Moving Charges and Magnetism explains how moving charges and electric currents create magnetic fields and how those fields, in turn, exert forces on the charges and currents. It covers the Lorentz force, circular and helical motion in a field, the cyclotron, the Biot-Savart and Ampere laws, fields of wires, loops and solenoids, the force between parallel currents, and the torque on a current loop that drives the moving coil galvanometer. It is one of the highest-yield Class 12 chapters for NEET, supplying both formula-based numericals and conceptual questions on direction and field geometry.
Key Concepts
1. Magnetic Force on a Moving Charge
Lorentz Force: F = qv × B = qvB sin θ
- F = force on charge (N)
- q = charge (C)
- v = velocity of charge (m/s)
- B = magnetic field (Tesla, T)
- θ = angle between v and B
Force is maximum when θ = 90° and zero when θ = 0° or 180° (charge moving parallel to field).
Direction: given by Fleming’s Left-Hand Rule or the right-hand cross product rule.
Important: Magnetic force does no work on the charge (F ⊥ v always). It only changes direction, not speed.
Motion of a Charged Particle in Magnetic Field
| Angle (θ) | Path |
|---|---|
| 0° or 180° | Straight line (no force) |
| 90° | Circle (radius r = mv/(qB)) |
| Between 0° and 90° | Helix (spiral) |
Radius of circular motion: r = mv/(qB)
Time period: T = 2πm/(qB) - independent of velocity!
2. Biot-Savart Law
The magnetic field dB due to a small current element Idl at a point P at distance r is:
dB = (μ₀/4π) × (Idl × r̂)/r²
where μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space)
Magnetic Field Due to Common Configurations
| Configuration | Magnetic Field |
|---|---|
| Centre of circular loop (radius R, current I) | B = μ₀I/(2R) |
| On axis of circular loop (at distance x) | B = μ₀IR²/[2(R² + x²)^(3/2)] |
| Infinite straight wire (at distance r) | B = μ₀I/(2πr) |
| Inside a solenoid (n turns/length) | B = μ₀nI |
| Inside a toroid (N total turns, radius r) | B = μ₀NI/(2πr) |
3. Ampere’s Circuital Law
The line integral of B around any closed loop equals μ₀ times the total current enclosed:
∮ B · dl = μ₀I_enclosed
Useful when there is symmetry (infinite wire, solenoid, toroid).
4. Force on a Current-Carrying Conductor
F = Il × B = BIl sin θ
Direction: Fleming’s Left-Hand Rule
Force Between Two Parallel Current-Carrying Conductors
F/l = μ₀I₁I₂/(2πd)
- Parallel currents (same direction): attract
- Anti-parallel currents (opposite direction): repel
Definition of Ampere: 1 Ampere is the current which, when flowing through two infinite parallel conductors 1 m apart in vacuum, produces a force of 2 × 10⁻⁷ N per metre of length.
5. Cyclotron
A device that accelerates charged particles to high energies using electric and magnetic fields.
- Particle moves in semicircles inside two D-shaped electrodes (dees)
- Magnetic field provides circular motion; Electric field accelerates at each half-turn
- Cyclotron frequency: ν = qB/(2πm) - independent of radius and speed
- Maximum energy: E = q²B²R²/(2m), where R is the radius of the dee
- Cannot accelerate electrons (too light - relativistic effects) or neutral particles
Important Definitions
| Term | Definition |
|---|---|
| Magnetic field (B) | Region where a moving charge or current experiences a force; unit: Tesla (T) |
| Lorentz force | Total force on a charge in E and B fields: F = qE + qv × B |
| Biot-Savart law | Gives the magnetic field due to a small current element |
| Ampere’s circuital law | ∮B·dl = μ₀I_enclosed around a closed loop |
| Cyclotron | Device that accelerates charged particles using crossed E and B fields |
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Solved Examples
Example 1
A proton moves with velocity 5 × 10⁶ m/s perpendicular to a magnetic field of 0.2 T. Find the radius of the circular path. (mp = 1.67 × 10⁻²⁷ kg)
Answer: r = mv/(qB) = (1.67 × 10⁻²⁷ × 5 × 10⁶)/(1.6 × 10⁻¹⁹ × 0.2) = 8.35 × 10⁻²¹/3.2 × 10⁻²⁰ = 0.26 m
Example 2
Find the magnetic field at the centre of a circular loop of radius 5 cm carrying 2 A current.
Answer: B = μ₀I/(2R) = (4π × 10⁻⁷ × 2)/(2 × 0.05) = 8π × 10⁻⁷/0.1 = 2.51 × 10⁻⁵ T
Example 3
Two parallel wires 10 cm apart carry currents of 5 A and 10 A in the same direction. Find the force per unit length.
Answer: F/l = μ₀I₁I₂/(2πd) = (4π × 10⁻⁷ × 5 × 10)/(2π × 0.1) = (200π × 10⁻⁷)/(0.2π) = 10⁻⁴ N/m (attractive)
Example 4
A solenoid of length 50 cm has 500 turns and carries 3 A. Find B inside.
Answer: n = 500/0.5 = 1000 turns/m. B = μ₀nI = 4π × 10⁻⁷ × 1000 × 3 = 3.77 × 10⁻³ T ≈ 3.8 mT
Important Questions for Board Exams
1-Mark
- State Biot-Savart law.
- What is the force on a charge moving parallel to a magnetic field?
3-Mark
- Derive the expression for magnetic field at the centre of a current-carrying circular loop.
- Using Ampere’s law, derive B inside a solenoid.
- Explain the working principle of a cyclotron.
5-Mark
- State Biot-Savart law. Derive the expression for B on the axis of a circular current loop.
- Derive the force per unit length between two parallel current-carrying conductors. Define the Ampere.
Quick Revision Points
- F = qvB sin θ; magnetic force ⊥ velocity → does no work
- Circular path: r = mv/(qB); T = 2πm/(qB) - independent of v
- Biot-Savart: dB = (μ₀/4π)(Idl sin θ)/r²
- Centre of loop: B = μ₀I/(2R); Straight wire: B = μ₀I/(2πr)
- Solenoid: B = μ₀nI; Toroid: B = μ₀NI/(2πr)
- Ampere’s law: ∮B·dl = μ₀I_enc
- F between wires: F/l = μ₀I₁I₂/(2πd); same direction → attract
- Cyclotron: ν = qB/(2πm); can’t accelerate electrons or neutrals
Previous: Ch 3 - Current Electricity
Next: Ch 5 - Magnetism and Matter
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Magnetic Force on a Moving Charge
A charge feels a magnetic force only when it moves across the field lines.
θ is the angle between v and B; a charge at rest feels no magnetic force.
- Maximum F = qvB when v ⊥ B; zero when v ∥ B
- Force ⊥ to both v and B → does no work, speed unchanged
- Direction from right-hand rule for q v × B (reverse for negative q)
The Lorentz Force
Combine the electric and magnetic effects into one expression.
For a straight wire of length L: F = B I L sin θ.
- Electric part qE acts even on a stationary charge
- Magnetic part q(v × B) acts only on a moving charge
- A current-carrying wire is just many moving charges → F = BIL sin θ
Circular Motion in a Field
Enter ⊥ to B and the force becomes centripetal, bending the path into a circle.
T and frequency are independent of speed and radius.
- r = √(2mK) / (qB), so r ∝ √K
- If v has a part along B → path is a helix
- Pitch = v cos θ · T (distance advanced per turn)
The Cyclotron
It accelerates ions using the fact that the time period stays constant.
R = dee radius; the alternating voltage runs at the cyclotron frequency f.
- Constant period lets the field flip in sync each half-circle
- Energy is gained only in the gap between the dees
- Cannot accelerate neutral particles or (relativistically) electrons well
Biot-Savart & Ampere’s Law
Two ways to find the field produced by a current.
μ0/4π = 10⁻7 T·m/A; Ampere’s law works for symmetric cases.
- Long straight wire: B = μ0 I / (2π a), so B ∝ 1/a
- Field circles the wire by the right-hand grip rule
- Currents outside the chosen loop add zero to ∮ B · dl
Loops, Solenoids & Toroids
Standard magnetic-field results you must recall instantly.
Solenoid uses n (turns per metre); toroid uses total N.
- On loop axis: B = μ0 N I R2 / [2(R2+x2)^(3/2)]
- Solenoid field is uniform inside, ~zero outside; half (μ0nI/2) at the end
- Toroid: B = μ0 N I / (2π r), field only inside the ring
Force Between Parallel Currents
Each wire sits in the other’s field and feels a force.
1 A gives 2 × 10⁻7 N/m for wires 1 m apart in vacuum.
- Parallel (same-direction) currents attract
- Antiparallel (opposite-direction) currents repel
- Force is mutual and equal even when I1 ≠ I2 (Newton’s 3rd law)
Torque, Magnetic Moment & Galvanometer
A current loop behaves as a magnetic dipole and twists in a field.
θ = angle between the loop’s normal and B.
- τ max when loop plane ∥ B; τ = 0 (stable) when plane ⊥ B
- Galvanometer (radial field): φ ∝ I, sensitivity = N A B / k
- Ammeter = shunt in parallel; voltmeter = high resistance in series
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Related Chapters in Class 12 Physics
- Current Electricity Class 12 Notes
- Magnetism and Matter Class 12 Notes
- Electromagnetic Induction Class 12 Notes
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Frequently Asked Questions
The Lorentz force is the total force on a charge q moving with velocity v in the presence of both an electric field E and a magnetic field B, given by F = q(E + v x B). The electric part qE acts even on a charge at rest, while the magnetic part q(v x B) acts only on a moving charge and is always perpendicular to its velocity.
When a charge enters perpendicular to a uniform field B, the magnetic force becomes centripetal and the path is a circle of radius r = mv / (qB), with time period T = 2 pi m / (qB). The time period and frequency depend only on the charge-to-mass ratio and B, not on the speed or the radius.
It is part of the Class 12 Magnetic Effects of Current unit and is firmly in the NEET syllabus. Magnetism as a whole usually contributes a few questions every year, and this chapter is high-yield because it mixes direct formula numericals (force, radius, fields, torque) with concept questions, so it is worth mastering fully.
The Biot-Savart law, dB = (mu0 / 4 pi) I dl sin(theta) / r squared, gives the field of a small current element and works for any geometry but needs integration. Ampere’s circuital law, the closed line integral of B dot dl equals mu0 times the enclosed current, is faster but only practical when the situation has high symmetry, such as a long straight wire, solenoid, or toroid.
Each wire sits in the magnetic field produced by the other and feels a force F per unit length equal to mu0 I1 I2 / (2 pi d). When the currents flow in the same direction the forces pull the wires together, so they attract; when the currents are antiparallel the forces push them apart, so they repel. This mutual force also defines the ampere.