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 :

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: