Question

A source of emf $E=10\text{V}$ and having negligible internal resistance is connected to a variable resistance. The resistance varies as shown in figure. The total charge that has passed through the resistor $R$ during the time interval from $t_1$ to $t_2$ is

• A. $40{\log }_{e}4$
• B. $30{\log }_{e}3$
• C. $20{\log }_{e}2$
• D. $10{\log }_{e}2$

Answer 1

The correct option is D $10{\log }_{e}2$

Let $R=at+b$
At $t=10\text{s},R=20\Omega$

$\Rightarrow 20=10a+b...\left(1\right)$

At $t=30\text{s},R=40\Omega$

$\Rightarrow 40=30a+b...\left(2\right)$

Solving these two equations, we get

$a=1\Omega s^{-1}$ and $b=10\Omega$

$\Rightarrow R=(t+10)$

Since, $i=\frac{E}{R}=\frac{10}{t+10}$

and $\Delta q=\underset{10}{\overset{30}{\int }}idt$

$\Rightarrow \Delta q=\underset{10}{\overset{30}{\int }}\left(\frac{10}{t+10}\right)dt$

$\therefore \Delta q=10{\log }_e\left(2\right)$
Hence, the correct answer is (D).