Is it correct to say: Assuming magnetic force does work on a moving charge and changes its kinetic energy?

F is always perpendicular to v, so F·v = 0. Magnetic force can change *direction* but never *speed* — only KE stays constant.

Is it correct to say: Forgetting the sin\theta factor in F = BIL sin\theta when wire is not perpendicular to B?

F = BIL only when wire ⊥ B. If parallel (\theta = 0), force is zero.

Is it correct to say: Using B = \mu_0 I/(2R) for a wire instead of a loop (or vice versa)?

Wire at distance r: B = \mu_0 I/(2\pi r). Circular loop centre: B = \mu_0 I/(2R). Note the factor \pi.

Is it correct to say: Applying Ampere's law without symmetry (irregular current distribution)?

Ampere's law is always true but only *useful* for high symmetry: infinite wire, solenoid, toroid, plane sheet.

Is it correct to say: Mixing up mass number and turn count when using B = \mu_0 nI vs B = \mu_0 NI/L for solenoid?

Small n = turns per metre; capital N = total turns. Both give same B via n = N/L.

Is it correct to say: Assuming cyclotron works for electrons at high energies?

At relativistic speeds mass increases, cyclotron frequency drops, particle drifts out of phase — this is the fundamental limitation.

Moving Charges and Magnetism

Lorentz force, Biot-Savart law, Ampere's law, solenoid, cyclotron, torque on a current loop — NCERT Class 12 Physics Ch 4

📐 13 formulas✏️ 3 examples🎯 6 practice⚖️ 6 marks🏫 CBSE📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→Biot-Savart derivation for the field at the centre of a circular loop is a repeat 3-mark question.
→Ampere's-law derivation of B inside a long solenoid appears almost every alternate year.
→Right-hand rule for direction — practise with wire, loop and solenoid till automatic.
→Cyclotron working + limitations (relativistic mass increase) is a 3–5 mark favourite.
→Never confuse permittivity \varepsilon_0 (electrostatics) with permeability \mu_0 (magnetism).

📊 Diagram

Charged particle moving in a magnetic field experiencing Lorentz force

📐 Formulas(13)

Lorentz Force★ Board fav

F = q (E + v \times B)

Symbol	Meaning
$q$	Charge (C)
$v$	Velocity of charge (m/s)
$B$	Magnetic field (T)

Biot–Savart Law★ Board fav

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

Field due to Long Straight Wire

B = \frac{μ _{0} I}{2 π r}

Field at Centre of Circular Loop★ Board fav

B = \frac{μ _{0} I}{2 R}

Field on Axis of Circular Loop

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 Inside a Solenoid

B = μ_{0} n I

Field Inside a Toroid

B = \frac{μ _{0} N I}{2 π r}

Force on Current-Carrying Conductor

F = I L \times B, F = B I L sin θ

Force Between Two Parallel Wires

\frac{F}{L} = \frac{μ _{0} I _{1} I _{2}}{2 π d}

Torque on a Current Loop

τ = m \times B, τ = N I A B sin θ

Cyclotron Frequency

f = \frac{q B}{2 π m}, T = \frac{2 π m}{q B}

Radius of Circular Motion

r = \frac{m v}{q B}

✏️ Solved Examples

1Solved Exampleeasy4 steps

A proton moves at 2×10⁶ m/s perpendicular to a magnetic field of 0.5 T. Find the magnetic force and the radius of its circular orbit. (m_p = 1.67×10⁻²⁷ kg, e = 1.6×10⁻¹⁹ C)

Perpendicular motion means \theta = 90°, sin\theta = 1

F = q v B = 1.6 \times 1 0^{- 19} \times 2 \times 1 0^{6} \times 0.5

2Solved Exampleboard5 steps

A solenoid 50 cm long has 500 turns and carries a current of 2 A. Find the magnetic field inside. If a coil of 20 turns and area 4 cm² is placed at its centre with axis parallel to the solenoid, find the flux linked.

Compute turns per unit length

n = \frac{N}{L} = \frac{500}{0.5} = 1000 turns/m

3Solved ExampleHOTS6 steps

In a cyclotron a deuteron (mass 3.34×10⁻²⁷ kg, charge 1.6×10⁻¹⁹ C) is accelerated to a maximum radius of 0.5 m with B = 1.5 T. Find (a) the cyclotron frequency, (b) the maximum kinetic energy of the deuteron in MeV.

Cyclotron frequency depends only on q, B, m

f = \frac{q B}{2 π m} = \frac{1.6 \times 1 0 ^{- 19} \times 1.5}{2 π \times 3.34 \times 1 0 ^{- 27}}

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Assuming magnetic force does work on a moving charge and changes its kinetic energy
✓F is always perpendicular to v, so F·v = 0. Magnetic force can change *direction* but never *speed* — only KE stays constant.
2
Forgetting the sin\theta factor in F = BIL sin\theta when wire is not perpendicular to B
✓F = BIL only when wire ⊥ B. If parallel (\theta = 0), force is zero.
3
Using B = \mu_0 I/(2R) for a wire instead of a loop (or vice versa)
✓Wire at distance r: B = \mu_0 I/(2\pi r). Circular loop centre: B = \mu_0 I/(2R). Note the factor \pi.
4
Applying Ampere's law without symmetry (irregular current distribution)
✓Ampere's law is always true but only *useful* for high symmetry: infinite wire, solenoid, toroid, plane sheet.
5
Mixing up mass number and turn count when using B = \mu_0 nI vs B = \mu_0 NI/L for solenoid
✓Small n = turns per metre; capital N = total turns. Both give same B via n = N/L.
6
Assuming cyclotron works for electrons at high energies
✓At relativistic speeds mass increases, cyclotron frequency drops, particle drifts out of phase — this is the fundamental limitation.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
A wire of length 2 m carries a current 5 A perpendicular to a field of 0.4 T. Find the force on the wire.
Q2
Field at the centre of a circular loop of radius 5 cm carrying 3 A?
Q3
Two long parallel wires 10 cm apart carry currents 5 A and 10 A in the same direction. Force per metre between them?
Q4
A solenoid has 400 turns/m and carries 3 A. Field inside?
Q5
An electron moves at 10⁷ m/s in a field of 10⁻³ T perpendicular to v. Radius of orbit? (m_e = 9.1×10⁻³¹ kg)
Q6
A rectangular coil of 100 turns, area 20 cm², carries 2 A. Torque in a field 0.5 T when plane parallel to field?

📝 Notes

Moving Charges and Magnetism — key concepts

A moving charge creates a magnetic field; a magnetic field exerts a force on moving charges. This chapter formalises both statements.

The two big laws

Biot–Savart law (dB from a current element) is analogous to Coulomb's law: it lets us calculate B due to any current distribution by integration.
Ampere's circuital law (\oint B·dl = \mu_0 I_{enc}) is analogous to Gauss's law: it gives B quickly when symmetry is high.

Use Biot–Savart for finite loops and arcs, Ampere for infinite wires, solenoids, toroids.

Direction rules

Straight wire: right-hand grip — thumb along I, fingers curl along B.
Circular loop: right-hand fingers curl along I, thumb gives B at centre.
Solenoid: right-hand fingers curl along I, thumb gives north pole.
Force on charge: right-hand for positive q — fingers point along v, curl to B, palm pushes in direction of F.

Magnetic force does no work

The Lorentz magnetic force is always perpendicular to velocity, so it can change direction but not speed. Hence, in a pure magnetic field, a charge's kinetic energy is conserved.

Cyclotron in one line

Radius r = mv/(qB) grows with speed, but time period T = 2\pi m/(qB) is independent of speed. So a fixed-frequency oscillator can keep accelerating the particle each half-turn — until relativity kicks in.

Quick sanity checks

Field inside a long solenoid is uniform; outside ≈ 0. This is why solenoids make good electromagnets.
Two parallel currents attract, antiparallel currents repel — remember this via analogy with springs pulling wires closer.
Torque on a current loop is maximum when its plane is parallel to B and zero when perpendicular (m along B).
Units: 1 T = 1 N/(A·m) = 1 Wb/m². 1 T is a huge field — Earth's field is only ≈ 5×10⁻⁵ T.