Wheatstone bridge

What is a Wheatstone bridge?

In plain words: the bridge has two routes from A to C, one through B and one through D. If both routes divide the voltage in the same ratio, then B and D sit at the same potential, and no current wants to cross between them. Finding that balance lets you measure an unknown resistance by comparison.

Balance condition

NCERT takes R₂ in AB, R₄ in BC, R₁ in AD and R₃ in DC. The currents are I₂ from A to B, I₄ from B to C, I₁ from A to D, I₃ from D to C, and I_g through G. The cell is taken to have no internal resistance (NCERT pages 100–101, Figure 3.18):

• In general all four resistors and the galvanometer carry currents. The case of interest is the balanced bridge, with I_g = 0 (page 100).
• With I_g = 0, Kirchhoff’s junction rule at D and B gives I₁ = I₃ and I₂ = I₄ (page 101).
• The loop rule for ADBA gives −I₁R₁ + 0 + I₂R₂ = 0, so I₁/I₂ = R₂/R₁ (NCERT Eq. (3.62)). For CBDC it gives I₂R₄ + 0 − I₁R₃ = 0, so I₁/I₂ = R₄/R₃ (NCERT Eq. (3.63), page 101).
• Hence the balance condition (NCERT Eq. 3.64(a), page 101):

• In words: at balance, B and D are at the same potential, so the potential drops across R₂ and R₁ are equal (each is IR by Ohm’s law), and so are those across R₄ and R₃ (reasoned from NCERT Eqs. (3.62)–(3.63)). The balance does not depend on the emf of the cell or on the resistance of the galvanometer.

Finding an unknown resistance

• Put the unknown resistance in the fourth arm, so R₄ is unknown. Keep known resistances R₁ and R₂ in the first and second arms, and vary R₃ (the standard arm) until the galvanometer shows null deflection (NCERT page 101).
• The bridge is then balanced, and (NCERT Eq. 3.64(b), page 101):

• A practical device that uses this principle is the metre bridge (NCERT spells it “meter bridge”, page 101).
• A null method like this needs no calibrated meter: the galvanometer only has to show when its current is zero (reasoned from the balance condition).

Solved examples

Common mistakes to avoid

Quick revision summary

A detector null establishes balance only with an energised bridge, positive finite arm resistances and a functioning connected galvanometer. A broken detector or disconnected source also gives zero detector current and is not a valid balance test.

• A Wheatstone bridge: four resistors in a diamond ABCD, cell across AC (battery arm), galvanometer across BD (galvanometer arm).
• Balanced (I_g = 0): I₁ = I₃, I₂ = I₄, and R₂/R₁ = R₄/R₃; B and D are at the same potential.
• Unknown resistance: R₄ = R₃ R₂/R₁, found by varying the standard arm R₃ for null deflection.
• The balance does not depend on the emf or the galvanometer; a balanced bridge’s middle arm can be removed.
• The metre bridge uses this principle; an unbalanced bridge needs Kirchhoff’s rules (NCERT Example 3.7).

Questions students ask

When is a Wheatstone bridge most sensitive?

A useful sensitive arrangement generally has the two arms of each route of the same order of resistance; sensitivity also depends on the source and detector resistances: a very unequal ratio, such as 1:100, makes a change in one arm shift the potential between the galvanometer corners far less (beyond the NCERT text, reasoned from Kirchhoff’s rules; the NCERT Laboratory Manual Physics Class XII raises it, Experiment 2, page 33). With a 3.0 V cell and a 40 Ω galvanometer, a 1 % change in a 200 Ω unknown drives about 31 µA with all four arms 200 Ω, but only about 6.6 µA with each route 2.0 Ω then 200 Ω. A larger emf or a more sensitive galvanometer also sharpens the null.

How do you find the equivalent resistance of an unbalanced Wheatstone bridge?

An unbalanced Wheatstone bridge cannot in general be reduced using series and parallel combinations alone, because its middle arm carries a current, so its equivalent resistance is found with Kirchhoff’s rules: put a potential difference V across A and C, find the current I it drives, and R_eq = V/I, as NCERT does for its cube of resistors (NCERT Example 3.5, page 99). For AB = 1 Ω, BC = 2 Ω, AD = 2 Ω and DC = 1 Ω with 1 Ω in BD, 7.0 V across AC gives, from the junction rule at B and D, potentials of 4.0 V at B and 3.0 V at D above C. So A supplies 3.0 + 2.0 = 5.0 A and R_eq = 7.0/5.0 = 1.4 Ω, with 1.0 A crossing from B to D.

Who invented the Wheatstone bridge?

The Wheatstone bridge was invented by Samuel Hunter Christie, who described the arrangement in 1833, and made widely known by Charles Wheatstone in 1843, which is how it came to carry Wheatstone’s name (beyond the NCERT text). Wheatstone described it, under the name differential resistance measurer, in his Bakerian Lecture (Philosophical Transactions of the Royal Society, volume 133, 1843), and there credited Christie with the first idea, from Christie’s paper in volume 123 of the same journal (1833). NCERT names the bridge but does not give its history (NCERT Section 3.13, page 100).

What are the applications of a Wheatstone bridge?

A Wheatstone bridge measures an unknown resistance by comparing it with known ones (NCERT Section 3.13, page 101), and through that anything that changes a resistance. The metre bridge uses it to find the resistivity of a wire, and the NCERT Laboratory Manual Physics Class XII suggests using it to follow how a wire’s resistance changes with temperature (Experiment 2, page 34), the principle of resistance thermometers such as the platinum one of NCERT Example 3.4 (page 92). Chemists use a bridge with an AC source to measure the conductivity of an ionic solution, since direct current would change the solution (NCERT Chemistry Part I (Class 12), Unit 2, pages 43–44).

Is the Wheatstone bridge in the NEET syllabus, and are unbalanced bridges included?

Yes, the Wheatstone bridge is in the NEET (UG) 2026 syllabus, which lists “Kirchhoff’s laws and their applications. Wheatstone bridge. Metre Bridge.” in Unit 12, Current Electricity, and the JEE Main syllabus lists the same three topics in its Unit 12. Neither says whether the bridge is balanced, but NCERT Class 12 Physics Part I solves an unbalanced bridge with Kirchhoff’s rules in Example 3.7 (page 102), right after the balance condition (Section 3.13, pages 100–101), so expect both kinds.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. A bridge balances with R₁ = 5 Ω, R₂ = 15 Ω and R₃ = 8 Ω. Find R₄.

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   R₄ = R₃R₂/R₁ = 8 × 15/5 = 24 Ω (NCERT Eq. 3.64(b), page 101).

2. Q2. What is meant by the null deflection of a bridge?

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   The galvanometer shows no deflection: no current flows through it, and the bridge is balanced (NCERT page 101).

3. Q3. Which arm of a Wheatstone bridge is called the battery arm?

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   The arm across which the source is connected, AC in NCERT’s figure (NCERT page 100).