explain how electric and magnetic fields can be used in velocity selection

Force on a Current‑Carrying Conductor

When a conductor carrying an electric current is placed in a magnetic field, the moving charge carriers experience the magnetic part of the Lorentz force. The resultant force on the whole conductor is given by

$$\mathbf{F}=I\,\mathbf{L}\times\mathbf{B}$$

where

• \(I\) is the current (A).
• \(\mathbf{L}\) is a vector of length equal to the segment of wire in the field, directed along the conventional current.
• \(\mathbf{B}\) is the magnetic flux density (T).

Derivation from the Lorentz Force on Charge Carriers

Consider a segment of wire of cross‑sectional area \(A\) containing charge carriers of charge \(q\) and number density \(n\). The drift velocity of the carriers is \(\mathbf{v}_d\). The magnetic force on a single carrier is

$$\mathbf{f}=q\,\mathbf{v}_d\times\mathbf{B}$$

The total number of carriers in the segment of length \(L\) is \(N=nAL\). Summing the forces gives

$$\mathbf{F}=N\mathbf{f}=nAL\,q\,\mathbf{v}_d\times\mathbf{B}$$

Since the current is \(I=nqA v_d\), we obtain the familiar result \(\mathbf{F}=I\mathbf{L}\times\mathbf{B}\).

Using Electric and Magnetic Fields for \cdot elocity Selection

A velocity selector allows only particles (or charge carriers) with a specific speed to pass undeflected. This is achieved by arranging uniform electric and magnetic fields perpendicular to each other and to the direction of particle motion.

Principle of Operation

For a particle of charge \(q\) moving with velocity \(\mathbf{v}\) through crossed fields \(\mathbf{E}\) and \(\mathbf{B}\) (with \(\mathbf{E}\perp\mathbf{B}\) and both perpendicular to \(\mathbf{v}\)), the net force is

$$\mathbf{F}=q\mathbf{E}+q\mathbf{v}\times\mathbf{B}$$

To achieve zero net force (no deflection), the electric and magnetic forces must cancel:

$$q\mathbf{E}+q\mathbf{v}\times\mathbf{B}=0\quad\Longrightarrow\quad \mathbf{E}=-\mathbf{v}\times\mathbf{B}$$

Since \(\mathbf{E}\) and \(\mathbf{B}\) are perpendicular, the magnitude condition simplifies to

$$E = vB\qquad\Longrightarrow\qquad v=\frac{E}{B}$$

Only particles with speed \(v=E/B\) travel straight through the selector; faster particles are deflected upward by the magnetic force, slower particles downward by the electric force.

Application to a Current‑Carrying Conductor

In a conductor the drift velocity of electrons is typically very small (≈ 10⁻⁴ m s⁻¹), so the magnetic force on the whole wire dominates. However, the same principle can be demonstrated using a beam of charged particles (e.g., electrons or ions) passing through a region where the conductor is placed. By adjusting \(\mathbf{E}\) and \(\mathbf{B}\) such that the selected velocity matches the drift velocity, the magnetic force on the moving carriers can be made to balance the electric force, effectively “nulling” the net force on the wire segment.

Practical Steps for a \cdot elocity Selector

1. Generate a uniform electric field \(\mathbf{E}\) between two parallel plates separated by distance \(d\) and held at potential difference \(V\) (so \(E=V/d\)).
2. Produce a uniform magnetic field \(\mathbf{B}\) using Helmholtz coils oriented perpendicular to \(\mathbf{E}\).
3. Direct the particle beam (or the current in a thin wire) perpendicular to both fields.
4. Adjust \(V\) or the coil current until the beam emerges undeflected; the selected speed is then \(v=E/B\).

Key Relationships

Quantity	Expression	Units
Magnetic force on conductor	$\mathbf{F}=I\mathbf{L}\times\mathbf{B}$	N
Electric force on charge	$\mathbf{F}=q\mathbf{E}$	N
Velocity for zero net force	$v=\dfrac{E}{B}$	m s⁻¹
Drift velocity in a wire	$v_d=\dfrac{I}{nqA}$	m s⁻¹

Summary

• The magnetic force on a current‑carrying conductor is $F=I L B\sin\theta$, derived from the Lorentz force on individual charge carriers.
• By arranging perpendicular electric and magnetic fields, a velocity selector transmits only particles with speed $v=E/B$.
• In the context of a conductor, the selector can be used to balance the magnetic force on moving carriers with an electric force, illustrating the equivalence of the two field effects.
• Understanding these principles underpins many A‑Level experiments, such as the Thomson mass‑to‑charge measurement and the operation of mass spectrometers.