Question

Explain with the help of a circuit diagram, how the value of an unknown resistance can be determined using a Wheatstone bridge?

Answer 1

Introduction

A Wheatstone bridge is an electrical circuit used for the precise measurement of an unknown electrical resistance. It operates on the principle of null deflection, meaning the measurement is taken when the current through a central galvanometer is zero. This makes the measurement highly accurate.

Circuit Diagram and Components

The circuit consists of the following components, as shown in the diagram:

• A Voltage Source (V): A battery or DC power supply that provides the electromotive force (EMF) for the circuit.
• Four Resistors: These are arranged in a diamond or bridge formation.
  • R₁ and R₂: Two known, fixed resistors. These are often called the "ratio arms."
  • R₃: A known, but adjustable resistor (a variable resistor or rheostat).
  • Rₓ: The unknown resistance that we want to measure.
• A Galvanometer (G): A very sensitive ammeter connected between the junction of R₁ and R₃, and the junction of R₂ and Rₓ. Its purpose is to detect any current flowing between these two points.

Fig. Wheatstone bridge

Principle of Operation

The core principle of the Wheatstone bridge is the balanced bridge condition.

When current flows from the voltage source, it splits into two parallel branches: one through R₁ and R₃, and the other through R₂ and Rₓ.

The bridge is said to be "balanced" when the potential (voltage) at the point between R₁ and R₃ is exactly equal to the potential at the point between R₂ and Rₓ. When these two points are at the same potential, there is no potential difference across the galvanometer. According to Ohm's law (I = V/R), if the voltage (V) is zero, the current (I) must also be zero.

Therefore, at the balanced condition, the galvanometer shows zero deflection (a null reading).

Procedure for Measurement

1. Setup: The circuit is connected as shown in the diagram with the unknown resistor Rₓ in place.
2. Power On: The voltage source is switched on. Current flows through the circuit.
3. Balancing the Bridge: Initially, the bridge is likely unbalanced, and the galvanometer will show a deflection, indicating current flow through it. The value of the variable resistor, R₃, is then carefully adjusted.
4. Achieving Null Point: The resistance of R₃ is varied until the galvanometer's pointer returns to the zero mark. This indicates that the bridge is now balanced, and no current is flowing through the galvanometer. This specific resistance value of R₃ is noted.

Derivation and Calculation

When the bridge is balanced, we can derive the relationship between the four resistors.

Let's label the junctions:
Let the current flowing through the R₁-R₃ branch be I₁.
Let the current flowing through the R₂-Rₓ branch be I₂.

At the balanced condition, the current through the galvanometer is zero. This means:

1. The voltage drop across resistor R₁ is equal to the voltage drop across resistor R₂.
   > I₁ R₁ = I₂ R₂ --- (Equation 1)

2. Since no current flows through the galvanometer, the current I₁ also flows through R₃, and the current I₂ also flows through Rₓ. Therefore, the voltage drop across resistor R₃ is equal to the voltage drop across resistor Rₓ.
   > I₁ R₃ = I₂ Rₓ --- (Equation 2)

Now, we can divide Equation 2 by Equation 1:

(I₁ R₃) / (I₁ R₁) = (I₂ Rₓ) / (I₂ R₂)

The currents I₁ and I₂ cancel out from the equation:

R₃ / R₁ = Rₓ / R₂

This is the Wheatstone bridge balance equation.

To find the value of the unknown resistance Rₓ, we rearrange the formula:

*Rₓ = R₂ (R₃ / R₁)**

Since the values of the ratio arms (R₁ and R₂) are known, and the value of the variable resistor (R₃) at the balance point has been recorded, the unknown resistance Rₓ can be calculated precisely.