Students will understand that a magnetic field can exert a mechanical force on a conductor carrying an electric current, and be able to predict the magnitude and direction of that force.
Key Concepts
The magnetic field $\mathbf{B}$ is a vector field that exerts forces on moving charges.
A current $I$ in a conductor represents a flow of charge carriers with drift velocity $\mathbf{v}_d$.
The Lorentz force on a single charge $q$ moving with velocity $\mathbf{v}$ in a magnetic field is
$$\mathbf{F}=q\,\mathbf{v}\times\mathbf{B}.$$
Summing the Lorentz forces on all charge carriers in a segment of wire yields the macroscopic force on the conductor.
Derivation of the Force on a Straight Conductor
Consider a straight segment of length $L$ carrying a current $I$ in a uniform magnetic field $\mathbf{B}$, making an angle $\theta$ with the field direction.
Current $I$ is defined as $I = nqA v_d$, where:
$n$ = number density of charge carriers (m$^{-3}$)
$q$ = charge of each carrier (C)
$A$ = cross‑sectional area of the conductor (m$^2$)
$v_d$ = drift speed (m s$^{-1}$)
The total number of carriers in the segment is $N = n A L$.
Each carrier experiences a magnetic force $q\mathbf{v}_d\times\mathbf{B}$.
Summing over all carriers gives the total force:
$$\mathbf{F}=N q \mathbf{v}_d\times\mathbf{B}= (nA L) q \mathbf{v}_d\times\mathbf{B}.$$
Substituting $I = nqA v_d$ and noting that $\mathbf{v}_d$ is parallel to the direction of the current $\mathbf{I}$,
$$\mathbf{F}= I L\,\hat{\mathbf{I}}\times\mathbf{B}.$$
Resulting Formula
The magnitude of the force on a straight conductor of length $L$ is
$$F = I L B \sin\theta,$$
where $\theta$ is the angle between the direction of the current and the magnetic field.
Direction of the Force – Fleming’s Left‑Hand Rule
To determine the direction of $\mathbf{F}$ without using the cross‑product, apply Fleming’s left‑hand rule:
First finger: direction of the magnetic field $\mathbf{B}$ (from north to south).
Second finger: direction of the current $I$ (conventional current, positive to negative).
Thumb: direction of the force $\mathbf{F}$ on the conductor.
Suggested diagram: A straight wire placed in a uniform magnetic field with the three fingers of Fleming’s left‑hand rule labelled.
Applications
Electric motors: Rotating armature coils experience forces that produce torque.
Railguns: Large currents in parallel rails generate a strong magnetic field that accelerates a projectile.
Galvanometers: Deflection of a moving coil in a magnetic field provides a measure of current.
Example Problem
Problem: A horizontal wire 0.30 m long carries a current of 5.0 A to the east. It is placed in a uniform magnetic field of magnitude 0.80 T directed vertically upward. Calculate the magnitude and direction of the force on the wire.
Identify the vectors:
$\mathbf{I}$ points east (positive $x$‑direction).
Since $\mathbf{I}$ and $\mathbf{B}$ are perpendicular, $\sin\theta = 1$.
Apply the formula:
$$F = I L B = (5.0\ \text{A})(0.30\ \text{m})(0.80\ \text{T}) = 1.2\ \text{N}.$$
Use the left‑hand rule: thumb points north (positive $y$‑direction). Hence the force is directed northward.
Summary Table
Quantity
Symbol
Expression
Units
Current
$I$
$I = nqAv_d$
A (ampere)
Magnetic field
$\mathbf{B}$
Vector quantity
T (tesla)
Force on a straight conductor
$\mathbf{F}$
$\mathbf{F}= I L\,\hat{\mathbf{I}}\times\mathbf{B}$
N (newton)
Magnitude of force
$F$
$F = I L B \sin\theta$
N
Key Take‑aways
A magnetic field exerts a force on moving charges; a current is a collective motion of many charges.
The force on a straight conductor is proportional to current, length of the conductor within the field, field strength, and the sine of the angle between current and field.
Direction is given by the vector cross‑product or Fleming’s left‑hand rule.
Understanding this principle underpins the operation of many electromagnetic devices.