Magnetic Forces on Moving Charges Revision Notes | A Level AQA Physics

1. Why Moving Charges Feel a Force

Picture a beam of electrons fired horizontally across a laboratory. Switch on a magnetic field pointing vertically downward — and the beam curves sideways. No contact, no electric field, yet a force appears from nowhere. That is the magnetic force on a moving charge, and it is the foundation of everything from particle accelerators to the aurora borealis.

The key insight is this: a stationary charge in a magnetic field feels nothing. The force only exists when the charge is moving, and it only has a component when that motion is not perfectly parallel to the field. This is fundamentally different from electric forces, which act on charges regardless of their motion.

Because the force is always perpendicular to the velocity, it cannot speed the particle up or slow it down — it can only change the direction of motion. This is why a charged particle in a uniform magnetic field (with velocity fully perpendicular to the field) travels in a circle at constant speed.

2. The Equation: $F = BQv$

We need a precise way to calculate the size of this force. Three quantities determine it: how strong the field is, how much charge the particle carries, and how fast it is moving. Combine them and you get the core equation for this topic.

Notice that $F$ is proportional to all three quantities. Double the field strength and the force doubles. Double the speed and the force doubles. A particle with twice the charge feels twice the force. This linear relationship makes the equation straightforward to apply — and straightforward to rearrange for any unknown.

One condition is baked into this version of the equation: the velocity must be perpendicular to the field. If the particle moves parallel to the field ($\theta = 0°$), then $\sin 0° = 0$ and the force is zero. The particle passes straight through undeflected.

3. Direction of the Force: Fleming's Left-Hand Rule

Knowing the magnitude of the force is only half the picture. Examiners regularly ask you to state or explain the direction — and getting this wrong costs marks even when your calculation is perfect. The tool you need is Fleming's left-hand rule.

• Hold your left hand with the thumb, index finger, and middle finger mutually perpendicular.
• Point your First (index) finger in the direction of the magnetic Field ($B$).
• Point your seCond (middle) finger in the direction of conventional Current — that is, the direction a positive charge would move.
• Your thuMb then points in the direction of the force (Motion) on the positive charge.

The rule is built around conventional current, which flows in the direction of positive charge movement. For a positive charge moving to the right in a field pointing into the page, the force points upward. Reverse the charge sign and the force reverses — a negative charge moving to the right experiences a downward force in the same field.

This reversal for negative charges is one of the most reliable sources of dropped marks in this topic. The physics is consistent: the force on a negative charge is always opposite to the force that would act on a positive charge moving in the same direction through the same field.

That worked example above shows exactly why the direction question catches students out — the reversal step for negative charges must be applied after, not during, the left-hand rule. Practise the rule on positive charges first, then reverse for electrons.

4. Worked Calculation: Finding the Magnetic Force

Theory is only useful once you can apply it under exam conditions. The calculation structure for $F = BQv$ is one of the more straightforward in this unit — but unit errors and charge-sign slips are common. The worked example below models the method examiners reward.

Worked example: Calculating the magnetic force on a proton

A proton moves at $3.0 \times 10^6$ m s$^{-1}$ perpendicular to a uniform magnetic field of flux density $0.25$ T. Calculate the magnetic force acting on the proton. (Charge on a proton: $Q = 1.6 \times 10^{-19}$ C)

1. $F = BQv$ — [M1] Selecting the correct equation is the first method mark. State it explicitly before substituting.
2. $B = 0.25$ T, $Q = 1.6 \times 10^{-19}$ C, $v = 3.0 \times 10^6$ m s$^{-1}$ — List every given quantity with its unit. This prevents substitution errors and shows the examiner your method clearly.
3. $F = 0.25 \times 1.6 \times 10^{-19} \times 3.0 \times 10^6$ — [M1] Substitution mark. All three values are in SI units, so no conversion is needed.
4. $0.25 \times 1.6 = 0.40$, then $0.40 \times 3.0 = 1.20$ — Work through the decimal multiplication first, keeping the powers of ten separate to avoid errors.
5. $10^{-19} \times 10^{6} = 10^{-19+6} = 10^{-13}$ — Add the exponents: $-19 + 6 = -13$. This is a common arithmetic slip — write it out explicitly.
6. $F = 1.20 \times 10^{-13}$ N $\approx 1.2 \times 10^{-13}$ N — [A1] Accuracy mark. Round to 2 significant figures to match the data given. Always include the unit — newtons (N).

Answer: $F = 1.2 \times 10^{-13}$ N

Notice how the powers of ten are handled as a separate step. Collapsing $10^{-19} \times 10^6$ in your head and writing the wrong exponent is one of the most common errors in this type of calculation. Writing the exponent addition explicitly — $-19 + 6 = -13$ — takes two seconds and eliminates the mistake.

5. Circular Motion in a Magnetic Field

Earlier we established that the magnetic force is always perpendicular to the velocity. This has a profound consequence: if a charged particle enters a uniform magnetic field with its velocity entirely perpendicular to the field, the force continuously redirects the particle without changing its speed. The result is uniform circular motion.

The magnetic force provides the centripetal force. Setting $BQv = \frac{mv^2}{r}$ and rearranging gives the radius of the circular path: $r = \frac{mv}{BQ}$. A faster particle, or one with greater mass, curves less (larger $r$). A stronger field or larger charge curves the path more tightly (smaller $r$). This principle underpins the design of cyclotrons and mass spectrometers.

Positive and negative charges entering the same field with the same velocity will curve in opposite directions — one clockwise, one anticlockwise — because the force on them is reversed. This is how physicists distinguish particle types in bubble chambers and cloud chambers.

6. Comparing Positive and Negative Charge Behaviour

AQA questions frequently present a diagram showing a particle entering a field and ask you to identify the charge sign from the direction of deflection, or to predict the deflection given the charge sign. A clear comparison of the two cases is worth having at your fingertips.

The magnitude of the force depends only on $|Q|$, not on the sign. An electron and a proton moving at the same speed through the same field experience forces of identical magnitude — but in opposite directions. This is a subtle but important distinction that examiners test.

7. Exam Technique and Common Mistakes

The two specification requirements for this topic — the equation $F = BQv$ and the direction of force on positive and negative charges — appear in calculation questions, describe/explain questions, and diagram-based questions. Each format has its own pitfalls.

For extended-response questions asking you to explain the circular motion of a charged particle in a magnetic field, the mark scheme typically rewards: stating that $F = BQv$ acts perpendicular to velocity; identifying this as the centripetal force; concluding that speed is constant but direction changes continuously; and noting that the radius depends on $\frac{mv}{BQ}$. Hitting all four points in a logical sequence is the route to full marks.

The multiple-select question above covers the full conceptual scope of the specification requirements for this topic. If you can identify all four correct statements and articulate why the incorrect one is wrong, you have the depth of understanding needed to answer any AQA question on magnetic forces on moving charges.