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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 = (μ0/4π) (I dl x r̂)/r2.

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

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Understand it

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.

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.

Formula and rule

dB = (μ0/4π)(I dl x r̂)/r2; magnitude dB = (μ0/4π)(I dl sin θ)/r2. μ0 is permeability of free space with value 4π 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.

See it worked out

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

Worked example

  1. 1For I = 5 A, dl = 2.0 × 10−3 m, r = 0.10 m, and θ = 90 degrees, dB = 10−7 × (5 × 2.0 × 10−3)/0.01 = 1.0 × 10−7 T.
  2. 2The field direction is given by dl x r̂.

Method you can reuse

  1. 1Mark current element and point of observation.
  2. 2Draw r from element to point.
  3. 3Write vector form first.
  4. 4Convert to magnitude using sin θ.
  5. 5Use symmetry for extended conductors.
  6. 6Integrate over the full current path if required.

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer
The direction of dB is decided by the cross product dl x r̂. It is perpendicular to both the current element direction and the line joining the element to the observation point.

Common wrong answer

Writing dB proportional to 1/r instead of 1/r2 confuses the field law with later integrated results for special geometries.

Don't mix this up

A common error is to ignore the angle between dl and r. The magnitude is dB = (μ0/4π)(I dl sin θ)/r2, so a current element pointing directly toward the observation point gives no contribution.

Exam tip

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

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 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.
  1. 1

    Enough for 1 mark

    Biot-Savart law gives the small magnetic field dB produced at a point by a small current element I dl, with dB = (μ0/4π) (I dl x r̂)/r2.

  2. 2

    Add for 2 marks

    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.

  3. 3

    Add for 3 marks

    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.

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