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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 mu0 times the net current enclosed by that path: integral B dot dl = mu0 I_enc.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    Ampere's circuital law states that the line integral of magnetic field around a closed path equals mu0 times the net current enclosed by that path: integral B dot dl = mu0 I_enc.

  2. 2

    Then show how to use it

    Inspect symmetry. Draw a suitable closed Amperian loop. Mark which current is enclosed. Write integral B dot dl = mu0 I_enc. Simplify the integral using constant B and direction relation. Solve for B and state field direction.

  3. 3

    Add one concrete 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(2pi r) = mu0 I, so B = mu0 I/(2pi r).

  4. 4

    Avoid this incomplete answer

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

Definition

Ampere's circuital law states that the line integral of magnetic field around a closed path equals mu0 times the net current enclosed by that path: integral B dot dl = mu0 I_enc.

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(2pi r) = mu0 I, so B = mu0 I/(2pi r).

Rule to remember

Ampere's law: closed integral B dot dl = mu0 I_enc. Long straight wire: B = mu0 I/(2pi r). Long solenoid interior: B = mu0 n I, where n is turns per metre. Toroid interior: B = mu0 N I/(2pi r). B is in tesla, I in ampere, r in metre, n in m^-1. Best used for steady currents with high symmetry.

Memory hook

Ampere's law counts current through the loop, not current near the loop.

Examples and method

Worked example

A long straight wire carries 8 A. Find B at r = 4 cm. B = mu0 I/(2pi r) = (4pi x 10^-7 x 8)/(2pi x 0.04) = 4.0 x 10^-5 T. The field circles the wire according to right-hand thumb rule.

Method to apply

Inspect symmetry. Draw a suitable closed Amperian loop. Mark which current is enclosed. Write integral B dot dl = mu0 I_enc. Simplify the integral using constant B and direction relation. Solve for B and state field direction.

Diagram support

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.

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.

Avoid common mistakes

Common confusion

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

Common wrong answer

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

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.

Quick check

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

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 2pi r.

Answer writing and exam use

1-mark answer

Ampere's circuital law states that the line integral of magnetic field around a closed path equals mu0 times the net current enclosed by that path: integral B dot dl = mu0 I_enc.

2-mark answer

Ampere's circuital law states that the line integral of magnetic field around a closed path equals mu0 times the net current enclosed by that path: integral B dot dl = mu0 I_enc. Ampere's law: closed integral B dot dl = mu0 I_enc. Long straight wire: B = mu0 I/(2pi r). Long solenoid interior: B = mu0 n I, where n is turns per metre. Toroid interior: B = mu0 N I/(2pi r). B is in tesla, I in ampere, r in metre, n in m^-1. Best used for steady currents with high symmetry. 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(2pi r) = mu0 I, so B = mu0 I/(2pi r).

3-mark answer

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. Ampere's law: closed integral B dot dl = mu0 I_enc. Long straight wire: B = mu0 I/(2pi r). Long solenoid interior: B = mu0 n I, where n is turns per metre. Toroid interior: B = mu0 N I/(2pi r). B is in tesla, I in ampere, r in metre, n in m^-1. Best used for steady currents with high symmetry. A long straight wire carries 8 A. Find B at r = 4 cm. B = mu0 I/(2pi r) = (4pi x 10^-7 x 8)/(2pi x 0.04) = 4.0 x 10^-5 T. The field circles the wire according to right-hand thumb rule. Asked in derivations for straight wire, solenoid, and toroid; also as conceptual questions on enclosed current and choice of Amperian loop. Using total current nearby instead of enclosed current gives wrong results, especially in multi-wire diagrams.
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