Is it correct to say: Using Biot–Savart law without noting the direction of dl × r̂?

Determine direction by right-hand rule. The magnitude alone is not enough for a full-mark answer.

Is it correct to say: Confusing μ₀ (permeability) with ε₀ (permittivity)?

μ₀ = 4π×10⁻⁷ T·m/A (magnetic); ε₀ = 8.854×10⁻¹² (electric). Different units and different laws.

Is it correct to say: Using B = μ₀I/(2R) with x ≠ 0?

That formula is only for x = 0 (centre). Use axial formula for other points.

Is it correct to say: Claiming magnetic force does work on a charge?

F is perpendicular to v, so W = F·v Δt = 0. Magnetic force never changes kinetic energy.

Is it correct to say: Ignoring the number of turns N in a coil formula?

For a coil of N turns, multiply the single-turn B by N (or replace I with NI as appropriate).

Is it correct to say: Deriving gyromagnetic ratio as e/m instead of e/(2m)?

Factor of 2 comes from area = πr² but current period involves 2πr. Retain the factor.

Magnetic Fields due to Electric Current

Biot-Savart law, Ampere's law, force on moving charges, cyclotron, gyromagnetic ratio — Maharashtra HSC Physics Ch 10

📐 12 formulas✏️ 3 examples🎯 5 practice⚖️ 6 marks🏫 Maharashtra State Board📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→Derivation of the magnetic field at the centre and on the axis of a circular current loop using the Biot–Savart law is a Maharashtra HSC favourite (4 marks).
→Cyclotron construction, principle, working with labelled diagram and expression for maximum kinetic energy is a very common 4-mark question.
→The concept of gyromagnetic ratio (g = e/2m for orbital motion) is unique to Maharashtra Board syllabus — do NOT skip it.
→Distinguish clearly between the units tesla (T) and gauss (G): 1 T = 10⁴ G. Board asks conversion in 1-mark MCQs.
→In numericals on force on a current-carrying conductor F = BIL sinθ, do not forget the angle θ between L and B.

📐 Formulas(12)

Biot–Savart Law★ Board fav

d B = \frac{μ _{0}}{4 π} \frac{I d l \times r ^}{r ^{2}}

Symbol	Meaning
$μ_{0}$	Permeability of free space = 4π×10⁻⁷ T·m/A
$I d l$	Current element (A·m)
$r$	Distance from element to field point (m)

Field at Centre of a Circular Loop★ Board fav

B = \frac{μ _{0} I}{2 R}

Field on Axis of a Circular Loop★ Board fav

B = \frac{μ _{0} I R ^{2}}{2 ( R ^{2} + x ^{2} ) ^{3/2}}

Ampere's Circuital Law★ Board fav

\oint B \cdot d l = μ_{0} I_{e n c}

Field due to a Long Straight Wire

B = \frac{μ _{0} I}{2 π r}

Field of a Solenoid

B = μ_{0} n I

Force on a Moving Charge (Lorentz Force)

F = q (v \times B)

Force on a Current-Carrying Conductor★ Board fav

F = I (L \times B), F = B I L sin θ

Cyclotron Frequency★ Board fav

f_{c} = \frac{q B}{2 π m}

Maximum Kinetic Energy in Cyclotron

K_{ma x} = \frac{q ^{2} B ^{2} R ^{2}}{2 m}

Magnetic Moment of Current Loop

m = I A = N I A

Gyromagnetic Ratio (orbital)★ Board fav

γ = \frac{m _{L}}{L} = \frac{e}{2 m _{e}}

✏️ Solved Examples

1Solved Exampleeasy3 steps

A long straight wire carries a current of 8 A. Find the magnetic field at a perpendicular distance of 4 cm from the wire.

Given values in SI units.

I = 8 A, r = 4 \times 1 0^{- 2} m

2Solved Exampleboard4 steps

A circular coil of 100 turns and radius 5 cm carries a current of 2 A. Find the magnetic field at (a) its centre, (b) a point 12 cm on its axis.

At centre, for N turns:

B_{c} = \frac{μ _{0} N I}{2 R} = \frac{4 π \times 1 0 ^{- 7} \times 100 \times 2}{2 \times 0.05}

3Solved ExampleHOTS4 steps

In a cyclotron with magnetic field 1.5 T and dee radius 0.6 m, protons are accelerated. Find (i) the cyclotron frequency, (ii) the maximum kinetic energy of the protons (m_p = 1.67×10⁻²⁷ kg).

Cyclotron frequency:

f_{c} = \frac{q B}{2 π m} = \frac{1.6 \times 1 0 ^{- 19} \times 1.5}{2 π \times 1.67 \times 1 0 ^{- 27}}

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Using Biot–Savart law without noting the direction of dl × r̂
✓Determine direction by right-hand rule. The magnitude alone is not enough for a full-mark answer.
2
Confusing μ₀ (permeability) with ε₀ (permittivity)
✓μ₀ = 4π×10⁻⁷ T·m/A (magnetic); ε₀ = 8.854×10⁻¹² (electric). Different units and different laws.
3
Using B = μ₀I/(2R) with x ≠ 0
✓That formula is only for x = 0 (centre). Use axial formula for other points.
4
Claiming magnetic force does work on a charge
✓F is perpendicular to v, so W = F·v Δt = 0. Magnetic force never changes kinetic energy.
5
Ignoring the number of turns N in a coil formula
✓For a coil of N turns, multiply the single-turn B by N (or replace I with NI as appropriate).
6
Deriving gyromagnetic ratio as e/m instead of e/(2m)
✓Factor of 2 comes from area = πr² but current period involves 2πr. Retain the factor.

🎯 Practice Yourself

🎯Practice Yourself5 questions

Q1
A wire carrying 5 A is placed perpendicular to a magnetic field of 0.4 T. Find the force per metre length.
Q2
Calculate the field at the centre of a circular loop of radius 10 cm carrying 4 A.
Q3
A solenoid has 500 turns/m and carries 3 A. Find B inside.
Q4
An electron moves at 2×10⁶ m/s perpendicular to B = 0.5 T. Find the radius of its circular motion.
Q5
Prove that the gyromagnetic ratio of an orbiting electron is e/(2m_e).

📝 Notes

Magnetic Fields due to Electric Current — HSC Overview

Chapter 10 of the Maharashtra Class 12 Physics textbook combines what NCERT splits between "Moving Charges and Magnetism" and part of the atomic-model discussion.

What Maharashtra Board emphasises differently

Gyromagnetic ratio (γ = e/2m_e) is a Maharashtra-specific concept — a 2-mark derivation is asked almost every year. NCERT does not treat this formally at the Class 12 level.
Cyclotron — construction, principle, limitations (relativistic mass increase, ion mass), and derivation of K_max — is a heavy 4-mark HSC question. Balbharati insists on labelling the dees, oscillator, and magnetic-field direction.
Derivation of magnetic field on the axis of a circular current loop appears more often in Maharashtra than in CBSE.
The magnetic moment of a revolving electron and its ratio to angular momentum is a Balbharati-specific formal treatment.

Study strategy

Master Biot–Savart law + right-hand rule first — all other formulas flow from it.
Cyclotron numericals reduce to just two formulas: f_c = qB/(2πm) and K_max = q²B²R²/(2m). Memorise both.
Distinguish clearly the units tesla (T), gauss (G), and weber/m². 1 T = 10⁴ G. The Maharashtra syllabus occasionally uses gauss for legacy problems.