Magnetic Field Due to Moving Charges Explained for School

An electric current is simply charge in motion, and every moving charge creates a magnetic field around it along with its electric field. The Biot–Savart law describes exactly how a small moving charge — or a short current element — contributes a tiny magnetic field at any point in space.

This idea is the seed of all electromagnetism. Fields from wires, coils, electromagnets and even the beam in a particle accelerator are built by adding the contributions of countless moving charges, which is why motors, MRI magnets and transformers work as predictably as they do.

Key Concepts

Biot–Savart law: a current element I dl produces field dB at distance r. Permeability of free space μ0: 4π × 10⁻⁷ tesla metre per ampere (T m/A). Direction: given by the right-hand rule — thumb along current, fingers curl along field lines. Superposition: total field is the vector sum of all element contributions. For a single point charge moving at velocity v, the field circles the line of motion and falls as 1/r².

Formulas

dB = (μ0/4π) × I dl sinθ / r² where dB in tesla (T), I in ampere (A), dl in metre (m), r in m and θ is angle between dl and the line joining element to field point. For a moving point charge: B = (μ0/4π) × q v sinθ / r², where q in coulomb (C) and v in m/s. Assumption: steady current, charges moving slowly compared with speed of light. Limit: the law in this simple form does not include time-varying electric effects of accelerating charges.

Worked Example

A proton (q = 1.6 × 10⁻¹⁹ C) moves at v = 1.0 × 10⁶ m/s. Find B at a point 1.0 × 10⁻¹⁰ m directly to its side (θ = 90°).

1. B = (μ0/4π) × q v / r², with μ0/4π = 1.0 × 10⁻⁷.
2. B = 1.0 × 10⁻⁷ × (1.6 × 10⁻¹⁹ × 1.0 × 10⁶)/(1.0 × 10⁻¹⁰)².
3. Numerator: 1.6 × 10⁻¹³. Denominator: 1.0 × 10⁻²⁰. Ratio = 1.6 × 10⁷.
4. B = 1.0 × 10⁻⁷ × 1.6 × 10⁷ = 1.6 T.

Close to a fast elementary charge the magnetic field can be surprisingly large, about 1.6 T, even though it vanishes far away because of the 1/r² fall.

Common Misconceptions

• A stationary charge also makes a magnetic field: No — only moving charges do; a stationary charge makes only an electric field.
• The field points along the motion: The magnetic field circles around the direction of motion, perpendicular to both v and r.
• Current and moving charge are unrelated: Current is just many charges drifting together; the same law governs both.

Real-World Applications

• Calculating fields of wires, loops and solenoids by summing current elements.
• Electron and ion beams in cathode-ray tubes, mass spectrometers and accelerators.
• Magnetic sensors and models of fields around moving charged particles in space.

Moving charge is the true source of magnetism; the Biot–Savart law quantifies its tiny field, and superposition of many charges builds every practical electromagnet we use.