Forces Due to Induced Currents

When a conductor moves through a magnetic field, free electrons inside it experience a force described by the Lorentz law, F = qv × B. This drives a current around any closed circuit the conductor forms — a phenomenon called motional induction. But induction is never a one-way street: that induced current, now flowing through a wire that sits inside the same magnetic field, itself experiences a magnetic force. This feedback between motion, induced current, and resulting force is the heart of every electric motor and generator ever built. The key insight is that the magnetic force on the induced current always opposes the motion that caused it. Pull a conducting rod to the right and the induced current flows in a direction such that the field pushes the rod back to the left. This is Lenz's Law in mechanical form, and it is a direct consequence of energy conservation — if the force aided the motion instead of opposing it, you could accelerate a conductor indefinitely and extract unlimited energy, violating thermodynamics. Nature forbids that. Understanding the forces on induced currents lets you calculate the mechanical power needed to drive a generator, predict the braking force in magnetic braking systems (used in roller coasters and MRI table stops), and design the torque characteristics of electric motors. In this article we build the full picture: from the microscopic Lorentz force on a single charge, up to the macroscopic braking force on a moving rod, with real numbers.

Key takeaways

Setting the Scene: The Sliding Rod
Motional EMF Drives a Current
The Induced Current Feels a Force
Energy Accounting
Generalising: Force on Any Current in a Field

Setting the Scene: The Sliding Rod

Picture two parallel conducting rails separated by a distance L, lying in the plane of this page, connected at one end by a resistor R. A conducting rod rests across the rails, free to slide. A uniform magnetic field B points perpendicularly out of the page everywhere in the rail region. This is the canonical 'sliding rod' setup, and it contains all the physics we need.

Motional EMF Drives a Current

Push the rod to the right with velocity v. Each free electron in the rod moves right with it, so it feels a Lorentz force F = qv × B directed along the rod (say, downward). Electrons pile up at the bottom, creating an EMF: ε = BLv. This EMF drives a current I = ε/R = BLv/R around the circuit. By the right-hand rule (or Lenz's Law), current flows downward through the rod, then around the external circuit through the resistor.

The Induced Current Feels a Force

Now apply F = IL × B to the current-carrying rod. The rod has length L, carries current I (downward), and sits in field B (out of page). Using F = BIL and the right-hand rule: force on the rod points to the LEFT — opposing the rightward push. Magnitude: F_brake = BIL = B(BLv/R)L = B²L²v/R. This is the magnetic braking force. It grows linearly with velocity, like a velocity-dependent friction.

Energy Accounting

The external agent pushing the rod must exert force F_push = B²L²v/R to maintain constant velocity v. Power delivered by the agent: P_mech = F_push · v = B²L²v²/R. Power dissipated in the resistor: P_elec = I²R = (BLv/R)²R = B²L²v²/R. They are equal — mechanical power in equals electrical power dissipated. No energy is created or destroyed; the magnetic field merely acts as the intermediary that converts mechanical work to electrical heat.

Generalising: Force on Any Current in a Field

For any straight conductor of length L carrying current I at an angle θ to a field B, the force magnitude is F = BIL sin θ, and its direction is given by F = IL × B (vector cross product). When θ = 90° the force is maximum; when the current is parallel to B, force is zero. This formula governs the torque in electric motors: a rectangular coil of N turns, area A, carrying current I in field B experiences maximum torque τ = NIAB, which drives rotation.

The Key Equation

F_{\text{brake}} = \frac{B^2 L^2 v}{R}

F_{\text{brake}} — Braking Force (N): Magnetic force on the rod opposing its motion, derived from F = BIL applied to the induced current.
B — Magnetic Flux Density (T (tesla)): Strength of the uniform external magnetic field perpendicular to the plane of motion.
L — Length of Rod (rail separation) (m): The length of the conducting rod bridging the two rails; determines both EMF and force.
v — Velocity of Rod (m s⁻¹): Speed at which the rod slides along the rails; higher speed means larger induced EMF and stronger braking.
R — Circuit Resistance (Ω (ohm)): Total resistance of the closed circuit; a smaller R allows more current and therefore a larger braking force.

Worked Example

Problem. A conducting rod of length 0.40 m slides along frictionless rails in a uniform magnetic field of 0.50 T directed perpendicular to the plane of the rails. The circuit has a total resistance of 2.0 Ω. An external agent pushes the rod at a constant velocity of 3.0 m s⁻¹. (a) Find the induced EMF and current. (b) Find the braking force on the rod. (c) Find the mechanical power the agent must supply and verify it equals the electrical power dissipated.

\varepsilon = 0.60\text{ V},\quad I = 0.30\text{ A},\quad F_{\text{brake}} = 0.060\text{ N},\quad P = 0.18\text{ W}

Answer. EMF = 0.60 V, current = 0.30 A, braking force = 0.060 N, mechanical power = electrical power = 0.18 W.

Figure

Diagram

Origins · 1 minHeinrich Friedrich Emil Lenz and how this was discovered

Heinrich Friedrich Emil Lenz

1804–1865 · Baltic German (Russian Empire)

Lenz formulated his famous law in 1834: the direction of an induced current is always such that its magnetic effect opposes the change causing it. Applied to forces, this means the magnetic force on an induced current always resists the motion of the conductor. Lenz's Law is essentially a statement of energy conservation in electromagnetic induction and is the foundation for understanding every braking force in electromagnetic systems. Lenz also made important contributions to the study of how electrical resistance varies with temperature — work that predated and laid groundwork for the modern understanding of conductivity. He carried out painstaking experiments in St. Petersburg through the Russian winter, using the cold environment to his advantage in studying temperature-dependent resistance.

Motional EMF — Motional EMF (ε = BLv) is the source of the induced current that then experiences the braking force — you cannot understand forces on induced currents without first understanding how those currents arise.
Lenz's Law — Lenz's Law predicts the direction of the braking force: the induced current always flows such that the resulting magnetic force opposes the motion, which is exactly what we calculated with F = BIL.
Force on a Current-Carrying Conductor — The formula F = BIL sin θ is applied directly to the induced current in the rod to find the braking force — this concept is the mechanical tool that completes the induction picture.
Faraday's Law of Induction — Faraday's Law (ε = −dΦ/dt) generalises motional EMF to any changing flux, making it the master equation governing all electromagnetic induction including the sources of forces studied here.