Force on a moving charge

Section: Magnetic Fields | Syllabus: Cambridge A Level Physics (9702)

Force on a Moving Charge

A single charge Q moving with velocity v is equivalent to a geometric current.

F = BQv sin θ Direction

For a positive charge, use Fleming's Left Hand Rule with the second finger in direction of v. For an electron (negative), the "current" is opposite to velocity.

Circular Motion in Magnetic Fields

If a charged particle enters a uniform field perpendicularly (θ = 90^):

The force F = BQv is constant in magnitude and always perpendicular to velocity.
This provides a centripetal force, causing circular motion at constant speed.

Centripetal Force = Magnetic Force

(mv^2/r) = BQv

Rearranging for radius: r = (mv/BQ)

Specific Charge

The specific charge (charge-to-mass ratio) is given by (Q/m). From the equation above, (Q/m) = (v/Br). This ratio is unique for different particles (e.g. protons vs electrons) and is used to identify them.

Velocity Selection

By applying perpendicular Electric (E) and Magnetic (B) fields, we can select particles of a specific velocity.

For straight line motion (zero deflection): Magnetic Force = Electric Force.

BQv = qE v = (E/B)

Application: Mass Spectrometer. After velocity selection, ions enter a region of uniform magnetic field only. The radius of curvature (r = mv/BQ) then depends only on mass (for same charge). Heavier ions travel in larger circles.

The Hall Effect

When current flows through a conductor in a magnetic field, charge carriers are pushed to one side by the magnetic force (F=Bqv), creating a potential difference called the Hall Voltage (V_H).

Derivation

Equilibrium is reached when Electric Force (F_E) balances Magnetic Force (F_B).

1. F_E = F_B qE = Bqv E = Bv.

2. V_H = E d (width). So V_H = Bvd.

3. Current I = nAqv = n(d · t)qv (where t is thickness/depth parallel to B-field).

4. Subst v = (I/n d t q) into V_H:

V_H = (BI/ntq)

This effect is used in Hall Probes to measure magnetic flux density B.

Hall Probe Sensitivity

Notice V_H ∝ 1/n. Semiconductors have a much smaller number density of charge carriers (n) than metals. Therefore, for the same current and field, semiconductors produce a much larger Hall Voltage, making them suitable for Hall Probes.

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Force on a moving charge