Magnetism and Magnetic Effects: PYQ Walkthrough (10)

easy • 2 min read
Tags Magnetism

Question

A circular coil of radius 0.1 m0.1\,\text{m} carries a current of 5 A5\,\text{A} and consists of 2020 turns. Find the magnetic field at the centre of the coil. Take μ0=4π×10−7 T m/A\mu_0 = 4\pi \times 10^{-7}\,\text{T m/A}. (Adapted from NEET 2021.)

Solution — Step by Step

For a circular coil with NN turns at its centre:

B=μ0NI2RB = \frac{\mu_0 N I}{2R}

where RR is the coil radius and II is the current per turn.

B=(4π×10−7)×20×52×0.1B = \frac{(4\pi \times 10^{-7}) \times 20 \times 5}{2 \times 0.1}

Numerator=4π×10−7×100=4π×10−5\text{Numerator} = 4\pi \times 10^{-7} \times 100 = 4\pi \times 10^{-5}

Denominator is 0.20.2. So:

B=4π×10−50.2=20π×10−5 TB = \frac{4\pi \times 10^{-5}}{0.2} = 20\pi \times 10^{-5}\,\text{T}

B=20π×10−5≈6.28×10−4 TB = 20\pi \times 10^{-5} \approx 6.28 \times 10^{-4}\,\text{T}

Final: B≈6.28×10−4B \approx 6.28 \times 10^{-4} T or about 0.63 mT, perpendicular to the plane of the coil (right-hand rule).

Why This Works

The factor NN in the formula is exact — every turn contributes the same field at the centre because they are coincident loops. This is why electromagnets use many turns: doubling NN doubles BB at no extra cost in current.

The 1/R1/R dependence (not 1/R21/R^2) comes from integrating the Biot-Savart law around a full circle. Surprisingly, the field at the centre falls only linearly with radius — much slower than for a magnetic dipole.

Alternative Method

For a single turn, the field is B1=μ0I/(2R)B_1 = \mu_0 I / (2R). Multiply by NN at the end:

B1=4π×10−7×50.2=π×10−5 TB_1 = \frac{4\pi \times 10^{-7} \times 5}{0.2} = \pi \times 10^{-5}\,\text{T}

B=20B1=20π×10−5 TB = 20 B_1 = 20\pi \times 10^{-5}\,\text{T}

Same result, less algebra in the substitution step.

Common Mistake

Students confuse the formula for a circular coil with the formula for a solenoid (B=μ0nIB = \mu_0 n I, where nn is turns per unit length). For a coil, NN is total turns; for a solenoid, nn is turns per metre. Different physical setups, different formulas.

Want to master this topic?

Read the complete guide with more examples and exam tips.

Go to full topic guide →

Try These Next

Ampere's Circuital Law — Field Inside a Solenoid

hard

Ampere's circuital law — find magnetic field inside a long solenoid

medium

Biot-Savart Law — Magnetic Field Due to Straight Wire

medium

Biot-Savart law — find magnetic field due to straight current-carrying wire

medium

Cyclotron — working principle, frequency, and limitations

medium