Moving Charges and Magnetism, Made Simple

Q: A proton moves exactly parallel to a uniform magnetic field B with speed v. What magnetic force does it feel?

The magnetic force is F = qvB sin(theta). When the velocity is parallel to B, the angle theta = 0 degrees, so sin(theta) = 0 and the force is zero. A magnetic force needs a component of velocity perpendicular to the field; it is maximum only when v is perpendicular to B.

A patient, from-scratch guide to how magnetic fields push moving charges and current-carrying wires, how currents create fields (Biot-Savart and Ampere's law), the solenoid, and the moving-coil galvanometer, with a fully worked numerical and clear SI units for CBSE Class 12 (and your JEE/NEET/CUET foundation).

By the PadhoDost Team · 📖 8 min read · Updated 4 August 2026

Part of Class 12 (CBSE) prep

🧠 Oersted's lucky accident

Padho, Dost! In 1820, a teacher named Oersted was demonstrating a circuit when he noticed a compass needle nearby twitch every single time he switched the current on. That little twitch revealed a huge secret: electricity and magnetism are do paisa of the same sikka. A moving charge stirs up a magnetic field around itself, and a charge moving through someone else's field feels a sideways push. This whole chapter is just those two ideas, explored carefully.

The push on a moving charge

Every charge that moves through a magnetic field feels a force. The complete rule is called the Lorentz force: a charge q moving with velocity v through an electric field E and a magnetic field B feels F = q(E + v x B). In this chapter we care about the magnetic part. Its size depends on how fast the charge moves and, crucially, on the angle between its motion and the field.

Magnetic force: F = q(v x B) -> magnitude F = q v B sin(theta). Here theta is the angle between v and B. The SI unit of B is the tesla (T); 1 T = 1 N per (A.m). The direction of F is always perpendicular to BOTH v and B.

⚠️ Common trap: a magnetic force can never change a charge's speed or kinetic energy, because the force is always perpendicular to the motion. It only bends the path. Since F = qvB sin(theta), the force is ZERO when the charge is at rest (v = 0) or moves parallel/anti-parallel to B (theta = 0 or 180 degrees). It is maximum when v is perpendicular to B (theta = 90 degrees).

🧠 The stone on a string

Imagine whirling a stone tied to a string (a gophan / sling) around your head. The string always pulls the stone inward, at a right angle to its motion. It never speeds the stone up or slows it down, it only bends the path into a circle. When a charge moves across a magnetic field, the magnetic force does exactly the same job as that string, which is why the charge travels in a circle.

Charge in a circle: the magnetic force supplies the centripetal force, so q v B = m v^2 / r -> radius r = m v / (q B). Time period T = 2 pi m / (q B), and frequency f = q B / (2 pi m). Notice T and f do NOT depend on the speed v (this fact runs the cyclotron).

Force on a current-carrying wire

A wire is just a pipe full of moving charges, so it too feels a force in a field: F = I(L x B) -> magnitude F = B I L sin(theta), where I is the current, L is the length of wire inside the field, and theta is the angle between the wire and B.

📝 Worked example: force on a wire

A straight wire of length L = 0.5 m carries a current I = 4 A.

It lies perpendicular (theta = 90 degrees) to a uniform field B = 0.25 T.

Formula: F = B I L sin(theta)

Substitute: F = 0.25 x 4 x 0.5 x sin(90 degrees)

sin(90 degrees) = 1, so F = 0.25 x 4 x 0.5 = 0.5

Direction: use the right-hand rule; F is perpendicular to both the wire and B.

Answer: the wire feels a force of 0.5 newton (N).

Where do magnetic fields come from?

Biot-Savart law (field from a tiny current element): dB = (mu0 / 4 pi) x (I dl sin(theta)) / r^2, with mu0 = 4 pi x 10^-7 T.m/A. Two famous results: long straight wire B = mu0 I / (2 pi r); centre of a circular loop B = mu0 I / (2 R) (multiply by N for N turns).

Ampere's circuital law (a shortcut for symmetric cases): the line integral of B around any closed loop equals mu0 times the enclosed current, (integral of B.dl) = mu0 I(enclosed). Applied to a long solenoid it gives the field inside as B = mu0 n I, where n = number of turns per metre. This field is strong and uniform, like that of a bar magnet.

💡 Direction tricks: for the field circling a wire, point your right thumb along the current and your curling fingers show the field's direction. For the force F = q(v x B) or F = I(L x B), use v x B for a positive charge; for an electron (negative charge) the force points the opposite way. Fleming's left-hand rule (thumb = force, forefinger = field, middle finger = current) is the handy exam version.

The moving-coil galvanometer

A galvanometer detects tiny currents. A coil of N turns and area A sits in a magnetic field B. When current I flows, the coil experiences a turning effect (torque) tau = N I A B. A spring pushes back, so the coil rotates until the spring's twist balances the torque. The pointer then rests at an angle phi that is directly proportional to the current: phi = (N A B / k) I, where k is the spring's torsion constant. That is why the scale reading tells you the current. Increasing N, A or B raises the current sensitivity (more deflection per ampere); dividing that by the coil resistance R gives the voltage sensitivity.

Situation	Formula	What it tells you
Charge moving in a field	F = q v B sin(theta)	Sideways push; zero when v is parallel to B
Wire carrying current	F = B I L sin(theta)	Force on a current-carrying conductor
Radius of circular path	r = m v / (q B)	More momentum means a bigger circle
Long straight wire's field	B = mu0 I / (2 pi r)	Field weakens as distance r grows
Centre of a circular loop	B = mu0 I / (2 R)	For N turns, multiply by N
Inside a long solenoid	B = mu0 n I	Uniform field; n = turns per metre
Galvanometer torque	tau = N I A B	Deflection phi is proportional to I

⚡ Quick check

A proton moves exactly parallel to a uniform magnetic field B with speed v. What magnetic force does it feel?

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