Two bars of radius ' r ' and ' 2 r ' are kept in contact as shown....

Two bars of radius '

r

' and '

2 r

' are kept in contact as shown. An electric current

I

is passed through the bars. Which one of the following is correct? Two bars of radius ' r ' and ' 2 r ' are kept in contact as shown. An electric current I is passed through the bars.

Two bars of radius ' r ' and ' 2 r ' are kept in contact as shown. An electric current I is passed through the bars.

Options

AHeat produced in bar $B C$ is 4 times the heat produced in bar $A B$
BElectric field in both halves is equal
CCurrent density across $A B$ is doubled that of across $B C$
DPotential difference across $A B$ is 4 times that of across $B C$

Correct answer

A. Heat produced in bar $B C$ is 4 times the heat produced in bar $A B$

Step-by-step solution

Current flowing through both the bars is equal. Now, the heat produced is given by

\begin{aligned}H & =I² R t \text{\text{ or } }H \propto R\text{\text{ or } } \frac{H_{A B}}{H_{B C}}=\frac{R_{A B}}{R_{B C}}\\ & =\frac{\left(\frac{1}{2 r}\right)^2}{\left(\frac{1}{r}\right)^2}\\ & =\frac{1}{4} \text{\text{ or } }H_{B C}=4 H_{A B} \end{aligned}

\left(\text{ as })R \propto \frac{1}{A} \propto \frac{1}{r^2}\right)

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