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Ampere's Circuital Law and Its Applications

Ampere's circuital law states that the line integral of magnetic field around a closed path equals μ0 times the net current enclosed by that path: integral B dot dl = μ0 Ienc.

In one line: Ampere's law counts current through the loop, not current near the loop.

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Understand it

The law is useful when the magnetic field has high symmetry, such as around a long straight wire, inside a long solenoid, or inside a toroid. The closed path used for integration is called an Amperian loop. The dot product means only the component of B along dl contributes.

Example

For a long straight wire, choose a circular Amperian loop of radius r around the wire. Since B is tangential and constant on the circle, B(2π r) = μ0 I, so B = μ0 I/(2π r).

Formula and rule

Ampere's law: closed integral B dot dl = μ0 Ienc.

Long straight wire: B = μ0 I/(2π r).

Long solenoid interior: B = μ0 n I, where n is turns per metre.

Toroid interior: B = μ0 N I/(2π r).

B is in tesla, I in ampere, r in metre, n in m−1.

Best used for steady currents with high symmetry.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

A long straight wire carries 8 A.

  1. 1Find B at r = 4 cm.
  2. 2B = μ0 I/(2π r) = (4π x 10−7 × 8)/(2π x 0.04) = 4.0 × 10−5 T.
  3. 3The field circles the wire according to right-hand thumb rule.

Method you can reuse

  1. 1Inspect symmetry.
  2. 2Draw a suitable closed Amperian loop.
  3. 3Mark which current is enclosed.
  4. 4Write integral B dot dl = μ0 Ienc.
  5. 5Simplify the integral using constant B and direction relation.
  6. 6Solve for B and state field direction.

Sketch it in your answer

Show the closed Amperian path, direction of dl, magnetic field direction, enclosed current, and excluded outside current if present. For solenoid, label turns per unit length and nearly uniform internal field.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is a circular Amperian loop chosen around a long straight current-carrying wire?

Think first, then show answer
A circular loop is chosen because the magnetic field has the same magnitude at all points at the same distance from the wire and is tangential to the circle. This makes integral B dot dl become B multiplied by 2π r.

Common wrong answer

Using total current nearby instead of enclosed current gives wrong results, especially in multi-wire diagrams.

Don't mix this up

Students sometimes include currents outside the Amperian loop in Ienc. Only net current passing through the chosen closed surface bounded by the loop is counted.

Exam tip

Choose the Amperian loop from symmetry. If B is not constant or not parallel to dl, the law is true but may not directly give B in a simple form.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Asked in derivations for straight wire, solenoid, and toroid; also as conceptual questions on enclosed current and choice of Amperian loop.
  1. 1

    Enough for 1 mark

    Ampere's circuital law states that the line integral of magnetic field around a closed path equals μ0 times the net current enclosed by that path: integral B dot dl = μ0 Ienc.

  2. 2

    Add for 2 marks

    Example: For a long straight wire, choose a circular Amperian loop of radius r around the wire. Since B is tangential and constant on the circle, B(2π r) = μ0 I, so B = μ0 I/(2π r).

  3. 3

    Add for 3 marks

    The law is useful when the magnetic field has high symmetry, such as around a long straight wire, inside a long solenoid, or inside a toroid.

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