P, Q, R, and S are four wires of resistances 3 Ω, 3 Ω, 3 Ω, and 4 Ω respectively. They are connected to form the four arms of a Wheatstone bridge circuit. The resistance with which S must be shunted in order that the bridge may be balanced is__________
Fill in the blank with the correct answer from the options given below.
Fill in the blank with the correct answer from the options given below.
- 14 Ω
7Ω
- 15 Ω
12 Ω
The Correct Option is D
Solution and Explanation
To solve this problem, we need to balance the Wheatstone bridge. The condition for a balanced Wheatstone bridge is that the ratio of the resistances in one branch should be equal to the ratio in the other branch.
In this case, we have four wires with resistances:
\(P = 3 Ω, Q = 3 Ω, R = 3 Ω, S = 4 Ω\)
The Wheatstone bridge condition is: \(\frac {P}{Q} = \frac {R}{(S_{parallel})}\)
Given a shunt resistance X that is in parallel with S, we need to find X such that:
\(\frac {3}{3} = \frac {3}{4X/(4+X)}\)
This simplifies to:
\(1= \frac {3(4+X)}{4X}\)
\(4X = 3(4+X)\)
\(4X = 12 + 3X\)
Subtract 3X from both sides to get:
\(X = 12\)
Hence, the shunt resistance required to balance the bridge is \(12 Ω\).
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