Question

The current that passes through the resistance $R_2$ nearest to the $V_0$ is

• A. $\frac{(k-1{)}^2}{k}\frac{V_0}{R_3}$
• B. $\frac{(k+1{)}^2}{k}\frac{V_0}{R_3}$
• C. $[k+\frac{1}{k^2}]\frac{V_0}{R_3}$
• D. $\left[\frac{k-1}{k^2}\right]\frac{V_0}{R_3}$

Answer 1

Answer: (D)

Given:

$V_1=\frac{V_0}{K},V_2=\frac{V_1}{K},V_3=\frac{V_2}{k},I=I_1+I_2$

$\frac{V_0-V_1}{R_1}=\frac{V_1-V_2}{R_1}+\frac{V_1-0}{R_2}$

$\frac{V_0-V_0/k}{R_1}=\frac{V_0/k-V_0/k^2}{R_1}+\frac{V_0/k}{R_2}$

$\frac{R_1}{R_2}=\frac{(k-1{)}^2}{k}$

Current in $R_1$ and $R_3$ will be the same:

$\frac{V_{n-1}-V_n}{R_1}=\frac{V_n}{R_3}$

$\frac{V_{n-1}-\frac{V_{n-1}}{k}}{R_1}=\frac{V_{n-1}}{k{R}_3}$

$R_1=R_3(k-1)$

Put the value of $R_1$, in the above expression, we get
$\frac{R_2}{R_3}=\frac{k}{k-1}$

Current in $R_2$ nearest to $V_0$, is
$I_2=\frac{V_1}{R_2}=\frac{V_0/k}{R_3\left[\frac{k}{k-1}\right]}=\left[\frac{k-1}{k^2}\right]\frac{V_0}{R_3}$