Problem 71 - Physics for Scientists and Engineers with Modern Physics

Question

A Wheatstone bridge is a type of "bridge circuit" used to make measurements of resistance. The unknown resistance to be measured, \(R_{x},\) is placed in the circuit with accurately known resistances \(R_{1}, R_{2},\) and \(R_{3}\) (Fig. \(\left.26-65\right) .\) One of these, \(R_{3},\) is a variable resistor which is adjusted so that when the switch is closed momentarily, the ammeter ( \(\mathrm{A}\) shows zero current flow. (a) Determine \(R_{x}\) in terms of \(R_{1}, R_{2},\) and \(R_{3}\) (b) If a Wheatstone bridge is "balanced" when \(R_{1}=630 \Omega, \quad R_{2}=972 \Omega\) and \(R_{3}=78.6 \Omega,\) what is the value of the unknown resistance?

Step-by-Step Solution

Answer

(a) \( R_x = \frac{R_1}{R_2} \times R_3 \); (b) \( R_x \approx 51.0 \, \Omega \).

Step 1: Analyze the Balanced Wheatstone Bridge Condition

In a Wheatstone bridge, the bridge is said to be balanced when the ratio of resistances in one arm is equal to the ratio in the other arm. This means that when the current through the ammeter is zero, the following condition holds:\[ \frac{R_1}{R_2} = \frac{R_x}{R_3} \] where \( R_x \) is the unknown resistance.

Step 2: Solve for the Unknown Resistance, \( R_x \)

To find \( R_x \) in terms of \( R_1, R_2, \) and \( R_3 \), solve the proportion:\[ \frac{R_x}{R_3} = \frac{R_1}{R_2} \]Cross-multiply to solve for \( R_x \):\[ R_x = \frac{R_1}{R_2} \times R_3 \]

Step 3: Plug in Known Values to Calculate \( R_x \)

Given that \( R_1 = 630 \, \Omega \), \( R_2 = 972 \, \Omega \), and \( R_3 = 78.6 \, \Omega \), substitute these values into the equation for \( R_x \):\[ R_x = \frac{630}{972} \times 78.6 \]Calculate \( R_x \):\[ R_x = \frac{630 \times 78.6}{972} \approx 51.0 \, \Omega \]

Key Concepts

Resistance Measurement

Resistance measurement is a fundamental concept used in various electrical circuits to determine the unknown resistance in a circuit. In the case of a Wheatstone bridge, this technique becomes quite precise due to its balanced conditions. The goal is to find the value of the unknown resistor, denoted as \( R_{x} \), which is part of the bridge circuit. Understanding the components of a Wheatstone bridge is crucial. These include four resistors; three are known, and one is unknown, which we're calculating.

To calculate \( R_{x} \), the balance equation of the Wheatstone bridge is used. When the bridge reaches a balanced state, no current flows through the ammeter. This zero current indicates that the potential drops are equal along both arms of the bridge.

• In one arm: \( R_1 \) and \( R_2 \)
• In the other arm: \( R_3 \) and \( R_x \)

This setup allows us to set up a proportion that directly relates \( R_{x} \) to the known resistances, leading to precise resistance measurements when the bridge is perfectly balanced.

Balanced Bridge Condition

A balanced Wheatstone bridge is a state where the current through the connected ammeter reads zero. This condition occurs when the ratio of the resistances in one of the bridge's arms is equal to the ratio in the other arm. In other words, if we denote the four resistors as \( R_1, R_2, R_3, \) and \( R_x \), the bridge is balanced when:

\[ \frac{R_1}{R_2} = \frac{R_x}{R_3} \]

Reaching this condition is critical for accurately measuring the unknown resistance \( R_x \). It relies on the principle that the voltage drop across one component should equal the voltage drop across its adjacent component in a crosswise manner. Achieving a balanced bridge doesn't just help in accurate measurements but also shows that the components are behaving as expected in a predictive and logical manner. This aspect provides an indirect check on the system's integrity.

• Equal ratios simplify to a straightforward calculation of \( R_x \).
• The absence of current flow is a reliable indicator of a balanced state.

In practice, when setting up the circuit, accurately balancing the bridge saves time and reduces error in the calculations.

Variable Resistor Adjustment

The variable resistor, often referred to as a rheostat or potentiometer in the Wheatstone bridge, plays a pivotal role. Its purpose is to adjust until the bridge reaches a balanced state. This component, denoted as \( R_3 \) in many cases, is manually tweaked during use to achieve the desired proportion where no current flows through the ammeter.

The adjustment process is straightforward yet crucial because having the bridge in a balanced condition is what allows for the accuracy of the measurement of \( R_{x} \). When you tweak the variable resistor, the resistance changes in small increments, and during this process:

• The balanced state is indicated by zero current flow through the ammeter.
• Reaching the balanced state means you can simplify the resistive calculations using the balance equation: \( \frac{R_1}{R_2} = \frac{R_x}{R_3} \).
• This adjustment offers high precision in laboratory settings and in field measurements.

Practically, carrying out these adjustments might take several attempts, but once the bridge is balanced, you'll have a precise method for determining an unknown resistor's value.