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Biot-Savart Law for Magnetic Field

Biot-Savart law gives the small magnetic field dB produced at a point by a small current element I dl, with dB = (mu0/4pi) (I dl x r_hat)/r^2.

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Student-friendly explanation

The magnetic field due to a current element depends directly on current I and element length dl, and inversely on the square of the distance r from the element. Its direction is perpendicular to the plane containing dl and r. The law is used by integrating contributions from all current elements in a wire or loop.

How to write this in exams

  1. 1

    Start with the exact idea

    Biot-Savart law gives the small magnetic field dB produced at a point by a small current element I dl, with dB = (mu0/4pi) (I dl x r_hat)/r^2.

  2. 2

    Then show how to use it

    Mark current element and point of observation. Draw r from element to point. Write vector form first. Convert to magnitude using sin theta. Use symmetry for extended conductors. Integrate over the full current path if required.

  3. 3

    Add one concrete example

    For a circular current loop, each small element produces a magnetic field at an axial point. The components perpendicular to the axis cancel in pairs, while axial components add.

  4. 4

    Avoid this incomplete answer

    Writing dB proportional to 1/r instead of 1/r^2 confuses the field law with later integrated results for special geometries.

Definition

Biot-Savart law gives the small magnetic field dB produced at a point by a small current element I dl, with dB = (mu0/4pi) (I dl x r_hat)/r^2.

Example

For a circular current loop, each small element produces a magnetic field at an axial point. The components perpendicular to the axis cancel in pairs, while axial components add.

Rule to remember

dB = (mu0/4pi)(I dl x r_hat)/r^2; magnitude dB = (mu0/4pi)(I dl sin theta)/r^2. mu0 is permeability of free space with value 4pi x 10^-7 T m A^-1, I is in ampere, dl and r are in metre, and dB is in tesla. Use for magnetic field due to a current distribution by integration.

Memory hook

Biot-Savart is a current-element law: small current piece, distance squared, and cross-product direction.

Examples and method

Worked example

For I = 5 A, dl = 2.0 x 10^-3 m, r = 0.10 m, and theta = 90 degrees, dB = 10^-7 x (5 x 2.0 x 10^-3)/0.01 = 1.0 x 10^-7 T. The field direction is given by dl x r_hat.

Method to apply

Mark current element and point of observation. Draw r from element to point. Write vector form first. Convert to magnitude using sin theta. Use symmetry for extended conductors. Integrate over the full current path if required.

Diagram support

Show a small current element I dl, observation point P, position vector r from element to P, angle theta between dl and r, and dB direction perpendicular to their plane.

How CBSE asks it

Asked as a statement of law, derivation of magnetic field on the axis of a circular loop, comparison with Coulomb's law, or a direction-based short answer.

Avoid common mistakes

Common confusion

A common error is to ignore the angle between dl and r. The magnitude is dB = (mu0/4pi)(I dl sin theta)/r^2, so a current element pointing directly toward the observation point gives no contribution.

Common wrong answer

Writing dB proportional to 1/r instead of 1/r^2 confuses the field law with later integrated results for special geometries.

Exam tip

In derivations, clearly show the vector direction and explain which components cancel by symmetry before writing the final field expression.

Quick check

What decides the direction of dB in Biot-Savart law?

The direction of dB is decided by the cross product dl x r_hat. It is perpendicular to both the current element direction and the line joining the element to the observation point.

Answer writing and exam use

1-mark answer

Biot-Savart law gives the small magnetic field dB produced at a point by a small current element I dl, with dB = (mu0/4pi) (I dl x r_hat)/r^2.

2-mark answer

Biot-Savart law gives the small magnetic field dB produced at a point by a small current element I dl, with dB = (mu0/4pi) (I dl x r_hat)/r^2. dB = (mu0/4pi)(I dl x r_hat)/r^2; magnitude dB = (mu0/4pi)(I dl sin theta)/r^2. mu0 is permeability of free space with value 4pi x 10^-7 T m A^-1, I is in ampere, dl and r are in metre, and dB is in tesla. Use for magnetic field due to a current distribution by integration. For a circular current loop, each small element produces a magnetic field at an axial point. The components perpendicular to the axis cancel in pairs, while axial components add.

3-mark answer

The magnetic field due to a current element depends directly on current I and element length dl, and inversely on the square of the distance r from the element. Its direction is perpendicular to the plane containing dl and r. The law is used by integrating contributions from all current elements in a wire or loop. dB = (mu0/4pi)(I dl x r_hat)/r^2; magnitude dB = (mu0/4pi)(I dl sin theta)/r^2. mu0 is permeability of free space with value 4pi x 10^-7 T m A^-1, I is in ampere, dl and r are in metre, and dB is in tesla. Use for magnetic field due to a current distribution by integration. For I = 5 A, dl = 2.0 x 10^-3 m, r = 0.10 m, and theta = 90 degrees, dB = 10^-7 x (5 x 2.0 x 10^-3)/0.01 = 1.0 x 10^-7 T. The field direction is given by dl x r_hat. Asked as a statement of law, derivation of magnetic field on the axis of a circular loop, comparison with Coulomb's law, or a direction-based short answer. Writing dB proportional to 1/r instead of 1/r^2 confuses the field law with later integrated results for special geometries.
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