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Wheatstone Bridge: Derive the Equation of the Balanced State Using Kirchhoff's Laws

A complete Class 12 Physics guide covering the Wheatstone Bridge principle, circuit diagram, step-by-step derivation using Kirchhoff's current and voltage laws, balanced condition formula, applications, and solved examples.

Physics 19 September, 2026 15 min read

Last updated: September 19, 2026

What is a Wheatstone Bridge?

Definition

A Wheatstone Bridge is an electrical circuit consisting of four resistances arranged in a quadrilateral (diamond) configuration, with a sensitive galvanometer connected across one diagonal and a battery (EMF source) connected across the other diagonal. It is used to measure an unknown resistance with high precision by balancing two legs of the bridge circuit.

The Wheatstone Bridge was invented by Samuel Hunter Christie in 1833 and later popularized by Sir Charles Wheatstone in 1843. It remains one of the most widely used circuits in physics laboratories and industrial measurement systems. In Hindi, it is commonly referred to as "Wheatstone Bridge kya hai" — a circuit that measures unknown resistance by comparing it with known standard resistances.

Wheatstone Bridge Principle

The principle of the Wheatstone Bridge is based on the concept of a null deflection — when the bridge is balanced, no current flows through the galvanometer. This occurs when the ratio of resistances in one branch equals the ratio in the other branch. The bridge essentially converts a resistance measurement into a null measurement, which is far more accurate than direct measurement methods.

The four arms of a Wheatstone Bridge have resistances labelled P, Q, R, and S. These four resistances form the four sides of the quadrilateral shape. The key advantage of this arrangement is that the measurement depends only on the ratio of resistances, not on the exact values of the source voltage or the galvanometer sensitivity.

Key Components of a Wheatstone Bridge

  • Four Resistances (P, Q, R, S) — form the four arms of the bridge. Three are typically known, and one is the unknown to be measured.
  • Galvanometer (G) — a sensitive current-detecting instrument connected between junctions B and D.
  • Battery / EMF Source (E) — provides the driving voltage, connected between junctions A and C.
  • Key or Switch — used to complete the circuit and make/break connections.

Wheatstone Bridge Diagram

The following diagram shows the standard Wheatstone Bridge circuit with four arm resistances P, Q, R, and S, a galvanometer (G) across BD, and a battery (E) across AC.

Wheatstone Bridge Circuit Diagram P Q R S G A B C D E + − I₁ I₂ Ig I → Arm 1 Arm 2 Arm 3 Arm 4

Figure 1: Wheatstone Bridge Circuit Diagram — The four arms P, Q, R, S form a diamond shape. The galvanometer G is connected between junctions B and D, and the battery E is connected between junctions A and C.

In the diagram above, current I enters the bridge at junction A from the battery. It splits into two branches: current I₁ flows through the arm containing resistance P toward junction B, and current I₂ flows through the arm containing resistance R toward junction D. A galvanometer current Ig may flow between B and D.

Kirchhoff's Laws — Foundation for the Derivation

Before deriving the Wheatstone Bridge balance equation, it is essential to understand Kirchhoff's two laws of electrical circuits. These laws, formulated by Gustav Kirchhoff in 1845, are fundamental tools for analyzing any complex electrical circuit. They are based on the principles of conservation of charge and conservation of energy.

Kirchhoff's First Law (KCL — Kirchhoff's Current Law)

Statement

Kirchhoff's First Law, also called the Junction Rule or Kirchhoff's Current Law (KCL), states: "The algebraic sum of all currents meeting at any junction (node) in an electrical circuit is zero."

Mathematical Expression

ΣI = 0   (at any junction)

Convention: Current entering the junction is taken as positive (+), and current leaving is taken as negative (−).

Kirchhoff's Current Law (KCL) at a Junction Junction I₁ (in) I₂ (in) I₃ (out) I₄ (out) I₁ + I₂ = I₃ + I₄   ⇒   I₁ + I₂ − I₃ − I₄ = 0

Figure 2: Kirchhoff's Current Law — The sum of currents entering a junction equals the sum of currents leaving it (conservation of charge).

Physical Basis

KCL is based on the law of conservation of electric charge. Since charge cannot accumulate at any junction in a steady-state circuit, whatever charge flows into a junction must flow out. This law applies to every junction in any circuit, no matter how complex.

Kirchhoff's Second Law (KVL — Kirchhoff's Voltage Law)

Statement

Kirchhoff's Second Law, also called the Loop Rule or Kirchhoff's Voltage Law (KVL), states: "The algebraic sum of all the potential differences (EMFs and voltage drops) around any closed loop in a circuit is zero."

Mathematical Expression

ΣV = 0   (around any closed loop)

Convention: EMFs in the direction of traversal are positive; voltage drops (IR) in the direction of current are positive when going from + to −.

Kirchhoff's Voltage Law (KVL) around a Loop R₁ R₂ R₃ E + − I → I ↓ ← I I ↑ Loop E − IR₁ − IR₂ − IR₃ = 0

Figure 3: Kirchhoff's Voltage Law — The sum of all EMFs and voltage drops around any closed loop equals zero (conservation of energy).

Physical Basis

KVL is based on the law of conservation of energy. As a charge moves around a complete loop and returns to its starting point, its total gain in potential energy (from EMF sources) must equal its total loss (through resistive voltage drops). The net energy change around any closed loop is zero.

Derivation of the Wheatstone Bridge Balance Condition Using Kirchhoff's Laws

We now derive the equation of the balanced state in a Wheatstone Bridge using Kirchhoff's laws. This is one of the most important derivations in Class 12 Physics (Current Electricity chapter).

Wheatstone Bridge — Derivation Diagram G A B C D P Q R S I₁ I₂ (I₁ − Ig) (I₂ + Ig) Ig Loop 1 (ABDA) Loop 2 (BCDB) E At balance: Ig = 0  →  No current through galvanometer

Figure 4: Wheatstone Bridge with current labels and two Kirchhoff loops marked for derivation. I₁ flows through arm P, I₂ flows through arm R, and Ig flows through the galvanometer.

Step-by-Step Derivation

1
Set up the circuit and label currents:

Let the total current from the battery be I. At junction A, the current divides into:

  • I₁ through arm AB (resistance P)
  • I₂ through arm AD (resistance R)

So by KCL at junction A: I = I₁ + I₂

Let Ig be the current through the galvanometer from B to D.

2
Apply KCL at Junction B:

Current entering B = I₁ (from A through P)

Current leaving B = Ig (through G to D) + (I₁ − Ig) through Q toward C

So current through Q = (I₁ − Ig)

3
Apply KCL at Junction D:

Current entering D = I₂ (from A through R) + Ig (from B through G)

Current leaving D through S toward C = (I₂ + Ig)

4
Apply the Balanced Condition:

When the bridge is balanced, the galvanometer shows null deflection, meaning:

Ig = 0

This means the potential at B equals the potential at D (VB = VD).

With Ig = 0:

  • Current through P and Q = I₁ (same current flows through both)
  • Current through R and S = I₂ (same current flows through both)
5
Apply KVL to Loop 1 (ABDA):

Traversing the closed loop A → B → D → A:

  • Voltage drop across P (A to B): +I₁P
  • Voltage drop across G (B to D): 0 (since Ig = 0)
  • Voltage drop across R (D to A, opposite to I₂): −I₂R

By KVL:   I₁P − I₂R = 0

I₁P = I₂R     — Equation (i)

6
Apply KVL to Loop 2 (BCDB):

Traversing the closed loop B → C → D → B:

  • Voltage drop across Q (B to C): +I₁Q
  • Voltage drop across S (C to D, opposite to I₂): −I₂S
  • Voltage drop across G (D to B): 0 (since Ig = 0)

By KVL:   I₁Q − I₂S = 0

I₁Q = I₂S     — Equation (ii)

7
Divide Equation (i) by Equation (ii):

(I₁P) / (I₁Q) = (I₂R) / (I₂S)

Cancelling I₁ from the left side and I₂ from the right side:

Wheatstone Bridge Balance Condition (Final Result)

P / Q = R / S

This is the Wheatstone Bridge balance equation derived using Kirchhoff's laws.

What This Equation Tells Us

When the Wheatstone Bridge is in the balanced state, the ratio of resistances in the two upper arms (P/Q) equals the ratio of resistances in the two lower arms (R/S). If any three of the four resistances are known, the fourth (unknown) resistance can be calculated precisely using this relationship. This is why the Wheatstone Bridge is such a powerful tool for resistance measurement.

Balance Condition of Wheatstone Bridge — Summary

The balanced condition of a Wheatstone Bridge can be expressed in several equivalent forms:

Equivalent Forms of the Balance Condition

P / Q = R / S

P × S = Q × R

PS = QR

Conditions at Balance

  1. No current flows through the galvanometer (Ig = 0).
  2. The potential difference across the galvanometer is zero (VB = VD).
  3. The galvanometer shows null deflection (pointer stays at zero).
  4. The product of opposite arm resistances are equal: P × S = Q × R.

Finding the Unknown Resistance

If S is the unknown resistance, then from the balance condition:

S = Q × R / P

By adjusting the known resistances P, Q, and R until the galvanometer reads zero, the unknown resistance S can be determined precisely.

Balance Condition: Visual Summary P Q Left Branch = R S Right Branch or PS = QR

Figure 5: The Wheatstone Bridge balance condition — the ratio of the left branch resistances equals the ratio of the right branch resistances.

Unbalanced Wheatstone Bridge Formula

When the bridge is not balanced (P/Q ≠ R/S), a current flows through the galvanometer. The voltage across the galvanometer terminals (B and D) in an unbalanced Wheatstone Bridge can be calculated as follows:

Unbalanced Bridge Voltage

VG = V × [ P/(P+Q) − R/(R+S) ]

Where V is the supply voltage and P, Q, R, S are the four arm resistances. Wait — note: this formula uses a slightly different arm convention. With the diamond layout: VBD = E × [R₁/(R₁+R₂) − R₃/(R₃+R₄)].

In the unbalanced state:

  • Current flows through the galvanometer (Ig ≠ 0).
  • The galvanometer deflects either to the left or right, depending on the direction of current.
  • The potential at B and D are not equal (VB ≠ VD).
  • The greater the imbalance (deviation from P/Q = R/S), the larger the galvanometer deflection.

This sensitivity to imbalance is what makes the Wheatstone Bridge useful in sensor circuits — even a small change in one resistance arm (due to temperature, strain, or pressure) produces a measurable voltage output.

Solved Examples & Numerical Problems

Example 1: Finding Unknown Resistance

Problem: The four arms of a Wheatstone Bridge have resistances as shown: P = 100 Ω, Q = 200 Ω, R = 150 Ω, and S is unknown. If the bridge is balanced, find the value of S.

Solution:

At balance: P/Q = R/S

100/200 = 150/S

1/2 = 150/S

S = 150 × 2 = 300 Ω

Example 2: Checking Balance Condition

Problem: A Wheatstone Bridge has P = 10 Ω, Q = 30 Ω, R = 20 Ω, S = 60 Ω. Is the bridge balanced?

Solution:

Check if P/Q = R/S:

P/Q = 10/30 = 1/3

R/S = 20/60 = 1/3

Since P/Q = R/S = 1/3, the bridge is balanced. No current flows through the galvanometer.

Example 3: Class 12 Board-Type Problem

Problem: In a Wheatstone Bridge, P = 5 Ω, Q = 10 Ω, and R = 15 Ω. The bridge is balanced. Calculate S. Also, if the positions of the galvanometer and the battery are interchanged, will the bridge still remain balanced?

Solution:

Part (a): At balance: P/Q = R/S

5/10 = 15/S ⇒ S = 15 × 10/5 = 30 Ω

Part (b): Yes. The Wheatstone Bridge balance condition (P/Q = R/S) depends only on the ratio of resistances, not on the positions of the galvanometer and battery. Interchanging them does not affect the balance condition. The bridge remains balanced because the product relationship PS = QR (5 × 30 = 10 × 15 = 150) still holds.

Example 4: Sensitivity Analysis

Problem: In a balanced Wheatstone Bridge with P = 100 Ω, Q = 100 Ω, R = 100 Ω, S = 100 Ω, the resistance S changes by 1% to 101 Ω. If the battery EMF is 5V, find the approximate voltage across the galvanometer.

Solution:

VG = E × [P/(P+R) − Q/(Q+S)]

VG = 5 × [100/(100+100) − 100/(100+101)]

VG = 5 × [0.5 − 100/201]

VG = 5 × [0.5 − 0.49751]

VG = 5 × 0.00249 ≈ 0.01245 V ≈ 12.45 mV

Applications of Wheatstone Bridge

The Wheatstone Bridge circuit has widespread applications in both laboratory measurements and modern industrial instrumentation. Here are the key applications, especially relevant for Class 12 board exams:

Applications of Wheatstone Bridge Wheatstone Bridge Resistance Precise measurement of unknown resistance Strain Gauge Measuring mechanical stress & deformation Temperature RTD-based precision temperature sensing Pressure Sensor Piezoresistive pressure transducers in industry Light Sensors & Gas Detection

Figure 6: Key applications of the Wheatstone Bridge in science and industry.

Detailed Applications

  1. Precise Resistance Measurement: The primary application — a Wheatstone Bridge is used to determine an unknown resistance by comparing it with standard known resistances. The null method provides accuracy superior to direct ammeter-voltmeter methods.
  2. Strain Gauge Measurement: Strain gauges (whose resistance changes with mechanical deformation) are placed in one arm of a Wheatstone Bridge to measure stress, strain, force, and weight in structural engineering.
  3. Temperature Sensing (RTDs): Resistance Temperature Detectors use the temperature-dependent resistance of platinum wire in a Wheatstone Bridge configuration for precision temperature measurement in laboratories and industries.
  4. Pressure Transducers: Piezoresistive sensors in a Wheatstone Bridge arrangement convert pressure changes into measurable electrical signals in automotive and aerospace applications.
  5. Light Intensity Measurement: Photoresistors (LDRs) placed in a Wheatstone Bridge arm enable precise light intensity measurement in photometry.
  6. Post Office Box: A practical laboratory instrument based on the Wheatstone Bridge principle used for resistance measurement in Class 12 physics practicals.
  7. Metre Bridge (Slide Wire Bridge): A simplified form of the Wheatstone Bridge used in school and college laboratories where a uniform wire replaces two of the four resistances.

Key Differences: Kirchhoff's Current Law vs Voltage Law

Feature Kirchhoff's First Law (KCL) Kirchhoff's Second Law (KVL)
Also Known As Junction Rule / Current Law Loop Rule / Voltage Law
Applies At Junctions (nodes) in a circuit Closed loops in a circuit
Physical Principle Conservation of electric charge Conservation of energy
Mathematical Form ΣI = 0 (at a junction) ΣV = 0 (around a loop)
What It Means Current in = Current out Sum of EMFs = Sum of IR drops
Used For Finding unknown currents at junctions Finding unknown voltages/EMFs in loops
Convention Entering current is +ve, leaving is −ve EMF in traversal direction is +ve, IR drop is +ve

Quick Revision Summary

What is a Wheatstone Bridge? Four-resistance diamond circuit with galvanometer & battery for measuring unknown resistance.
Balance Condition P/Q = R/S  or  PS = QR
At Balance Ig = 0, VB = VD (null deflection)
Kirchhoff's 1st Law ΣI = 0 at any junction (charge conservation)
Kirchhoff's 2nd Law ΣV = 0 around any loop (energy conservation)

Lab Experiments & Virtual Demos

Reinforce your understanding of the Wheatstone Bridge, Kirchhoff's laws, and related concepts with hands-on and virtual laboratory experiments. Click on any experiment to try the interactive demo.

Wheatstone Bridge & Carey Foster Bridge Experiments

Carey Foster Bridge

Measurement of Electrical Resistance of Copper Wire Using Carey Foster Bridge

Determine the unknown resistance of a copper wire specimen using the Carey Foster Bridge — a modified Wheatstone Bridge with a slide wire.

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Carey Foster Bridge

Measurement of Electrical Resistance of Brass Specimen Using Carey Foster Bridge

Use the Carey Foster Bridge to measure the electrical resistance of a brass specimen and verify the balance condition P/Q = R/S.

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Carey Foster Bridge

Measurement of Electrical Resistance of Aluminum Rod Using Carey Foster Bridge

Measure the resistance of an aluminum rod using the Carey Foster Bridge and understand how the slide wire enables precise null-point detection.

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Carey Foster Bridge

Measurement of Specific Heat Capacity of Water Using Carey Foster Bridge

A combined experiment using the Carey Foster Bridge to measure resistance changes for determining the specific heat capacity of water.

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Carey Foster Bridge

Measurement of Thermal Conductivity of Copper Wire Using Carey Foster Bridge

Use resistance measurements on the Carey Foster Bridge to determine the thermal conductivity of a copper wire specimen.

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Potentiometer Experiments (Resistance Measurement)

Potentiometer

Measurement of Electrical Resistance of Copper Wire Using Potentiometer

Use a potentiometer — which works on the Wheatstone Bridge principle — to precisely measure the resistance of a copper wire.

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Potentiometer

Measurement of Electrical Resistance of Brass Specimen Using Potentiometer

Determine the electrical resistance of brass using the potentiometer method, applying Kirchhoff's voltage law around the measurement loop.

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Potentiometer

Measurement of Electrical Resistance of Glass Prism Using Potentiometer

Explore how the potentiometer's null-point method can measure the very high resistance of a glass prism.

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Potentiometer

Measurement of Electrical Resistance of Quartz Using Potentiometer

Measure the electrical resistance of quartz using potentiometer-based comparison with standard resistance values.

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Kirchhoff's Laws & Ohm's Law Verification

Kirchhoff's Laws

Verification of Kirchhoff's Laws Using Multimeter and Breadboard

Verify both Kirchhoff's Current Law (KCL) and Voltage Law (KVL) experimentally using a multimeter and breadboard circuit.

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Kirchhoff's Laws

Verification of Kirchhoff's Laws Using Standard Laboratory Setup

A full laboratory experiment to verify Kirchhoff's junction and loop rules using a standard lab circuit board with ammeters and voltmeters.

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Virtual Simulation

Verification of Kirchhoff's Laws Using Virtual Simulation

An interactive virtual lab simulation to verify Kirchhoff's current and voltage laws without physical equipment.

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Ohm's Law

Verification of Ohm's Law Using Multimeter and Breadboard

Verify the linear relationship V = IR using a breadboard circuit, multimeter, and variable resistors.

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Ohm's Law

Verification of Ohm's Law Using Standard Laboratory Setup

A standard lab experiment to verify Ohm's Law by plotting the V-I characteristic graph for a metallic conductor.

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Virtual Simulation

Verification of Ohm's Law Using Virtual Simulation

An interactive virtual experiment to verify Ohm's Law by changing voltage and resistance values in a simulated circuit.

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Related Physics Experiments

Faraday's Law

Verification of Faraday's Law of Induction Using Multimeter and Breadboard

Verify Faraday's law of electromagnetic induction by measuring induced EMF using a multimeter in a coil-magnet setup.

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Faraday's Law

Verification of Faraday's Law of Induction Using Virtual Simulation

An interactive simulation demonstrating how changing magnetic flux induces an EMF in a coil, verifying Faraday's law.

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Biot-Savart Law

Verification of Biot-Savart Law Using Standard Laboratory Setup

Verify the Biot-Savart law by measuring the magnetic field at various points around a current-carrying conductor.

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Virtual Simulation

Verification of Biot-Savart Law Using Virtual Simulation

An interactive virtual lab to explore the magnetic field produced by current elements and verify the Biot-Savart law.

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About These Lab Experiments

These experiments are available as interactive demonstrations on our platform. The Carey Foster Bridge experiments directly apply the Wheatstone Bridge principle, while the Kirchhoff's Laws verification experiments demonstrate the foundational laws used in the derivation above. Virtual simulations allow you to perform the experiments without physical equipment — ideal for revision and practice before board exams.

Frequently Asked Questions (FAQ)

A Wheatstone Bridge is an electrical circuit with four resistances (P, Q, R, S) arranged in a diamond shape, with a galvanometer connected across one diagonal and a battery across the other. It is primarily used to determine an unknown electrical resistance with high precision by balancing two legs of the bridge. When the bridge is balanced, P/Q = R/S, and the unknown resistance can be calculated from the three known values. It is also used in strain gauges, temperature sensors, and pressure transducers.

To derive the balance condition: (1) Label the four arm resistances P, Q, R, S and currents I₁ through P and I₂ through R. (2) Apply the balanced condition: galvanometer current Ig = 0. (3) Apply Kirchhoff's Voltage Law to loop ABDA to get I₁P = I₂R. (4) Apply KVL to loop BCDB to get I₁Q = I₂S. (5) Divide the two equations to eliminate I₁ and I₂, giving the final result: P/Q = R/S.

Kirchhoff's First Law (KCL), also known as the Junction Rule, states that the algebraic sum of all currents at any junction in an electrical circuit is zero: ΣI = 0. In simpler terms, the total current flowing into a junction equals the total current flowing out. It is based on the conservation of electric charge. For example, if three wires meet at a junction with currents of 2A and 3A flowing in, then 5A must flow out.

Kirchhoff's Second Law (KVL), also known as the Loop Rule, states that the algebraic sum of all potential differences around any closed loop in a circuit is zero: ΣV = 0. This means the total EMF in a loop equals the total voltage drops across the resistances. It is based on the conservation of energy. This law allows us to write equations for any closed path in a complex circuit and solve for unknown voltages and currents.

The Wheatstone Bridge formula for the balanced condition is: P/Q = R/S (equivalently, P × S = Q × R), where P, Q, R, and S are the four arm resistances. For an unbalanced bridge, the galvanometer voltage is VG = V × [R₁/(R₁+R₂) − R₃/(R₃+R₄)]. The unknown resistance can be found as S = QR/P when the bridge is balanced.

The four arms of a Wheatstone Bridge are the four resistances that form the sides of the diamond-shaped (quadrilateral) circuit. They are typically labelled P, Q, R, and S. Arms P and Q form one branch (say left/top and left/bottom), while R and S form the other branch. In a measurement setup, one arm contains the unknown resistance, one contains a variable (adjustable) resistance, and the other two contain fixed known resistances (ratio arms).

The principle of the Wheatstone Bridge is based on the null measurement method. When the ratio of resistances in one pair of opposite arms equals the ratio in the other pair (P/Q = R/S), the bridge is balanced and no current flows through the galvanometer. This null condition is independent of the battery EMF and galvanometer sensitivity, making it a highly accurate method for resistance measurement. The principle relies on the fact that at balance, the potentials at the two middle junctions (B and D) are equal.

Balanced Wheatstone Bridge: P/Q = R/S is satisfied. No current flows through the galvanometer (Ig = 0). The potentials at B and D are equal. Used for precise resistance measurement.

Unbalanced Wheatstone Bridge: P/Q ≠ R/S. A current flows through the galvanometer. There is a voltage difference between B and D. The degree of deflection indicates how far the bridge is from balance. Used in sensor applications where the change in one resistance (due to temperature, strain, etc.) is converted to a measurable voltage output.

Wheatstone Bridge ek electrical circuit hai jisme chaar resistances (P, Q, R, S) ko ek heere (diamond) ke aakaar mein jodte hain. Ek galvanometer ko do junction (B aur D) ke beech joda jaata hai, aur ek battery ko doosre do junction (A aur C) ke beech. Jab bridge balanced hota hai (P/Q = R/S), tab galvanometer mein koi current nahi bahta. Iska upyog angyaat (unknown) resistance ko bahut accurately nikalne ke liye kiya jaata hai. Ye Kirchhoff ke niyamon par aadharit hai.

Yes. The balance condition P/Q = R/S remains valid even if the galvanometer and battery are interchanged (the galvanometer is placed across AC and the battery across BD). This is because the balance condition depends only on the ratio of the four arm resistances, not on which diagonal has the galvanometer and which has the battery. However, the sensitivity of the bridge (ability to detect small changes) may differ after interchange.

Related Topics in Physics

Ohm's Law Kirchhoff's Laws Wheatstone Bridge Metre Bridge Potentiometer Resistance Drift Velocity Potential Gradient Coulomb's Law Biot-Savart Law Ampere's Circuital Law Faraday's Law Transformer Current Electricity Class 12 Post Office Box