Apply the relationships for the magnetic force on a moving charge F = q v B sin(theta) and on a current-carrying conductor F = B I L sin(theta), including the right-hand rule, circular motion of charged particles in uniform magnetic fields, and forces between parallel conductors

What this dot point is asking

QCAA wants you to apply the two magnetic force relationships ($\vec{F} = q \vec{v} \times \vec{B}$ for moving charges, $\vec{F} = I \vec{L} \times \vec{B}$ for current-carrying conductors), use the right-hand rule to determine the direction of the force in 3D, derive the radius and period of a charged particle's circular motion in a uniform field, and compute the force per unit length between two parallel conductors. This dot point underpins IA1 short response on velocity selectors and mass spectrometers, and feeds the EA Paper 2 derivations on motor and generator action.

The answer

Magnetic force on a moving charge

A charge $q$ moving with velocity $\vec{v}$ through a magnetic field $\vec{B}$ experiences a force:

where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$. The force is perpendicular to both $\vec{v}$ and $\vec{B}$, given in direction by $\vec{F} = q \vec{v} \times \vec{B}$.

Key consequences:

• A charge moving parallel to $\vec{B}$ ($\theta = 0°$ or $180°$) experiences zero force.
• A charge moving perpendicular to $\vec{B}$ ($\theta = 90°$) experiences the maximum force $F = q v B$.
• The force is always perpendicular to the velocity. The magnetic force does no work on the charge. The speed is unchanged; only the direction changes.

Right-hand rule for a moving positive charge

Point your right hand's fingers in the direction of $\vec{v}$, curl them toward $\vec{B}$; the thumb points in the direction of $\vec{F}$ on a positive charge. For a negative charge (electron), reverse the direction.

Equivalent: point the right palm so that fingers extend along $\vec{B}$ and the thumb along $\vec{v}$; the force on a positive charge pushes out of the palm.

Circular motion of a charged particle

A charge moving perpendicular to a uniform field experiences a constant-magnitude force always perpendicular to its velocity. The trajectory is a circle. Setting magnetic force equal to centripetal force:

The period of the circular motion:

Importantly, $T$ is **independent of $v$**: a fast and a slow charge of the same mass and charge complete the same circle in the same time. This is the basis of the cyclotron, where an oscillating electric field at a fixed frequency accelerates charged particles to high speeds.

If the velocity has a component parallel to $\vec{B}$, that component is unaffected and the path becomes a helix.

Try it: Lorentz force calculator. Enter charge, speed, field and angle to get force magnitude, radius and period.

Magnetic force on a current-carrying conductor

A wire of length $L$ carrying current $I$ in a magnetic field $\vec{B}$ experiences a force:

where $\theta$ is the angle between the current direction and $\vec{B}$. The direction is given by $\vec{F} = I \vec{L} \times \vec{B}$ (right-hand rule with fingers along the current, curled toward $\vec{B}$, thumb gives $\vec{F}$).

This is exactly the per-charge force $qvB$ summed over all charge carriers in the wire. The two relationships ($F = qvB$ and $F = BIL$) are the same physics in two presentations.

Force between parallel conductors

Two long parallel wires carrying currents $I_1$ and $I_2$ separated by distance $d$ exert a force per unit length on each other:

with $\mu_0 / (2 \pi) = 2.0 \times 10^{-7}$ T m / A.

• Currents in the same direction attract.
• Currents in opposite directions repel.

The qualitative reasoning: wire 1 produces a field at wire 2 that wraps around it (right-hand grip rule). Wire 2 sits in this field carrying its own current, so $\vec{F} = I \vec{L} \times \vec{B}$ acts on it. Working out the cross product gives the same-direction-attract result.

This force was historically used to define the ampere.

Combining electric and magnetic fields: the velocity selector

A charged particle passing through perpendicular $\vec{E}$ and $\vec{B}$ fields experiences forces $qE$ and $qvB$. These can be made to cancel for a specific velocity:

Particles with this exact speed pass straight through; others are deflected. Velocity selectors are the entry stage of a mass spectrometer.

How this appears in IA1 and IA2

IA1 data test. Expect a velocity-selector or mass-spectrometer geometry with diagrams and a question on the radius of curvature, or a current-balance stimulus measuring the force between two parallel conductors. Right-hand-rule direction questions are routine.

IA2 student experiment. A common IA2 design measures the force on a current-carrying conductor between the poles of a permanent magnet as a function of current (or wire length), and extracts the field strength $B$ from the slope. Strong reports linearise $F$ vs $I$ and report $B$ with uncertainty.

Common traps

Forgetting the $\sin\theta$ factor. Maximum force occurs at $\theta = 90°$; parallel motion gives zero force.

Using the right-hand rule for a negative charge as if it were positive. Reverse the thumb direction (or equivalently use the left hand) for an electron.

Claiming the magnetic force changes the kinetic energy. It does not. Magnetic force is always perpendicular to velocity, so it does no work; the speed is constant in a uniform field.

Mixing up $F = qvB$ and $F = BIL$ contexts. $qvB$ is for a single moving charge; $BIL$ is for a wire carrying current. Do not double-count.

Getting the parallel-conductor force direction backward. Same direction attracts, opposite direction repels. (The opposite is true for electric charges, which can confuse students.)

Treating the period $T = 2 \pi m / (q B)$ as depending on $v$. It does not. This is why the cyclotron works at a fixed frequency.

In one sentence

A moving charge in a magnetic field experiences a force $F = q v B \sin\theta$ perpendicular to both $\vec{v}$ and $\vec{B}$ (which forces a charged particle to circle with radius $r = m v / (q B)$ and period $T = 2 \pi m / (q B)$), and a current-carrying conductor experiences $F = B I L \sin\theta$ in the same geometry, with parallel currents attracting and antiparallel currents repelling.