Question

The figure shows a graph of the current in a discharging circuit of a capacitor through a resistor of resistance $10\Omega$

• A. the initial potential difference across the capacitor is 100 V
• B. the capacitance of the capacitor is $\frac{1}{10ln2}F$
• C. the total heat produced in the circuit will be $\frac{500}{ln2}J$
• D. all of the above

Answer 1

The correct option is D all of the above

Initial voltage , $V_0=i_{0}R=10\left(10\right)=100V$ from graph $i_0=10$
The equation of current at time t is $i\left(t\right)=i_0{e}^{-t/RC}$
or $e^{t/RC}=\frac{i_0}{i}\Rightarrow t/RC=ln\left(i_0/i\right)$
or $C=\frac{t}{Rln\left(i_0/i\right)}$
From the graph, $i_0=10,$ at $t=2,i=2.5$
so $C=\frac{2}{10ln\left(10/2.5\right)}=\frac{1}{5ln4}=\frac{1}{5ln{2}^2}=\frac{1}{10ln2}F$
As the energy from the source 50 % stored in capacitor and 50 % loss so
Heat produced $=$ energy stored in capacitor $=\frac{1}{2}C{V}_0^2=\frac{1}{2}\times \frac{1}{10ln2}\times (100{)}^2=\frac{500}{ln2}J$