Question

A graph between current and time during charging of a capacitor by a battery in series with a resistor is shown. The graphs are drawn for two circuits. Here $R_1,R_2,$$C_1,C_2$ and $V_1,V_2$ are the values of resistances, capacitances and EMF of cells in the two circuits. If only two parameters out of resistance, capacitance, emf are different in two circuits. What will be the correct option?

• A. $V_1=V_2,R_1>R_2,C_1>C_2$
  
• B. $V_1>V_2,R_1>R_2,C_1=C_2$
  
• C. $V_1<V_2,R_1<R_2,C_1=C_2$
  
• D. $V_1<V_2,R_1=R_2,C_1<C_2$
  

Answer 1

The correct option is C $V_1<V_2,R_1<R_2,C_1=C_2$


The current at any time $t$ is given by,$$I=I_0{e}^{-t/RC}$$As, $I_0=\frac{V}{R}$

$\Rightarrow I=\frac{V}{R}{e}^{-t/RC}$

Taking $\ln$ on both sides we have,

$\ln I=\ln \left(\frac{V}{R}\right)+\frac{-t}{RC}\ln e$

$\Rightarrow \ln I=\ln \left(\frac{V}{R}\right)-\frac{t}{RC}.....\left(1\right)$

For the plotting of $\ln I$ vs $t$, it can be inferred from equation (1) that it will be a straight line with, $\text{slope |m|}=\frac{-1}{RC}$ and $\text{Intercept}=\ln \left(\frac{V}{R}\right)$

From the given graph,

${\text{(Intercept)}}_1={\text{(Intercept)}}_2$

and ${\text{|(slope)}}_1|>{\text{|(slope)}}_2|$

It is only possible if $R_1<R_2$ and $V_1<V_2$ such that $\ln \left(\frac{V}{R}\right)$ remains unaffected & $|m_2|<|m_1|$ is achieved.

$\Rightarrow V_1<V_2,R_1<R_2,C_1=C_2$
(c) is the correct answer.