Force on moving charges

Stand near a wire carrying current next to a bar magnet, or watch a beam of electrons bend inside an old cathode-ray tube, and you are watching the same force at work: a moving charge in a magnetic field gets pushed sideways. This sideways push, the magnetic part of the Lorentz force, is stranger than the electric force you already know — it depends on how fast and in which direction the charge moves, it always acts at right angles to that motion, and it never speeds up or slows down the charge, only steers it. Scale the idea up from a single charge to the river of charges inside a current-carrying wire, and you get the force that makes electric motors turn and the force between two current-carrying wires that is used, by international agreement, to define the ampere itself. This article builds the Lorentz force law, applies it to a wire in a field, and uses it to find the force between parallel wires.

The core insight

A charge $q$ moving with velocity $\vec v$ through a magnetic field $\vec B$ feels a force $\vec F = q\vec v\times\vec B$ that is always perpendicular to $\vec v$ , so a current-carrying wire in a field feels $\vec F = I\vec l\times\vec B$ , and two parallel current-carrying wires exert a force per unit length on each other of $\mu_0 I_1 I_2/2\pi d$ .

The core idea

The Lorentz force. A charge $q$ moving with velocity $\vec v$ through a region with electric field $\vec E$ and magnetic field $\vec B$ feels a total force that is the sum of an electric push and a magnetic push:

\vec F = q\vec E + q(\vec v\times\vec B)

where: $q$ is the charge (with sign), $\vec E$ is the electric field, $\vec v$ is the charge's velocity, and $\vec B$ is the magnetic field. This article focuses on the magnetic part, $\vec F_B = q\vec v\times\vec B$ .

Magnitude and direction. The size of the magnetic force is $F_B = qvB\sin\theta$ , where $\theta$ is the angle between $\vec v$ and $\vec B$ ; its direction follows the right-hand rule for $\vec v\times\vec B$ (for a positive charge — reverse the direction for a negative one). Two consequences follow directly from "always perpendicular to $\vec v$ ":

The magnetic force does zero work, because work needs a force component along the displacement, and $\vec F_B$ has none. It can only turn the velocity, never change its magnitude — so a charge moving through a uniform field at constant speed traces a circle (if $\vec v\perp\vec B$ ) or a helix (if $\vec v$ has a component along $\vec B$ , since that component feels no force at all).
If $\vec v$ is parallel to $\vec B$ ( $\theta=0$ ), $\sin\theta=0$ and the force vanishes entirely — a very common point where students forget the $\sin\theta$ factor.

Figure

A positive charge moving with velocity v to the right in a magnetic field pointing into the page follows a circular arc, with the magnetic force always drawn perpendicular to v and pointing toward the centre of the circle

Force on a current-carrying conductor. A straight wire of length $l$ carrying current $I$ in a field $\vec B$ feels

\vec F = I\vec l\times\vec B, \qquad F = BIl\sin\theta

where: $\vec l$ points along the direction of conventional current flow and $\theta$ is the angle between the wire and $\vec B$ . This is simply the sum of the Lorentz forces on all the moving charges that make up the current, worked out below.

Force between two parallel wires. Two long parallel wires a distance $d$ apart, carrying currents $I_1$ and $I_2$ , each sit in the other's magnetic field and so exert a force on each other, per unit length:

\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}

where: $\mu_0 = 4\pi\times10^{-7}\ \text{T}\,\cdot\,\text{m/A}$ is the permeability of free space and $d$ is the wire separation. Like currents attract, unlike currents repel. This relation defines the ampere: it is the current that, flowing in two infinite parallel wires $1\ \text{m}$ apart in vacuum, produces exactly $2\times10^{-7}\ \text{N}$ of force per metre of length between them.

Derivation AForce on a Current-Carrying Wire, from the Lorentz Force on Its Charge Carriers

The goal is to show that the familiar wire formula $F = BIl\sin\theta$ is not a separate law but simply the Lorentz force $qvB\sin\theta$ on a single charge, added up over every charge carrier drifting through a straight segment of wire — a chain of substitutions using only the definition of current and no new physics.

Step 1. Picture a straight wire segment of length $l$ , cross-sectional area $A$ , carrying current $I$ in a field $\vec B$ at angle $\theta$ to the wire. Suppose there are $n$ free charge carriers per unit volume, each of charge $q$ , drifting with speed $v_d$ along the wire. The number of carriers inside the segment is $N = n\,(A l)$ , since $Al$ is the segment's volume.

Step 2. Each carrier moves with drift velocity $v_d$ through $\vec B$ , so each feels a Lorentz force of magnitude $F_1 = q v_d B\sin\theta$ , directed perpendicular to both $v_d$ and $B$ (same direction for every carrier, since they all drift the same way). The total force on the segment is just $N$ copies of this single-carrier force, because forces on all the carriers point the same way and simply add: $F = N F_1 = (nAl)(q v_d B\sin\theta)$ .

Step 3. Regroup the factors as $F = (nqv_d A)\,l\,B\sin\theta$ . The bracketed quantity $nqv_dA$ is exactly the definition of current, $I = nqv_dA$ (the charge per unit time flowing past a cross-section), because in time $t$ a length $v_dt$ of carriers of charge density $nq$ passes through area $A$ , giving charge $nq(v_dt)A$ and hence current $nqv_dA$ .

Step 4. Substitute $I$ back in: $F = I\,l\,B\sin\theta$ , exactly the wire-force formula. Nothing new was assumed beyond the Lorentz force on one charge and the definition of current — the wire law is that single-charge law, summed over the wire's own population of moving charges. Check: setting $\theta=90°$ recovers the maximum case $F=BIl$ used directly in worked problems; setting $\theta=0$ (wire along $\vec B$ ) correctly gives $F=0$ , matching the parallel-velocity case for a single charge.

Worked example

A proton ( $q=1.6\times10^{-19}\ \text{C}$ ) moves at $2\times10^6\ \text{m/s}$ perpendicular to a magnetic field of magnitude $0.5\ \text{T}$ . Find the magnetic force on it, then use the same wire-force idea to find the force per unit length between two long parallel wires $0.1\ \text{m}$ apart carrying $5\ \text{A}$ and $8\ \text{A}$ in the same direction.

Step 1. Since $\vec v\perp\vec B$ , $\sin\theta=1$ , so $F=qvB=(1.6\times10^{-19})(2\times10^6)(0.5)$ .

Step 2. Multiply the digits, $1.6\times2\times0.5=1.6$ , and the powers of ten, $10^{-19}\times10^6=10^{-13}$ , giving $F=1.6\times10^{-13}\ \text{N}$ , perpendicular to both $\vec v$ and $\vec B$ by the right-hand rule. This force supplies the centripetal force that curves the proton into a circular arc, exactly as in the figure above.

Step 3. Now the wires. Use $\dfrac{F}{L}=\dfrac{\mu_0 I_1 I_2}{2\pi d}$ with $\mu_0=4\pi\times10^{-7}\ \text{T}\,\cdot\,\text{m/A}$ , $I_1=5\ \text{A}$ , $I_2=8\ \text{A}$ , $d=0.1\ \text{m}$ : $\dfrac{F}{L}=\dfrac{4\pi\times10^{-7}\times5\times8}{2\pi\times0.1}$ .

Step 4. Cancel $\pi$ and simplify: $\dfrac{F}{L}=\dfrac{4\times10^{-7}\times40}{0.2}=\dfrac{1.6\times10^{-5}}{0.2}=8\times10^{-5}\ \text{N/m}$ . Since the currents flow in the same direction, this force is attractive.

Final answer: the proton feels $1.6\times10^{-13}\ \text{N}$ , and the two wires attract each other with $8\times10^{-5}\ \text{N}$ per metre of length.

Common traps

Dropping the $\sin\theta$ factor and writing $F=qvB$ or $F=BIl$ even when $\vec v$ or the wire is not perpendicular to $\vec B$ — if either is parallel to $\vec B$ , the force is zero.
Believing the magnetic force changes a charge's speed — it only changes direction, since it is always perpendicular to velocity and does zero work.
Applying the right-hand rule for $\vec v\times\vec B$ without reversing it for a negative charge.
Mixing up the parallel-wire rule — same-direction currents attract, opposite-direction currents repel — a frequently reversed fact.

Ask the tutor: Want a few extra right-hand-rule drills, or help visualising why circular motion in a magnetic field never changes a charge's speed?

If the magnetic force does no work, how does a motor actually do useful work and spin things?

The magnetic force on each charge inside the motor's wire never speeds up that individual charge — it only pushes it sideways. But the wire itself is a rigid conductor, and that sideways push on the charges is transmitted, through the rigid structure of the wire, into a sideways push on the whole coil, which is free to rotate. The coil turning against a resisting load (like a fan blade) is where the actual mechanical work happens, driven by the electrical energy source that keeps pushing current through the wire against the induced back-effects — not by the magnetic force doing work on individual charges.

How do I quickly tell whether the force between two parallel wires is attractive or repulsive?

Use the right-hand rule twice. First find the magnetic field created by wire 1 at the location of wire 2 (curl your right-hand fingers around wire 1 in the direction of its current; your thumb points along the current, fingers show the field direction at wire 2). Then apply $\vec F=I\vec l\times\vec B$ to wire 2 in that field. Doing this carefully always confirms the shortcut rule: same-direction currents pull together, opposite-direction currents push apart. Memorise the shortcut, but know you can always rebuild it from the two right-hand rules if you forget which way it goes.

Does the formula F = BIl sin θ apply to a wire of any shape, or only a straight one?

As derived here, $F=BIl\sin\theta$ is for a straight segment of wire of length $l$ in a uniform field. For a curved or bent wire, you would need to break it into many tiny straight segments, apply the same idea to each ( $d\vec F = I\,d\vec l\times\vec B$ ), and add up the pieces — the underlying Lorentz-force reasoning is identical, only the bookkeeping changes. The syllabus here only expects you to work with straight segments and parallel straight wires, so you will not need to carry out that summation.

Force on moving charges

The core idea

Worked example

Common traps

Before this

Where this leads