Magnetic Force on a Current-carrying Conductor

Magnetic Force on a Current-Carrying Conductor

A current in a wire is simply a stream of moving charges. Since a magnetic field exerts a force on each moving charge, it also exerts a net force on the entire current-carrying wire.

Derivation for a Straight Rod

Consider a straight rod of uniform cross-sectional area $A$ and length $l$, carrying a steady current $I$. Let the number density of mobile charge carriers (e.g., electrons) be $n$. The total number of mobile carriers in the rod is $n l A$.

Each carrier has charge $q$ and an average drift velocity $\mathbf{v}_d$. In an external magnetic field $\mathbf{B}$, the force on a single carrier is $q \mathbf{v}_d \times \mathbf{B}$. Therefore, the total force on all carriers in the rod is:

Now, the current density $\mathbf{j}$ is given by $\mathbf{j} = n q \mathbf{v}_d$, and the current $I$ is $I = |\mathbf{j}| A$. Substituting these relations:

Since $\mathbf{j} A$ has the same direction as the current, we define a vector $\mathbf{l}$ of magnitude $l$ pointing in the direction of the current. This gives the final result:

• $I$: Current in the conductor (scalar).
• $\mathbf{l}$: Vector of length $l$ in the direction of the current.
• $\mathbf{B}$: External magnetic field (not the field produced by the wire itself).

The magnitude of the force is $F = I l B \sin \theta$, where $\theta$ is the angle between $\mathbf{l}$ and $\mathbf{B}$.

For an Arbitrary Shape …