C
CraftExam
medium difficulty9 min

Magnetic Field on the Axis of a Circular Current Loop

The magnetic field at a point on the axis of a circular loop of radius R carrying current I is B = μ0 I R2/[2(R2 + x2)3/2], directed along the axis of the loop.

In one line: For a loop, sideways parts cancel; axis parts collect.

Concept Practice Coming Soon

Understand it

  • Each current element of the loop produces a magnetic field at the axial point.
  • By symmetry, components perpendicular to the axis cancel, while components along the axis add.
  • At the centre of the loop, x = 0, so the expression becomes B = μ0 I/(2R).
  • For N turns, the field is multiplied by N.

Example

A circular coil produces a strong field near its centre. Increasing the number of turns or current increases the field, while increasing radius decreases the centre field.

Formula and rule

On-axis field: B = μ0 I R2/[2(R2 + x2)3/2].

At centre: B = μ0 I/(2R).

For N turns: B = μ0 N I R2/[2(R2 + x2)3/2].

R and x are in metre, I in ampere, B in tesla.

Use only for points lying on the axis of a circular loop.

See it worked out

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

Worked example

A circular coil of radius 0.10 m has 50 turns and carries 2 A.

  1. 1At the centre, B = μ0 N I/(2R) = (4π x 10−7 × 50 × 2)/(0.20) = 6.28 × 10−4 T.
  2. 2The field is along the axis as decided by right-hand thumb rule.

Method you can reuse

  1. 1Choose an element dl.
  2. 2Write dB using Biot-Savart law.
  3. 3Resolve dB into axial and transverse components.
  4. 4Cancel transverse components by symmetry.
  5. 5Integrate the axial component around the loop.
  6. 6Put x = 0 for centre if required.

Sketch it in your answer

Draw a circular loop in a plane, axis through centre O, point P at distance x from O, radius R, current direction, dB components, and resultant B along the axis.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why do only axial components of dB add for a circular current loop?

Think first, then show answer
Only axial components add because for every current element, an opposite element produces an equal transverse component in the opposite direction. These transverse components cancel, while the components along the axis are in the same direction.

Common wrong answer

Using B = μ0 I/(2π R) at the centre confuses the circular loop result with the long straight wire formula.

Don't mix this up

Students often substitute diameter in place of radius or forget the power 3/2 in (R2 + x2)3/2.

Exam tip

In derivation questions, state the symmetry cancellation clearly. This is often where marks are awarded before the final formula.

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

Commonly asked as a derivation, a centre-field numerical, or a conceptual question on why transverse components cancel.
  1. 1

    Enough for 1 mark

    The magnetic field at a point on the axis of a circular loop of radius R carrying current I is B = μ0 I R2/[2(R2 + x2)3/2], directed along the axis of the loop.

  2. 2

    Add for 2 marks

    Example: A circular coil produces a strong field near its centre. Increasing the number of turns or current increases the field, while increasing radius decreases the centre field.

  3. 3

    Add for 3 marks

    By symmetry, components perpendicular to the axis cancel, while components along the axis add.

Practice

Concept practice is coming soon

Login to access concept-level MCQs and weak-concept practice as they become available.

10 MCQs5 MinutesInstant Results
Login

Help improve this page

Found something confusing, incorrect, or missing?

Open Learning Map