11- Magnetic field and moving charges

What is the Lorentz Force?

The force exerted on a moving charge by a magnetic field,

• q is the charge

• v⃗ is velocity

• B⃗ is the magnetic field.

How is the direction of the Lorentz force determined?

By the right-hand rule:

• Thumb points in direction of v⃗ × B⃗

• If q is positive, F⃗ is in the same direction as v⃗ × B⃗

• If q is negative, F⃗ is in the opposite direction

Why can't the Lorentz force change a particle’s speed?

Because it is always perpendicular to velocity v⃗, meaning it cannot do work or change kinetic energy—only the direction of motion.

What is a Hall Probe?

• A device that measures magnetic fields using the Hall effect

• A voltage difference (Hall voltage, V_H) is generated due to charge deflection by a magnetic field.

How does a Hall Probe work?

• A current I flows through a semiconductor slab placed perpendicular to B⃗.

• The Lorentz force deflects charge carriers, creating a charge separation and an internal electric field E⃗_H that balances the force.

What is the Hall voltage equation?

• k_H = Hall coefficient (material-dependent)

• I = Current through the sample

• B = Magnetic field strength

What is the force on a charge in simultaneous electric and magnetic fields?

How can the net force on a charge be zero in combined E⃗ and B⃗ fields?

If E⃗ and B⃗ are perpendicular, the charge moves at a velocity v where electric and magnetic forces cancel, leading to F⃗ = 0.

What experimental setup did J.J. Thomson use to discover the electron?

• A cathode ray tube with a beam of charged particles passing through crossed electric (E⃗) and magnetic (B⃗) fields

• This allowed him to control and measure their deflection.

What were the key steps in Thomson’s experiment to discover and electron?

1. Set E = 0 and B = 0 and note the position of the spot on screen S due to undeflected beam.

2. Turn on E⃗ and measure the resulting beam deflection.

3. Maintaining E⃗ , now turn on B⃗ and adjust its value until the beam returns to the undeflected position ⇒ with the forces in opposition, they can be made to cancel