Solving Wheatstone Bridge Problems

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 includes the unknown component. The circuit is said to be balanced when no current flows through the galvanometer connected across the two midpoints.

The condition for a Wheatstone bridge to be balanced is given by the ratio of resistances in adjacent arms being equal. Let the four arms of the bridge have resistances P, Q, R, and S. The balance condition is:

\(\frac{P}{Q} = \frac{R}{S}\)

In this problem, we are given the resistances of the four arms:

• Resistance P = 3 \(\Omega\)
• Resistance Q = 3 \(\Omega\)
• Resistance R = 3 \(\Omega\)
• Original Resistance S = 4 \(\Omega\)

The bridge is not balanced with the original S = 4 \(\Omega\) because \(\frac{P}{Q} = \frac{3}{3} = 1\) and \(\frac{R}{S} = \frac{3}{4}\). We are told that resistance S must be shunted by another resistance (\(R_{shunt}\)) in order for the bridge to be balanced. Shunting a resistance means connecting another resistance in parallel with it.

Let the effective resistance of the arm containing S after shunting be \(S_{eff}\). For the bridge to be balanced with the given values of P, Q, and R, the ratio must satisfy:

\(\frac{P}{Q} = \frac{R}{S_{eff}}\)

Substituting the given values:

\(\frac{3}{3} = \frac{3}{S_{eff}}\)

\(1 = \frac{3}{S_{eff}}\)

This implies that the required effective resistance for the fourth arm (\(S_{eff}\)) must be 3 \(\Omega\).

Now, we need to find the resistance \(R_{shunt}\) that, when connected in parallel with the original S = 4 \(\Omega\), results in an equivalent resistance of \(S_{eff} = 3\) \(\Omega\). The formula for the equivalent resistance (\(R_{eq}\)) of two resistors \(R_1\) and \(R_2\) connected in parallel is:

\(\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}\)

In our case, \(R_{eq} = S_{eff} = 3\) \(\Omega\), \(R_1 = S_{original} = 4\) \(\Omega\), and \(R_2 = R_{shunt}\). Substituting these values into the parallel resistance formula:

\(\frac{1}{3} = \frac{1}{4} + \frac{1}{R_{shunt}}\)

To find \(R_{shunt}\), we can rearrange the equation:

\(\frac{1}{R_{shunt}} = \frac{1}{3} - \frac{1}{4}\)

Find a common denominator for the fractions on the right side, which is 12:

\(\frac{1}{R_{shunt}} = \frac{4}{12} - \frac{3}{12}\)

\(\frac{1}{R_{shunt}} = \frac{4 - 3}{12}\)

\(\frac{1}{R_{shunt}} = \frac{1}{12}\)

Taking the reciprocal of both sides gives the value of \(R_{shunt}\):

\(R_{shunt} = 12\) \(\Omega\)

Therefore, a resistance of 12 \(\Omega\) must be shunted (connected in parallel) with the 4 \(\Omega\) resistance S in order to balance the Wheatstone bridge.

Wheatstone Bridge Problem Summary

Let's summarize the steps involved:

1. Identify the resistances of the four arms P, Q, R, and the original S.
2. State the balance condition for a Wheatstone bridge: \(\frac{P}{Q} = \frac{R}{S_{eff}}\).
3. Use the balance condition and the values of P, Q, and R to calculate the required effective resistance (\(S_{eff}\)) for the arm containing S.
4. Understand that shunting means connecting a resistance (\(R_{shunt}\)) in parallel with the original resistance (S).
5. Use the formula for parallel resistances: \(\frac{1}{S_{eff}} = \frac{1}{S_{original}} + \frac{1}{R_{shunt}}\).
6. Solve the equation for \(R_{shunt}\).

Resistance Calculation Details

The calculation involved subtracting fractions and finding the reciprocal:

\(\frac{1}{R_{shunt}} = \frac{1}{3} - \frac{1}{4}\)

\(\frac{1}{R_{shunt}} = \frac{1 \times 4}{3 \times 4} - \frac{1 \times 3}{4 \times 3}\)

\(\frac{1}{R_{shunt}} = \frac{4}{12} - \frac{3}{12}\)

\(\frac{1}{R_{shunt}} = \frac{1}{12}\)

\(R_{shunt} = 12\) \(\Omega\)

Revision Table: Key Concepts

Additional Information: Practical Wheatstone Bridge

In a practical Wheatstone bridge, one or more of the resistances (P, Q, R, S) are variable. By adjusting a variable resistor, the bridge is brought into balance, indicated by zero current through a galvanometer. Knowing the values of the other three resistances (usually two fixed ratio arms and one known variable resistance), the unknown resistance can be accurately calculated using the balance condition formula.

Wheatstone bridges are sensitive instruments capable of detecting small changes in resistance, making them useful in sensors (like strain gauges) where physical changes are converted into resistance changes.