Question

In a Wheatstone bridge, all four arms have equal resistance $r$. If the resistance of the galvanometer arm is also $r$, the equivalent resistance of the combination as seen by the battery is

• A. $\frac{r}{4}$
• B. $\frac{r}{2}$
• C. $r$
• D. $2r$

Answer 1

The correct option is C

$r$

Step 1: Given Data

Let the Wheatstone bridge consist of four resistances $R_1,R_2,R_3$ and $R_4$ as shown in the given figure.

It also consists of a galvanometer or resistance $R_g$.

From the question we have,

$R_1=R_2=R_3=R_4=R_g=r$

Step 2: Calculate the equivalent resistance

When a Wheatstone bridge is in a balanced condition, no current flows through the coil and the potential difference across the galvanometer is zero.

Since $R_1$ and $R_2$ are in series,

$R_{CAD}=R_1+R_2$

Since $R_3$ and $R_4$ are in series,

$R_{CBD}=R_3+R_4$

Assume the battery resistance $r_i=0$ we can write,

$R_{CAD}$ and $R_{CBD}$ are in parallel.

$\therefore \frac{1}{R_{eq}}=\frac{1}{R_1+R_2}+\frac{1}{R_3+R_4}$

$R_{eq}=\frac{\left(R_1+R_2\right)\left(R_3+R_4\right)}{\left(R_1+R_2\right)+\left(R_3+R_4\right)}$

$=\frac{\left(r+r\right)\left(r+r\right)}{\left(r+r\right)+\left(r+r\right)}$

$=\frac{4r^2}{4r}$

$=r$

Hence, the correct answer is option (C).