Force on Current-Carrying Conductor

When an electric current flows through a conductor placed inside a magnetic field, the conductor experiences a mechanical force. This phenomenon — known as the motor effect — is one of the most practically important results in electromagnetism, forming the operating principle behind electric motors, loudspeakers, galvanometers, and many other devices we rely on every day. The force arises because a current is simply a flow of charged particles, and moving charges in a magnetic field experience a magnetic (Lorentz) force. When billions of electrons drift through a wire simultaneously, their individual tiny forces add up to a macroscopic push or pull on the entire conductor. The direction of this force depends on both the direction of the current and the orientation of the magnetic field, and it is always perpendicular to both — a consequence that makes it ideal for converting electrical energy into rotational or linear mechanical motion. Understanding the force on a current-carrying conductor requires combining two key ideas: the Lorentz force on individual charge carriers and the macroscopic description of current in terms of charge flow. Together they yield a clean, predictive equation that lets engineers design precise electromagnetic machines and that lets physicists explore the deep connection between electricity and magnetism.

Key takeaways

Start with the Lorentz Force on a Single Charge
Relate Drift Velocity to Current
Sum the Forces Over All Carriers in a Segment
Interpreting the Magnitude
Applying the Right-Hand Rule
Extending to a Curved or Non-Uniform Conductor

Start with the Lorentz Force on a Single Charge

A single charge $q$ moving with velocity $\vec{v}$ through a magnetic field $\vec{B}$ feels a force $\vec{F} = q\vec{v} \times \vec{B}$ . The magnitude is $F = qvB\sin\theta$ , where $\theta$ is the angle between the velocity vector and the field. The direction is given by the right-hand rule: point fingers along $\vec{v}$ , curl them toward $\vec{B}$ , and the thumb points in the direction of $\vec{F}$ for a positive charge.

Relate Drift Velocity to Current

In a conductor of cross-sectional area $A$ with $n$ free electrons per unit volume, each drifting at speed $v_d$ , the conventional current is $I = nqv_dA$ . This means we can express the drift speed as $v_d = I/(nqA)$ . The direction of conventional current is opposite to electron flow, so we define a current direction $\hat{\ell}$ along conventional current.

Sum the Forces Over All Carriers in a Segment

Consider a short segment of wire of length $L$ aligned along unit vector $\hat{\ell}$ . The number of charge carriers in this segment is $N = nAL$ . Each carrier has charge $q$ and drifts with velocity $\vec{v}_d = v_d\hat{\ell}$ . The total force on the segment is $\vec{F} = N(q\vec{v}_d \times \vec{B}) = (nAL)(qv_d\hat{\ell} \times \vec{B})$ . Substituting $nqv_dA = I$ gives the fundamental result: \vec{F} = I\vec{L} \times \vec{B} where $\vec{L} = L\hat{\ell}$ is the length vector pointing in the direction of conventional current.

Interpreting the Magnitude

The scalar magnitude of the force is $F = BIL\sin\theta$ , where $\theta$ is the angle between the wire direction and the magnetic field. Maximum force occurs when the wire is perpendicular to the field ( $\theta = 90°$ , $\sin\theta = 1$ ), giving $F = BIL$ . Zero force occurs when the wire is parallel to the field ( $\theta = 0°$ ), which makes physical sense — charge carriers moving along the field lines feel no magnetic force.

Applying the Right-Hand Rule

To find the direction of the force on the conductor: point the fingers of your right hand in the direction of conventional current $I$ , then curl them toward the magnetic field $\vec{B}$ . Your thumb points in the direction of the force. Alternatively, use Fleming's Left-Hand Rule (popular in engineering): the First finger points in the direction of the Field ( $\vec{B}$ ), the seCond finger in the direction of conventional Current, and the thuMb gives the direction of Motion (force). Both methods give identical results.

Extending to a Curved or Non-Uniform Conductor

For a wire that curves or passes through a non-uniform field, divide it into infinitesimal segments $d\vec{\ell}$ . The force on each element is $d\vec{F} = I\,d\vec{\ell} \times \vec{B}$ , and the total force is found by integration: $\vec{F} = I\int d\vec{\ell} \times \vec{B}$ . For a straight wire in a uniform field this integral simply returns $I\vec{L} \times \vec{B}$ , confirming our earlier result.

The Key Equation

\vec{F} = I\vec{L} \times \vec{B}

\vec{F} — Magnetic Force (N (newton)): The mechanical force exerted on the conductor by the magnetic field; always perpendicular to both the current direction and the field.
I — Electric Current (A (ampere)): The conventional current flowing through the conductor; larger current means larger force.
\vec{L} — Length Vector (m (metre)): A vector of magnitude equal to the length of the conductor segment, pointing in the direction of conventional current flow.
\vec{B} — Magnetic Flux Density (T (tesla)): The external magnetic field in which the conductor is placed; measures the strength and direction of the field.
\theta — Angle (° or rad (dimensionless)): The angle between the direction of conventional current and the direction of the magnetic field.

Worked Example

Problem. A straight copper wire of length 0.25 m carries a current of 4.0 A. It is placed horizontally in a uniform magnetic field of flux density 0.30 T directed vertically upward. The wire is oriented so that the current flows due east. Calculate (a) the magnitude of the magnetic force on the wire, and (b) the direction of this force.

F = BIL\sin\theta = (0.30)(4.0)(0.25)(1) = 0.30\text{ N, directed south}

Answer. The magnetic force has magnitude 0.30 N and acts horizontally toward the south.

Figure

Diagram

Origins · 1 minAndré-Marie Ampère and how this was discovered

André-Marie Ampère

1775–1836 · French

Ampère founded the science of electrodynamics after hearing of Ørsted's discovery in 1820. Within weeks he derived the mathematical law describing the force between two parallel current-carrying conductors — a direct application of the force on a current in a field — and showed that magnetism itself could be explained by circulating electric currents. His work unified electricity and magnetism decades before Maxwell's full theory. Ampère was largely self-taught; he learned Latin specifically so he could read the mathematical works of Euler and Bernoulli. After the tragic execution of his father during the French Revolution, he suffered a deep depression but channelled his grief into an extraordinary burst of scientific creativity.

Lorentz Force on Moving Charges — The force on a current-carrying conductor is derived directly by summing the Lorentz forces on all individual charge carriers inside the wire, so understanding F = qv × B is the microscopic foundation of F = IL × B.
Magnetic Flux Density and Field Lines — The magnitude of the force F = BIL sin θ depends directly on the magnetic flux density B; understanding what B represents and how to determine its direction is essential for applying the force equation.
Operation of the DC Electric Motor — The DC motor is the most important application of the force on a current-carrying conductor: a current-carrying coil in a magnetic field experiences a torque F = NBIL (where N is turns and A is area) that drives continuous rotation.
Force Between Two Parallel Current-Carrying Conductors — Two parallel wires each generate a magnetic field that exerts a force on the other wire's current, combining Ampère's law for B with F = BIL. This interaction historically defined the ampere as the SI unit of current.
Electric Current and Drift Velocity — The derivation F = BIL relies on the relation I = nqv_d A connecting current to the microscopic drift of charge carriers, making drift velocity a key bridge between the electrical and mechanical descriptions.