Why does a charged particle move in a circle in a magnetic field?

The force is always perpendicular to its velocity, so it changes direction continuously without changing speed — the condition for uniform circular motion.

Does a magnetic field do work on a charge?

No. Work needs a force component along the motion, and the magnetic force has none.

Why reverse Fleming’s rule for electrons?

The rule uses conventional current, the direction positive charge would move. Electrons move opposite to that.

Why is the period independent of speed?

Faster particles travel in larger circles in exact proportion, since r v, so the time per revolution 2 r/v stays constant.

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7.2ElectricityStretch

Magnetic Fields and Forces

A current-carrying wire in a magnetic field feels a force, and so does a single charged particle moving through one. Both are the same effect — magnetic fields push on moving charge — and because the force is always perpendicular to the motion, it can bend a particle into a perfect circle.

40 min Topic 7 · Electric and Magnetic Fields

What you'll be able to do

Describe magnetic flux density and its unit, the tesla
Use $F = B I l sin θ$ and Fleming’s left-hand rule
Use $F = B q v sin θ$ for a moving charge
Explain and calculate the circular motion of charged particles in a field

Magnetic flux density

The strength of a magnetic field is its $magnetic flux density$ $B$ , measured in teslas. A field of 1 T exerts a force of 1 N on a 1 m wire carrying 1 A at right angles to the field.

Field lines run from north to south outside a magnet. Around a straight current-carrying wire they form concentric circles, with direction given by the right-hand grip rule.

Typical values span a wide range: the Earth’s field is about $5 \times 1 0^{- 5}$ T, a fridge magnet around 0.01 T, an MRI scanner several tesla.

1 T = 1 N A^{- 1} m^{- 1}

The tesla.

Tip — Magnetic fields exert forces on moving charges and currents, not on stationary charges. A charge at rest in a magnetic field feels nothing.

Force on a current-carrying wire

A wire of length $l$ carrying current $I$ at angle $θ$ to a field $B$ experiences a force $F = B I l sin θ$ . The force is maximum when the wire is perpendicular to the field and zero when parallel.

The direction is perpendicular to both the wire and the field, given by $Fleming’s left-hand rule$ : first finger for Field, second finger for Current, thumb for the resulting Motion (force).

This is the $motor effect$ , the principle behind electric motors and loudspeakers.

F = B I l sin θ

Force on a wire.

F = 0.12 \times 4.0 \times 0.25 \times sin 3 0^{\circ}

= 0.12 \times 0.5

Answer

0.060 N

Force on a moving charge

A current is a flow of charges, so each moving charge must feel a force. For a charge $q$ moving at speed $v$ at angle $θ$ to the field, $F = B q v sin θ$ .

It follows from $F = B I l$ : in time $t$ , charge $q$ travels $l = v t$ and constitutes a current $I = q / t$ , so $B I l = B (q / t) (v t) = B q v$ .

The direction again comes from Fleming’s left-hand rule, using conventional current. For a negative charge such as an electron, the force is in the $opposite$ direction to that for a positive charge.

F = B q v sin θ

Force on a moving charge.

F = 0.40 \times 1.60 \times 1 0^{- 19} \times 2.0 \times 1 0^{6}

Answer

1.3 \times 1 0^{- 13} N

Tip — For electrons, point the second finger opposite to their direction of motion, since conventional current is the flow of positive charge.

Circular paths

The magnetic force on a charge is always perpendicular to its velocity, so it does no work and cannot change the speed — only the direction. A particle moving perpendicular to a uniform field therefore travels in a $circle$ .

The magnetic force provides the centripetal force: $B q v = m v^{2} / r$ , giving $r = \frac{m v}{B q} = \frac{p}{B q}$ .

Faster or heavier particles curve less; stronger fields or larger charges curve them more. This is the principle of the mass spectrometer, which separates ions by their radius of curvature, and of the cyclotron, where the time for a half-circle is independent of speed.

r = \frac{m v}{B q}

Radius of the circular path.

T = \frac{2 π m}{B q}

Period — independent of speed.

r = \frac{9.11 \times 1 0 ^{- 31} \times 5.0 \times 1 0 ^{6}}{2.0 \times 1 0 ^{- 3} \times 1.60 \times 1 0 ^{- 19}}

= \frac{4.555 \times 1 0 ^{- 24}}{3.2 \times 1 0 ^{- 22}}

Answer

r = 1.4 \times 1 0^{- 2} m

, about 1.4 cm.

Equation recap

F = B I l sin θ

Force on a wire.

F = B q v sin θ

Force on a charge.

r = m v / B q

Circular path radius.

T = 2 π m / B q

Period in a field.

Common mistakes to avoid

Saying a stationary charge feels a magnetic force.

Only moving charges (or currents) experience magnetic forces.

Claiming the magnetic force speeds a particle up.

It is perpendicular to velocity, so it changes direction only.

Using the right-hand rule for the motor effect.

Fleming’s

left

-hand rule gives the force.

Forgetting mT or µT conversions.

Convert to teslas before substituting.

Key takeaways

Magnetic flux density $B$ is measured in teslas.
A wire in a field feels $F = B I l sin θ$ , direction by Fleming’s left-hand rule.
A moving charge feels $F = B q v sin θ$ ; negative charges are forced the opposite way.
The force is perpendicular to velocity, so it does no work.
Charges perpendicular to a uniform field move in circles of radius $r = m v / B q$ .

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Magnetic Fields and Forces

Magnetic flux density

Force on a current-carrying wire

Force on a moving charge

Circular paths

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