Source: Physics 212 Spring 2022, University of Illinois at Urbana-Champaign
Tags: motional EMF, sliding bar, moving conductor, magnetic force on current, Lenz's law direction, rail problems, moving loop, PHYS 212, electromagnetism
Difficulty: Intermediate Prerequisites: Part 1 (Faraday's Law Fundamentals), cross product (v⃗ × B⃗), magnetic force on a charge (F = qv⃗ × B⃗), Ohm's law.
When a conductor moves through a magnetic field, the free charges inside it experience a magnetic force that pushes them along the conductor. This creates a voltage (motional EMF) and, if the conductor is part of a closed circuit, a current flows. This is the microscopic explanation for Faraday's law in cases where the area of a loop changes. It also leads to a braking force on the moving conductor, which is the basis for magnetic braking in trains and roller coasters. You need to be comfortable with cross products and force on a moving charge before working through this material.
A conductor moving through a magnetic field develops a voltage across it (motional EMF = vBL for a straight bar). If the bar is part of a circuit, current flows, and that current in the magnetic field creates a force that opposes the bar's motion. This is Lenz's law seen from the force perspective.
Motional EMF
The EMF induced in a conductor moving through a magnetic field. For a straight conductor of length L moving with velocity v perpendicular to a uniform field B: ε = vBL.
Think of it as the voltage a moving wire "generates" because the magnetic force separates positive and negative charges to opposite ends of the wire.
Magnetic force on a moving charge
F⃗ = qv⃗ × B⃗. A charge moving through a magnetic field experiences a force perpendicular to both its velocity and the field. This is what drives charge separation in a moving conductor.
Magnetic force on a current-carrying wire
F⃗ = IL⃗ × B⃗, where I is the current, L⃗ is the length vector in the direction of current flow, and B⃗ is the field. Once a motional EMF drives a current, that current itself experiences a force in the magnetic field.
Magnetic braking
The phenomenon whereby a conductor moving through a magnetic field experiences a retarding force. The induced current creates a force (via F = IL × B) that always opposes the motion. This is Lenz's law expressed as a mechanical force.
Consider a metal bar of length L moving with velocity v⃗ through a uniform magnetic field B⃗.
The free electrons inside the bar are carried along at velocity v⃗.
Each electron experiences a force F⃗ = (−e)v⃗ × B⃗.
This force pushes electrons towards one end of the bar and leaves a positive charge deficit at the other end.
The charge separation creates an electric field inside the bar that grows until the electric force balances the magnetic force.
The resulting potential difference across the bar is the motional EMF: ε = vBL (when v⃗, B⃗, and L⃗ are mutually perpendicular).
This is the classic motional EMF setup:
A conducting bar slides along two parallel rails connected by a resistor (or just connected at one end).
A uniform magnetic field fills the region.
As the bar moves, the area of the circuit changes, so the flux changes, and an EMF is induced.
Finding the current direction (two equivalent methods):
Method 1 (Faraday/Lenz): work out whether the flux is increasing or decreasing, then apply Lenz's law.
Method 2 (force on charges): use F⃗ = qv⃗ × B⃗ on a positive charge in the bar to find which way positive charges are pushed. That direction is the direction of conventional current in the bar.
Both methods give the same answer. Use whichever feels more natural for the problem at hand.
Setup: Bar slides in the −x⃗ direction, B⃗ = B₀ẑ (out of page), U-shaped wire completes the circuit.
Method 1 (Lenz): The enclosed area shrinks as the bar moves left. Flux Φ = BA is decreasing. Lenz's law says the induced current opposes this, so it creates a B field out of the page inside the loop, requiring counterclockwise current. In the bar: current flows from bottom to top (i.e. in the +y⃗ direction).
Method 2 (force on charges): A positive charge in the bar moves with v⃗ = −v x⃗. The force is F⃗ = qv⃗ × B⃗ = q(−v x⃗) × (B₀ ẑ) = −qvB₀ (x⃗ × ẑ). Since x⃗ × ẑ = −ŷ, we get F⃗ = +qvB₀ ŷ. Positive charges are pushed in the +y⃗ direction (bottom to top). Same answer.
Setup: Bar slides left (−x⃗ direction), B⃗ points into the page (−z⃗ direction), U-shaped wire with a resistor.
Q16: Force on a positive charge in the bar. F⃗ = qv⃗ × B⃗ = q(−v x⃗) × (−B ẑ) = qvB (x⃗ × ẑ) = qvB(−ŷ) = −qvB ŷ. The force is in the −y⃗ direction. Answer: (b) −y.
Q17: Current direction through the resistor. Positive charges in the bar are pushed downward (−y⃗), so conventional current in the bar flows downward. Following the circuit around, current flows downward through the bar, along the bottom rail, up through the resistor, and along the top rail back to the bar. Through the resistor: current flows upward. Answer: (a) upwards.
Q18: Force on the current in the bar. The current in the bar flows in the −y⃗ direction. F⃗ = IL⃗ × B⃗ = I(−L ŷ) × (−B ẑ) = ILB(ŷ × ẑ) = ILB x⃗. The force is in the +x⃗ direction, opposing the bar's leftward (−x⃗) motion. This is Lenz's law as a force. Answer: (a) +x.
A square loop with a resistor moves at constant upward velocity (+z⃗ direction) in a uniform magnetic field B⃗ out of the page (+z⃗ direction). Wait, let us read the problem carefully: the loop moves upward (+y⃗) and B⃗ is out of the page (+z⃗).
Key insight: The loop is entirely within a uniform field, and the loop is not changing shape. The flux Φ = BA cos θ is constant because B, A, and θ are all constant.
Therefore: dΦ/dt = 0, and there is no induced EMF or current.
Q19: Force on a negative charge in the leading edge. The leading edge (top side) moves in the +y⃗ direction. For a negative charge: F⃗ = (−e)(v ŷ) × (B ẑ) = −evB(ŷ × ẑ) = −evB x⃗. The force is in the −x⃗ direction. Answer: (b) −x.
But notice: the force on charges in the top and bottom edges pushes them sideways (in ±x⃗), and the forces on charges in the left and right edges cancel out by symmetry. No net current flows around the loop. Charges redistribute slightly, but no steady current circulates.
Q20: Current through the resistor. No induced current. Answer: (c).
This is a crucial exam trap. A moving loop in a uniform field has no induced current, even though individual charges feel a force. The forces on opposite sides of the loop cancel.
When a rectangular loop moves through a region of magnetic field with sharp boundaries, the behaviour splits into phases:
Entering the field: Only part of the loop is in the field. The enclosed flux is increasing. An EMF is induced.
Fully inside the field: The entire loop is in a uniform field. The flux is constant (even though the loop moves). No EMF.
Exiting the field: The enclosed flux is decreasing. An EMF is induced (opposite direction to the entering phase).
The induced EMF during entry or exit is ε = Bvw, where v is the speed, w is the width of the loop edge crossing the boundary, and B is the field strength.
The force required to maintain constant velocity during entry/exit is F = BIw = B²w²v/R (opposing the motion), where R is the loop resistance.
Quantity | Formula | Notes |
|---|---|---|
Motional EMF (straight bar) | ε = vBL | v, B, L mutually perpendicular |
Force on a moving charge | F⃗ = qv⃗ × B⃗ | Use for finding current direction in the bar |
Force on a current-carrying wire | F⃗ = IL⃗ × B⃗ | Use for finding braking force |
Induced current | I = ε / R = vBL / R | Ohm's law applied to the loop |
Force to maintain constant velocity | F = B²L²v / R | Derived from F = BIL with I = vBL/R |
EMF during coil entry/exit | ε = Bvw | w = width of edge crossing the field boundary |
Magnetic braking is used in roller coasters, high-speed trains (eddy current brakes), and laboratory balances. No physical contact is needed, so there is no wear. The braking force automatically increases with speed, providing smooth deceleration. Electromagnetic induction in moving conductors is also the principle behind eddy current testing for cracks in metal parts, used extensively in aerospace maintenance.
Students often think a loop moving through a uniform field always has an induced current. It does not. If the loop is entirely within the uniform field, the flux is constant and there is no EMF. The current only flows while the loop is partially in, partially out.
A common error with cross products is getting the sign wrong. Write out v⃗ × B⃗ component by component rather than guessing. The cyclic rule helps: x⃗ × ŷ = ẑ, ŷ × ẑ = x⃗, ẑ × x⃗ = ŷ.
Students sometimes forget that the magnetic force on the bar (F = BIL) always opposes the motion. If your answer says the magnetic force accelerates the bar, something has gone wrong, as that would violate conservation of energy.
When a non-rectangular loop (like the two-square loop in Q22–26) enters a field, students sometimes use the wrong width. Only the edge currently crossing the field boundary matters.
⚠️ The "loop fully inside a uniform field" trap is extremely common on exams. Always check whether the flux is changing before computing an EMF.
⚠️ Cross-product direction problems (force on charges in a moving bar) appear frequently. Practise until you can do them quickly and reliably.
⚠️ The three-phase behaviour of a coil entering, traversing, and exiting a field region is a favourite problem type. Know when the EMF is nonzero and which direction the current flows in each phase.
⚠️ The force required to maintain constant velocity equals the magnetic braking force. This often appears as a quantitative problem.
True or false: A conducting bar moving through a uniform magnetic field always produces an EMF across its ends.
Fill in the blank: The motional EMF for a bar of length L moving at speed v through a field B is ε = ______.
True or false: A square loop moving at constant velocity entirely within a uniform magnetic field has an induced current.
Fill in the blank: The magnetic force on a current-carrying bar in a field always ______ the bar's motion.
True or false: When a loop exits a field region, the induced current flows in the same direction as when it was entering.
Answers: 1. True (there is always a potential difference, though current only flows if there is a closed circuit). 2. vBL. 3. False. 4. Opposes. 5. False (opposite direction).
Q: A metal bar of length 0.5 m moves at 3 m/s perpendicular to a 0.8 T magnetic field. What is the motional EMF?
A: ε = vBL = (3)(0.8)(0.5) = 1.2 V.
Q: In the sliding-bar setup, if the bar moves to the right in a field pointing out of the page, which direction does current flow in the bar?
A: Using F⃗ = qv⃗ × B⃗ with v⃗ = +x⃗ and B⃗ = +ẑ: F⃗ ∝ x⃗ × ẑ = −ŷ. Positive charges are pushed in the −y⃗ direction (downward). Current in the bar flows downward.
Q: A rectangular loop enters a region of magnetic field pointing into the page, moving to the right. The leading edge (right side) is inside the field, the trailing edge (left side) is still outside. Is the induced current clockwise or counterclockwise?
A: Flux into the page is increasing. By Lenz's law, the induced current opposes this increase by creating a field out of the page inside the loop. Right-hand rule: counterclockwise.
Q: Once the same loop is entirely inside the field region and still moving to the right, what is the induced current?
A: Zero. The flux is constant because B and A are both constant.
Motional EMF connects to the Hall effect: in both cases, a magnetic force on moving charges creates a transverse voltage. The braking force on a moving conductor is the basis for understanding eddy currents, which appear in any bulk conductor exposed to a changing flux. Eddy currents are revisited when studying transformers and AC circuits, where they represent an energy loss mechanism.
motional EMF, sliding bar on rails, magnetic braking, eddy current brake, Lenz's law force, cross product direction, F = qvB, F = BIL, coil entering magnetic field, loop in uniform field, no induced current, PHYS 212, electromagnetic induction, university physics