describe magnetic fields around magnets, current-carrying wires and solenoids; apply the right-hand rule to determine the directions of fields and forces; apply $F = qvB$ for a charged particle moving perpendicular to a uniform magnetic field, including circular motion of the particle

What this dot point is asking

VCAA wants you to describe magnetic fields around magnets, current-carrying wires and solenoids, apply the right-hand rule to find field and force directions, and **apply $F = qvB$** to a charged particle moving perpendicular to a uniform field (including the resulting circular motion).

The answer

The magnetic field model

A magnetic field $B$ is a vector at each point in space, measured in tesla (T). Field lines run from the north pole of a magnet to its south pole externally (and from south to north inside the magnet, forming closed loops). They never start or end on a charge; magnetic monopoles do not exist.

Field shapes

Bar magnet. Field lines emerge from the north pole, curve around, and enter the south pole. The field is strongest near the poles.

Straight current-carrying wire. The field forms concentric circles around the wire. The direction is given by the right-hand grip rule: thumb in the direction of conventional current, fingers curl in the direction of $B$.

Solenoid. A long coil of wire carrying current. Inside the solenoid the field is uniform and parallel to the axis (like a bar magnet's interior). Outside, the field resembles that of a bar magnet, with one end acting as north and the other as south. Apply the right-hand grip rule with fingers curling in the direction of current to find which end is north.

The strength of a solenoid's field can be increased by increasing the current, increasing the turns per unit length, or inserting a ferromagnetic core.

Force on a moving charge

A charged particle moving with velocity $v$ through a magnetic field $B$ experiences a force:

where $\theta$ is the angle between $v$ and $B$. For $v$ perpendicular to $B$ (the VCE case), $F = qvB$.

The direction is given by the right-hand slap rule for a positive charge:

• Fingers point in the direction of velocity $v$.
• Curl them toward the field $B$.
• The thumb points in the direction of the force $F$.

For a negative charge, the force is in the opposite direction.

If $v$ is parallel or anti-parallel to $B$, the force is zero. The magnetic force never does work on a charged particle (it is always perpendicular to $v$); it changes the direction of motion, not the speed.

Circular motion of a charged particle in a uniform IMATH_21

When a charged particle moves perpendicular to a uniform $B$, the constant magnitude of $F = qvB$ supplies the centripetal force:

So the radius of the circular path is:

and the period is:

The period is independent of speed. Faster particles travel in larger circles at the same period.

If $v$ has a component along $B$ as well as across it, the path is a helix (the parallel component is unaffected by the field; the perpendicular component drives circular motion).

Comparing the three field models

Worked example with numbers

An electron travels at $2.0 \times 10^7$ m/s perpendicular to a $0.50$ T magnetic field. Find the radius and period of its circular path.

$r = \frac{m v}{q B} = \frac{9.1 \times 10^{-31} \times 2.0 \times 10^7}{1.6 \times 10^{-19} \times 0.50} = \frac{1.82 \times 10^{-23}}{8.0 \times 10^{-20}} = 2.3 \times 10^{-4}$ m.

$T = \frac{2 \pi m}{q B} = \frac{2 \pi \times 9.1 \times 10^{-31}}{1.6 \times 10^{-19} \times 0.50} = 7.1 \times 10^{-11}$ s.

Try it: Lorentz force calculator - enter charge, speed and field strength and get $F$, $r$ and $T$.

Common traps

Forgetting that the magnetic force does no work. $F$ is always perpendicular to $v$, so the speed of a charged particle in a pure magnetic field is constant.

Using the wrong hand. VCAA uses the right hand for positive charges and conventional current. For a negative charge (electron), apply the right-hand rule and reverse the direction.

Confusing field direction with force direction. Field lines show the direction $B$ points, not the direction of force on a charge.

Mixing up grip and slap rules. Use the grip rule for the field around a wire (curl fingers around the current). Use the slap rule for the force on a moving charge (or current).

Treating a parallel $v$ and $B$ as producing a force. When $v$ is parallel to $B$, $\sin\theta = 0$ and $F = 0$.

In one sentence

Magnetic fields loop from north to south around magnets, in concentric circles around wires (right-hand grip rule), and are uniform inside solenoids; a charge moving perpendicular to $B$ feels $F = qvB$ (right-hand slap rule), travelling in a circle of radius $r = mv/qB$ and period $T = 2\pi m / qB$.