In the previous sections, we learned that a magnetic field exerts a force on an individual moving charge. In most practical situations, however, charges move through a metal wire or another conductor rather than freely through space. Because these charges are confined within the conductor, the magnetic force acting on each charge is transmitted to the conductor itself. As a result, an entire wire carrying current can experience a measurable force.

This principle is the foundation of many technologies, including electric motors, loudspeakers, and magnetic actuators used in medical devices such as infusion pumps and surgical instruments. In each case, electrical energy is converted into mechanical motion through the interaction between an electric current and a magnetic field.

To derive the magnetic force on a current-carrying wire, we begin with the magnetic force on a single moving charge:

where [latex]q[/latex] is the charge, [latex]v_d[/latex] is the drift velocity of the charge carriers, [latex]B[/latex] is the magnetic field strength, and [latex]\theta[/latex] is the angle between the direction of motion and the magnetic field.

If the magnetic field is uniform over a length [latex]l[/latex] of wire, every charge carrier in that section experiences a force in the same direction. Therefore, the total magnetic force is the sum of the forces on all the moving charges:

where [latex]N[/latex] is the number of charge carriers within the portion of wire inside the magnetic field.

The number of charge carriers can be written as

where [latex]n[/latex] is the number of charge carriers per unit volume and [latex]V[/latex] is the volume of the wire exposed to the field. Since the volume of this section is

where [latex]A[/latex] is the wire's cross-sectional area, the force becomes

Rearranging the terms gives

From our earlier study of electric current, we know that

where [latex]I[/latex] is the electric current. Substituting this relationship into the previous expression leads to one of the fundamental equations of electromagnetism:

This equation gives the magnitude of the magnetic force acting on a straight wire of length [latex]l[/latex] carrying a current [latex]I[/latex] in a uniform magnetic field [latex]B[/latex]. The force is greatest when the current is perpendicular to the magnetic field ([latex]\theta=90^\circ[/latex]) and becomes zero when the current flows parallel to the field.

Sometimes it is useful to know the force acting on each meter of wire. Dividing the previous equation by the wire length gives

The direction of the magnetic force is determined using Right-Hand Rule 1 (RHR-1). Point your right thumb in the direction of the conventional current, extend your fingers in the direction of the magnetic field, and your palm points in the direction of the magnetic force acting on the wire.

The magnetic force acting on current-carrying conductors provides a direct way to convert electrical energy into mechanical motion. Electric motors are the most familiar example, using loops of wire that rotate when placed in magnetic fields. Another interesting application is magnetohydrodynamics (MHD), which uses magnetic forces to move electrically conducting fluids without any moving mechanical parts.

In an MHD pump, a strong magnetic field is applied across a tube while an electric current passes through the conducting fluid. According to the magnetic force equation, the fluid experiences a force parallel to the tube and begins to flow. Since there are no moving blades or valves, MHD pumps are particularly useful for transporting hot or chemically reactive liquids, such as the liquid sodium used as a coolant in some advanced nuclear reactors.

Researchers have also investigated MHD technology for biomedical applications. For example, experimental artificial heart pumps have explored using magnetic forces to move blood without mechanical impellers, potentially reducing wear and improving long-term reliability. However, the strong magnetic fields required can affect living cells and tissues, making this application technically challenging.

MHD propulsion has also been proposed for submarines. Instead of using a rotating propeller, an MHD drive accelerates seawater directly using magnetic forces, producing thrust with much less mechanical noise. A quieter propulsion system could make submarines more difficult to detect. Although this concept has attracted considerable research interest, current MHD propulsion systems remain relatively heavy and inefficient compared with conventional propellers.