Question

In a wheat stone's bridge, three reststance \(P, Q\) and \(R\) connected in three arm a and the fourth arm is formed by two resistances \(\mathrm{S}_{1}\) and \(\mathrm{S}_{2}\) connected in parallel The condition for bridge to be balanced will be. (A) \((\mathrm{P} / \mathrm{Q})=\left\\{\mathrm{R} /\left(\mathrm{S}_{1}+\mathrm{S}_{2}\right)\right\\}\) (B) \((\mathrm{P} / \mathrm{Q})=\left\\{(2 \mathrm{R}) /\left(\mathrm{S}_{1}+\mathrm{S}_{2}\right)\right\\}\) (C) \((\mathrm{P} / \mathrm{Q})=\left[\left\\{\mathrm{R}\left(\mathrm{S}_{1}+\mathrm{S}_{2}\right)\right\\} /\left\\{\mathrm{S}_{1} \mathrm{~S}_{2}\right\\}\right]\) (D) \((\mathrm{P} / \mathrm{Q})=\left[\left\\{\mathrm{R}\left(\mathrm{S}_{1}+\mathrm{S}_{2}\right)\right\\} /\left\\{2 \mathrm{~S}_{1} \mathrm{~S}_{2}\right\\}\right]\)

Short answer

The short answer is: \( \frac{P}{Q} = \frac{R(S_1+S_2)}{S_1S_2} \).

Step-by-step solution

Recall Wheatstone Bridge principle

A Wheatstone bridge is balanced when the ratio of the resistances in the bridge is equal, that is, \(\frac{P}{Q} = \frac{R}{S}\), where S is the equivalent resistance of the parallel combination of S1 and S2.

Calculate equivalent resistance of S1 and S2 in parallel

The equivalent resistance S of two resistances connected in parallel can be found using the formula: \[\frac{1}{S} = \frac{1}{S_1}+\frac{1}{S_2}\] We can solve this equation for S to find the equivalent resistance: \[S = \frac{S_1S_2}{S_1+S_2}\]

Substitute equivalent resistance into Wheatstone Bridge principle

Now, substitute the equivalent resistance S into the Wheatstone bridge balance condition: \(\frac{P}{Q} = \frac{R}{S}\) Substitute S with the equivalent resistance calculated in the previous step: \[\frac{P}{Q} = \frac{R}{\frac{S_1S_2}{S_1+S_2}}\]

Simplify the expression

Finally, simplify the expression by multiplying both sides by \((S_1+S_2)\): \[\frac{P}{Q} = \frac{R(S_1+S_2)}{S_1S_2}\] This expression shows the condition for the bridge to be balanced, which matches option (C).

Key concepts

Balanced Bridge Condition

The concept of a balanced Wheatstone bridge is essential for understanding how this type of electrical circuit can be used to measure unknown resistance. The balance condition is achieved when there is no current flowing through the galvanometer, an instrument used to detect current. To reach this state, the ratio of the resistances in the two bridge arms must be equal. In mathematical terms, if you have resistance values labeled as \(P\), \(Q\), and \(R\), with \(S\) being the equivalent resistance of components \(S_1\) and \(S_2\) in parallel, then the bridge is balanced when:

• \(\frac{P}{Q} = \frac{R}{S}\)

This equality allows us to determine unknown resistances within the circuit by adjusting the bridge until it reaches this state of balance. Understanding this principle is fundamental to applying the Wheatstone bridge in practical scenarios.

Parallel Resistance Calculation

When dealing with resistive elements connected in parallel, such as \(S_1\) and \(S_2\) in our Wheatstone bridge example, it is crucial to calculate their equivalent resistance accurately. In a parallel configuration, the total or equivalent resistance \(S\) can be calculated using the formula:

• \(\frac{1}{S} = \frac{1}{S_1} + \frac{1}{S_2}\)

Solving for \(S\) gives us:

• \(S = \frac{S_1 S_2}{S_1 + S_2}\)

This formula lets us replace the parallel pair with a single resistance value in theoretical calculations, simplifying the equations needed to analyze the circuit. It is essential to simplify circuits in this way to make manual or computational analysis manageable and accurate.

Resistance Ratio Principle

The resistance ratio principle is an underlying concept in many circuit analysis techniques, especially in the context of the Wheatstone bridge. It refers to the comparison of ratios of resistances on different arms of the bridge to determine balance.

• In a balanced Wheatstone bridge, the relationship is expressed as: \(\frac{P}{Q} = \frac{R(S_1+S_2)}{S_1S_2}\)

This principle provides a practical way to derive an unknown resistance in a circuit when the bridge is balanced. Simplification of this principle will often involve algebraic manipulation to achieve the right conditions for a balance, as shown in the equation above. Understanding and applying this principle is key to leveraging the Wheatstone bridge for delicate and accurate resistance measurements in various applications.