Question

If the Wheatstone bridge shown in the figure is balanced, then the value of R is

• A. 50 Ω
• B. 200 Ω
• C. 300 Ω
• D. 100 Ω

Hint:

For a balanced Wheatstone bridge, the ratio of resistances in adjacent arms is equal. That is, if the resistances are arranged in a diamond shape P, Q, R, S clockwise, and the galvanometer is between the junction of P-Q and R-S, then $\frac{P}{R} = \frac{Q}{S}$ (or $\frac{P}{Q} = \frac{R}{S}$ depending on specific labeling, but usually cross-multiplied products are equal: $P \times T = S \times Q$). In the given diagram's common labeling, the balance condition is $\frac{P}{S} = \frac{Q}{T}$.

Answer 1

The Correct Option is D

Step 1: Understand the Concept of a Balanced Wheatstone Bridge
A Wheatstone bridge is a circuit used to measure an unknown electrical resistance by balancing two legs of a bridge circuit, one leg of which contains the unknown component. When the bridge is balanced, no current flows through the galvanometer (or the detector arm), meaning the potential difference between points B and D (in the given figure) is zero. The condition for a balanced Wheatstone bridge is that the ratio of resistances in the two arms of the bridge is equal.
Referring to the given diagram, let the resistances be:
$P = 200 \, \Omega$ (resistance between A and B)
$Q = R$ (resistance between B and C)
$S = 100 \, \Omega$ (resistance between A and D)
$T = 50 \, \Omega$ (resistance between D and C)
The balance condition is given by the formula: \[ \frac{P}{S} = \frac{Q}{T} \] Step 2: Substitute the Given Values into the Balance Condition
Substitute the values of the known resistances and the unknown resistance R into the formula: \[ \frac{200 \, \Omega}{100 \, \Omega} = \frac{R}{50 \, \Omega} \] Step 3: Solve for the Unknown Resistance (R)
Simplify the left side of the equation: \[ 2 = \frac{R}{50 \, \Omega} \] Now, solve for R: \[ R = 2 \times 50 \, \Omega \] \[ R = 100 \, \Omega \] Step 4: Verify the Result
To verify, substitute $R = 100 \, \Omega$ back into the balance condition: \[ \frac{P}{S} = \frac{200 \, \Omega}{100 \, \Omega} = 2 \] \[ \frac{Q}{T} = \frac{100 \, \Omega}{50 \, \Omega} = 2 \] Since $\frac{P}{S} = \frac{Q}{T} = 2$, the bridge is indeed balanced with $R = 100 \, \Omega$. Step 5: Analyze the Options
\begin{itemize} \item Option (1): 50 $\Omega$. Incorrect. \item Option (2): 200 $\Omega$. Incorrect. \item Option (3): 300 $\Omega$. Incorrect. Option (4): 100 $\Omega$. Correct, as it matches our calculated value of R. \end{itemize}