Magnetic Force on a Charge

The magnetic force on an isolating moving charge, such an electron, is given by the equation:

F = BQv sinθ

Where:
- F = force on the charge (N)
- B = magnetic flux density (T)
- Q = charge of the particle (C)
- v = speed of the charge (m s^-1)
- θ = angle between charge’s velocity and magnetic field (degrees)

Force on isolated moving charge, downloadable AS & A Level Physics revision notes

The force on an isolated moving charge is perpendicular to its motion and the magnetic field B

Equivalent to the force on a wire, if the magnetic field B is perpendicular to the direction of the charge’s velocity, the equation simplifies to:

$F = B Q v$

According to Fleming’s left hand rule:
- When an electron enters a magnetic field from the left, and if the magnetic field is directed into the page, then the force on it will be directed downwards

The equation shows:
- If the direction of the electron changes, the magnitude of the force will change too

The force due to the magnetic field is always perpendicular to the velocity of the electron
- Note: this is equivalent to circular motion

Fleming’s left-hand rule can be used again to find the direction of the force, magnetic field and velocity
- The key difference is that the second finger representing current I (direction of positive charge) is now the direction of velocity v of the positive charge

Direction of Magnetic Force, downloadable AS & A Level Physics revision notes

The direction of the magnetic force F on positive and negative particles in a B field in and out of the page

Worked example

An electron is moving at 5.3 × 10⁷ m s^-1 in a uniform magnetic field of flux density 0.2 T.

Calculate the force on the electron when it is moving at 30° to the field, and state the factor it increases by compared to when it travels perpendicular to the field.

Answer:

Step 1: Write out the known quantities

Speed of the electron, v = 5.3 × 10⁷ m s^-1
Charge of an electron, Q = 1.60 × 10^-19 C
Magnetic flux density, B = 0.2 T
Angle between electron and magnetic field, θ = 30^°

Step 2: Write down the equation for the magnetic force on an isolated particle

F = BQv sinθ

Step 3: Substitute in values, and calculate the force on the electron at 30^°

F = (0.2) × (1.60 × 10^-19) × (5.3 × 10⁷) × sin(30) = 8.5 × 10^-13 N

Step 4: Calculate the electron force when travelling perpendicular to the field

F = BQv = (0.2) × (1.60 × 10^-19) × (5.3 × 10⁷) = 1.696 × 10^-12 N

Step 5: Calculate the ratio of the perpendicular force to the force at 30^°

Calculating Magnetic Force on a Moving Charge Worked Example equation

Therefore, the force on the electron is twice as strong when it is moving perpendicular to the field than when it is moving at 30^° to the field

Exam Tip

Remember not to mix this up with F = BIL!

F = BIL is for a current carrying conductor
F = Bqv is for an isolated moving charge (which may be inside a conductor)

It is important to note that when the moving charge is travelling along the field direction - precisely with or against the field lines - then there is no magnetic force on that charge! There is only a force if the charge if not parallel to the field.

Magnetic Force on a Charge (HL IB Physics)

Revision Note