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 :

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 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 =(2 ρ/π l) ln (( b / a )) (B) R =(ρ/π l) ln (( b / a )) (C) R =(ρ/2 π l)(( b / a )) (D) R =(ρ/2 π l) ln (( b / a ))

Tardigrade Admin · Accepted Answer

Maximum power in external resistance is generated when it is equal to internal resistance of battery.

< b r / > < b r / > d R = ρ \frac{d r}{2 π r l} < b r / > \int d R = \frac{ρ}{2 π l} \int_{a}^{b} \frac{1}{r} d r < b r / > R = \frac{ρ}{2 π l} [Ln (r)]_{a}^{b} < b r / > R = \frac{ρ}{2 π l} Ln (\frac{b}{a}) < b r / > r = R < b r / > < b r / >

for max heat transfer

$R = \frac{2 ρ}{π l} ln (\frac{b}{a})$

$R = \frac{ρ}{π l} ln (\frac{b}{a})$

$R = \frac{ρ}{2 π l} (\frac{b}{a})$

$R = \frac{ρ}{2 π l} ln (\frac{b}{a})$

R=πl2ρ​ln(ab​)

R=πlρ​ln(ab​)

R=2πlρ​(ab​)

R=2πlρ​ln(ab​)

Maximum power in external resistance is generated when it is equal to internal resistance of battery. <br/><br/>dR=ρ2πrldr​<br/>∫dR=2πlρ​∫ab​r1​dr<br/>R=2πlρ​[Ln(r)]ab​<br/>R=2πlρ​Ln(ab​)<br/>r=R<br/>​<br/> for max heat transfer

$R = \frac{2 ρ}{π l} ln (\frac{b}{a})$

$R = \frac{ρ}{π l} ln (\frac{b}{a})$

$R = \frac{ρ}{2 π l} (\frac{b}{a})$

$R = \frac{ρ}{2 π l} ln (\frac{b}{a})$