Question 403: Current Electricity

Question

In the circuit shown, the resistance r is a variable resistance. If for r =
fR,
the heat generation in r is maximum then the value of
f
is :
![](https://app-content.cdn.examgoal.net/fly/@width/image/1l80yzs1u/e1ba1aab-6e23-4801-b858-73e1f1b3178a/325e7a20-33d0-11ed-bcbd-5353ee2ab2b1/file-1l80yzs1v.png?format=png)

Accepted Answer

Here equivalent resistance
R
eq
= R + ${{r 	imes R} \over {r + R}}$
=  R + ${{f{R^2}} \over {fR + R}}$
=   R + ${{fR} \over {f + 1}}$
=   ${{\left( {2f + 1} ight)R} \over {\left( {f + 1} ight)}}$
Circuit current,
${m I}$ = ${V \over {{{\mathop{m R}
olimits} _{eq}}}}$
=   ${{V\left( {f + 1} ight)} \over {{\mathop{m R}
olimits} \left( {2f + 1} ight)}}$
Current in the resistance r,
I = ${{m I} \over {f + 1}}$ = ${V \over {R\left( {2f + 1} ight)}}$
Heat generated in r,
H = ${m I}_2^2\,r$
=  ${{{V^2}f} \over {R{{\left( {2f + 1} ight)}^2}}}$
For maximum H,
${{dH} \over {df}}$ = 0
$ \Rightarrow $   ${{{V^2}} \over R}\left[ {{1 \over {{{\left( {2f + 1} ight)}^2}}} - {{4f} \over {{{\left( {2f + 1} ight)}^3}}}} ight]$ = 0
$ \Rightarrow $   2f + 1 = 4f
$ \Rightarrow $   2f = 1
$ \Rightarrow $   f = ${1 \over 2}$

Answer

${1 \over 4}$

Answer

${1 \over 2}$

Answer

${3 \over 4}$

Answer

1

Current Electricity

In the circuit shown, the resistance r is a variable resistance. If for r = fR, the heat generation in r is maximum then the value of f is :

Choose the correct answer: