Line Current ($I$)
Charge flowing down a thin wire.
$$ \mathbf{I} = \lambda \mathbf{v} $$
$\lambda$: Linear charge density (C/m)
The Lorentz Force Law
1. The Phenomenon: Forces on Moving Charges
In electrostatics, we learned that a stationary charge $q$ produces an electric field $\mathbf{E}$, and another charge $Q$ feels a force $\mathbf{F} = Q\mathbf{E}$. This is simple and intuitive.
But what happens when charges move? Consider two parallel wires. The wires are electrically neutral (they have equal amounts of positive and negative charge), so there should be no electrostatic force between them. Yet, when we run a current through them, they physically move.
If the currents flow in the same direction, the wires attract. If they flow in opposite directions, they repel. Since the net charge is zero, this cannot be an electric force. It is a Magnetic Force.
2. The "Chicken and Egg" Problem
We know a force exists, but how do we describe the field that causes it? In gravity and electrostatics, the force vector points in the direction of the field lines. Magnetism is not so kind.
We cannot simply define the magnetic field $\mathbf{B}$ by placing a stationary charge and measuring the force, because stationary charges feel no magnetic force. The charge must be moving.
Operational Definition: Probing the Field
Imagine we are in a room with a hidden magnetic field. We have a "test charge" $q$ and a gun to fire it at velocity $\mathbf{v}$. How do we find $\mathbf{B}$?
- If we fire the charge along the direction of the field, we observe zero force. This tells us the direction of $\mathbf{B}$.
- If we fire the charge perpendicular to that line, we observe maximum force. This allows us to calculate the magnitude.
- Crucially, the force is always perpendicular to both the velocity and the field.
Use the simulation below to act as the experimenter. Try to deduce the hidden field direction by firing particles.
3. The Lorentz Force Law
The peculiar relationship between velocity, field, and force—where the force is orthogonal to the motion—is mathematically described by the cross product.
If a charge $Q$ moves with velocity $\mathbf{v}$ in the presence of both an electric field $\mathbf{E}$ and a magnetic field $\mathbf{B}$, the total force is:
$$ \mathbf{F} = q[\mathbf{E} + (\mathbf{v} \times \mathbf{B})] $$
The magnetic component, $\mathbf{F}_{mag} = q(\mathbf{v} \times \mathbf{B})$, has profound implications:
- Orthogonality: The force is always perpendicular to the velocity vector.
- No Work: Since work is the dot product of force and displacement ($W = \int \mathbf{F} \cdot d\mathbf{l}$), and $\mathbf{F}$ is perpendicular to motion ($d\mathbf{l}$), magnetic forces do no work. They can change the direction of a particle, but never its speed.
4. Cyclotron Motion
Because the magnetic force steers the particle without speeding it up, a particle moving perpendicular to a uniform magnetic field will travel in a perfect circle.
By equating the magnetic force ($qvB$) to the centripetal force required for circular motion ($mv^2/R$), we find the Cyclotron Radius ($R$):
$$ qvB = \frac{mv^2}{R} \implies R = \frac{mv}{qB} $$
If the particle has some velocity parallel to the field lines, that component remains unchanged (since $\mathbf{v}_{\parallel} \times \mathbf{B} = 0$). The combination of constant forward motion and circular rotation creates a helix.
5. From Charges to Currents
In the laboratory, we rarely deal with single electrons fired into a vacuum. We deal with trillions of electrons flowing through wires. We call this Current.
Since magnetic force acts on moving charge, it acts on current-carrying wires. We model this flow in three dimensions:
Surface Current ($\mathbf{K}$)
Charge flowing across a sheet/ribbon.
$$ \mathbf{K} = \sigma \mathbf{v} $$
$\sigma$: Surface charge density (C/m²)
Volume Current ($\mathbf{J}$)
Charge flowing through a 3D volume (e.g., inside a thick wire).
$$ \mathbf{J} = \rho \mathbf{v} $$
The total current is the flux through the cross-section $A$: $I = JA$.