}{2 π ℓ} ln \frac{b}{a}

JEE Main · 2020 · 03 Sep 2020 · Shift 1 · Single correct option

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JEE Main · 2020 · 03 Sep 2020 · Shift 1 · Current Electricity

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