1Magnetic Force on a Moving Charge

A stationary charge feels nothing from a magnetic field. Only moving charge does — and the force it feels is stranger than the electric force: it depends on velocity, and it always acts perpendicular to both the velocity and the field.

Lorentz Force (full, both fields)
F = q E + q (v × B)
The magnetic part has magnitude F = qvB sin θ, where θ is the angle between v and B. Because the force is always perpendicular to velocity, a magnetic field alone can never speed a charge up or slow it down — it can only change direction.
Common Slip
Since magnetic force is always perpendicular to velocity, it does zero work. Students often forget this and try to apply work-energy theorem incorrectly to a pure magnetic force.

2Motion in a Magnetic Field

If a charged particle enters a uniform field exactly perpendicular to it, the magnetic force provides exactly the centripetal force needed for uniform circular motion.

Radius of Circular Path
r = m v / (q B)
Larger momentum → bigger circle; stronger field or charge → tighter circle.
Period and Frequency
T = 2πm / (qB)
Notice speed doesn’t appear — every particle of the same q/m ratio takes the same time to complete a loop, regardless of how fast it’s going. This is exactly what makes the cyclotron work.

3Motion in Combined Electric and Magnetic Fields

Set up perpendicular E and B fields so their forces on a charge exactly cancel, and only particles moving at one specific speed pass through undeflected — the principle behind a velocity selector.

Selected Velocity
v = E / B
Any particle moving faster or slower gets deflected off the straight path and is filtered out.

4The Cyclotron

A cyclotron accelerates charged particles to high energy using a magnetic field to keep them circling, and a small oscillating electric field to give them a “kick” of energy every time they cross the gap between the two D-shaped chambers (dees). Because the period is speed-independent (from Section 2), the oscillator can run at a fixed frequency — the cyclotron frequency — even as the particle speeds up and spirals outward.

Maximum Kinetic Energy
KE_max = q²B²r²_max / (2m)
r_max is the radius of the outermost orbit, limited by the physical size of the dees.

5Force on a Current-Carrying Conductor

A current is just many moving charges, so a current-carrying wire in a magnetic field feels a force too — the sum of the tiny forces on every charge carrier inside it.

Force on a Straight Wire
F = I L × B
Magnitude: F = BIL sin θ, where θ is the angle between the wire (current direction) and B. Direction found using the right-hand rule — this is the basic principle behind electric motors.

6Biot–Savart Law

Just as Coulomb’s Law gives the field of a point charge, the Biot–Savart Law gives the magnetic field due to a tiny current element — the starting point for calculating the field of any current distribution.

Field Due to a Current Element
dB = (μ₀ / 4π) · I dl × r̂ / r²
μ₀ = 4π × 10⁻⁷ T·m/A is the permeability of free space. Notice the strong resemblance to Coulomb’s Law — same 1/r² falloff, with μ₀/4π playing the role k does for electric fields.

7Magnetic Field on the Axis of a Circular Current Loop

Integrating the Biot–Savart Law around a full circular loop of radius R gives the field at any point on the axis, a distance x from the centre:

On-Axis Field
B = μ₀ I R² / [2(R² + x²)^3/2]
At the Centre (x = 0)
B = μ₀ I / (2R)
The maximum field along the axis — it falls off as you move away from the centre in either direction.

8Ampere’s Circuital Law

Ampere’s Law is to magnetism what Gauss’s Law is to electrostatics — a shortcut that lets you find B algebraically for highly symmetric current distributions, without integrating Biot–Savart directly.

Ampere’s Law
∮ B · dl = μ₀ I_enclosed
The line integral of B around any closed loop equals μ₀ times the current passing through that loop — independent of the loop’s exact shape.
Application: Long Straight Wire
B = μ₀ I / (2π r)
Field circles the wire, falling off as 1/r — this is the single most-used result from this section.

9Solenoid and Toroid

Solenoid

Field Inside a Long Solenoid
B = μ₀ n I
n = number of turns per unit length. The field inside is uniform and parallel to the axis; outside, it’s essentially zero for an ideal (long, tightly-wound) solenoid — a close magnetic analogue of the field between parallel capacitor plates.

Toroid

Field Inside a Toroid
B = μ₀ N I / (2π r)
N = total number of turns, r = distance from the toroid’s central axis. Field is confined entirely within the toroid’s core — zero outside and in the central hole.

10Force Between Two Parallel Currents

Each current-carrying wire creates a field, and that field exerts a force on the other wire’s current — the basis for the SI definition of the ampere itself.

Force per Unit Length
F / L = μ₀ I₁ I₂ / (2π d)
d = separation between the wires. Currents in the same direction attract; currents in opposite directions repel — the reverse of what many students initially expect from analogy with like/unlike charges.

11Torque on a Current Loop; Magnetic Dipole Moment

A current loop in a magnetic field behaves like a tiny magnetic dipole — it feels a torque that tries to align it with the field, exactly analogous to an electric dipole in an electric field.

Magnetic Moment of a Loop
m = N I A
N = number of turns, A = area of the loop, direction given by the right-hand rule (curl fingers along current direction, thumb points along m).
Torque
τ = m × B, magnitude τ = N I A B sin θ
Maximum when the loop’s plane is parallel to B (θ = 90° between m and B); zero when m is aligned with B. This is the working principle of every electric motor and moving-coil meter.

12Moving Coil Galvanometer

A galvanometer detects small currents by suspending a coil in a magnetic field; current through the coil produces a torque (Section 11), rotating a pointer against a restoring spring until the torques balance.

• Current sensitivity is increased by increasing N, A, or B, or decreasing the spring constant — deflection per unit current.
• To convert a galvanometer into an ammeter, a small resistance (shunt) is connected in parallel, diverting most of the current around the coil.
• To convert a galvanometer into a voltmeter, a large resistance is connected in series, limiting the current through the coil for a given voltage.

Formula Summary

Lorentz ForceF = q(E + v×B)
Circular Radiusr = mv/(qB)
Cyclotron PeriodT = 2πm/(qB)
Velocity Selectorv = E/B
Force on WireF = BIL sinθ
Biot–SavartdB = (μ₀/4π) I dl×r̂/r²
Loop Centre FieldB = μ₀I/2R
Ampere’s Law∮B·dl = μ₀I
Long Wire FieldB = μ₀I/2πr
SolenoidB = μ₀nI
ToroidB = μ₀NI/2πr
Force Between WiresF/L = μ₀I₁I₂/2πd
Magnetic Momentm = NIA
Torqueτ = NIAB sinθ

Solved Examples

Example 1 · Circular Motion

A proton moving at 2×10⁶ m/s enters a 0.5 T magnetic field perpendicular to its velocity. Find the radius of its circular path. (m_p = 1.67×10⁻²⁷ kg, q = 1.6×10⁻¹⁹ C)

Solution: r = mv/(qB) = (1.67×10⁻²⁷ × 2×10⁶) / (1.6×10⁻¹⁹ × 0.5) = (3.34×10⁻²¹) / (8×10⁻²⁰) ≈ 0.0418 m ≈ 4.2 cm.

Example 2 · Force Between Parallel Wires

Two long parallel wires, 2 cm apart, carry currents of 5 A and 10 A in the same direction. Find the force per unit length between them, and state whether it’s attractive or repulsive.

Solution: F/L = μ₀I₁I₂/(2πd) = (2×10⁻⁷ × 5 × 10) / 0.02 = (1×10⁻⁵) / 0.02 = 5×10⁻⁴ N/m.

Since both currents flow in the same direction, the force is attractive.

Quick Check

1. The magnetic field at the centre of a circular current loop of radius R carrying current I is:
✓ B = μ₀I/2R (correct)
B = μ₀I/4πR
B = μ₀I/R²
B = μ₀IR
2. Two long parallel wires carrying current in opposite directions will:
Attract each other
✓ Repel each other (correct)
Feel no force
Rotate only
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