Question

A capacitor of capacitance $2\mu F$ is charged to a potential difference of 5 V. Now the charging battery is disconnected and the capacitor is connected in parallel to a resistor of $5\Omega$ and another unknown resistor of resistance R as shown in figure. If the total heat produced in $5\Omega$ resistance is $10\mu J$, then the unknown resistance R is equal to

• A. $10\Omega$
• B. $15\Omega$
• C. $\left(10/3\right)\Omega$
• D. $7.5\Omega$

Answer 1

The correct option is C $\left(10/3\right)\Omega$

We know that total heat produced $H=$ energy stored in capacitor $=\frac{1}{2}C{V}^2=\frac{1}{2}\times 2\times {10}^{-6}\times (5{)}^2=25\mu J$
Now this should distribute in inverse ratio of resistors, as they are in parallel.$(P=V^2/R)$ .
Thus, $\frac{H_5}{H_R}=\frac{R}{5}$
or $H_5=\frac{R}{R+5}H$
or $10=\frac{R}{R+5}\left(25\right)$
or $2=\frac{R}{R+5}\left(5\right)$
or $2R+10=5R\Rightarrow R=\frac{10}{3}\Omega$