20.1.5 Force on a Moving 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)

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 = BQv

• 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

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^°

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