Question

In a Wheatstone bridge (see fig.), resistances P and Q are approximately equal. When $R=400\Omega$, the bridge is equal. When $R=400\Omega$, the bridge is balanced. On inter-changing P and Q, the value of R, for balance, is $405\Omega$. The value of X is close to

• A. $403.5ohm$
• B. $404.5ohm$
• C. $401.5ohm$
• D. $402.5ohm$

Answer 1

The correct option is D

$402.5ohm$

Step 1: Given

Initial resistance, $R_1=400ohm$

Final resistance, $R_2=405ohm$

Step 2: Balance the bridge with initial resistance

We can calculate the unknown resistance in a Wheatstone bridge by balancing the two legs of the bridge circuit.

Therefore,

$\frac{P}{Q}=\frac{R_1}{X}-\left(1\right)$

Step 3: Balance the bridge with the final resistance

After interchanging P and Q,

$\frac{Q}{P}=\frac{R_2}{X}-\left(2\right)$

Step 4: Solve the equations

Dividing equations (1) by (2)

$$\begin{gathered} \frac{\frac{P}{Q}}{{\displaystyle \frac{Q}{P}}}=\frac{\frac{R_1}{X}}{{\displaystyle \frac{R_2}{X}}} \\ 1=\frac{R_1{R}_2}{X^2} \end{gathered}$$

Step 5: Solve for X.

$$\begin{gathered} X^2=R_1{R}_2 \\ X=\sqrt{400\times 405} \\ =402.5ohm \end{gathered}$$

Therefore, option D is the correct option.