Force on a Current-Carrying Conductor

4.5.4 Determine the direction of the force on beams of charged particles in a magnetic field

Key Concepts

When a current-carrying conductor is placed in a magnetic field, it experiences a force. The direction of this force can be determined using the right-hand rule.

The Magnetic Force Equation

The magnitude of the magnetic force (F) on a charge (q) moving with a velocity (v) in a magnetic field (B) is given by:

$$ \mathbf{F} = q(\mathbf{v} \times \mathbf{B}) $$

where 'x' represents the cross product.

For a current-carrying conductor, the charge carriers are typically electrons (negative charge, -e) or positive ions (positive charge, +e). The force experienced by a single charge is:

$$ \mathbf{F} = |q|(\mathbf{v} \times \mathbf{B}) $$

The magnitude of the force is also given by:

$$ F = |q|vB \sin{\theta} $$

where θ is the angle between the velocity vector and the magnetic field vector.

Right-Hand Rule

The right-hand rule is used to determine the direction of the magnetic force:

1. Point the fingers of your right hand in the direction of the velocity (v) of the charge carriers.
2. Curl your fingers towards the direction of the magnetic field (B).
3. Your thumb will point in the direction of the magnetic force (F) on a positive charge.
4. For a negative charge, the force will be in the opposite direction to that indicated by your thumb.

Force on a Conductor

For a conductor carrying a current, the charge carriers (electrons) are deflected by the magnetic field. This deflection results in a net force on the conductor.

The force on the conductor is the sum of the forces on all the individual charge carriers.

The direction of the net force on the conductor can be determined using the right-hand rule, where the velocity is the drift velocity of the electrons.

Applications

The principle of the force on a current-carrying conductor is used in various applications, including:

• Electric Motors: The force on the current-carrying conductor is used to produce rotational motion in electric motors.
• Magnetic Levitation: Opposing currents can be used to create repulsive magnetic forces that can levitate objects.
• Mass Spectrometry: Charged particles are deflected by magnetic fields to determine their mass-to-charge ratio.

Example Problem

A current of 2 A flows through a straight wire of length 0.5 m placed in a uniform magnetic field of 0.4 T. The magnetic field is perpendicular to the wire. Determine the force on the wire.

Solution:

1. Calculate the charge of a single electron, $q = -1.6 \times 10^{-19} C$.
2. Determine the drift velocity of the electrons. The drift velocity is related to the current by: $v_{drift} = \frac{I}{n A}$, where I is the current, n is the charge carrier density, and A is the cross-sectional area of the wire. We need to assume a value for n and A to proceed. Let's assume n = $10^{28} m^{-3}$ and A = $10^{-6} m^2$. Then, $v_{drift} = \frac{2 \times 10^{-3} A}{10^{28} m^{-3} \times 10^{-6} m^2} = 2 \times 10^{-2} m/s$.
3. The magnitude of the force is: $F = |q|v_{drift}B \sin{90^\circ} = |(-1.6 \times 10^{-19} C)(2 \times 10^{-2} m/s)(0.4 T)(1)| = 1.28 \times 10^{-20} N$.

Table Summary

Concept	Description
Magnetic Force Equation	$$ \mathbf{F} = q(\mathbf{v} \times \mathbf{B}) $$
Right-Hand Rule	Used to determine the direction of the magnetic force.
Force on a Conductor	Net force on the conductor due to the deflection of charge carriers.
Applications	Electric motors, magnetic levitation, mass spectrometry.