Short on time? This page is the recap. For the diagrams, derivations and worked examples, use the full study guide. This topic is in the calculus-based course.
Recap
- A changing flux induces a current in a conducting loop. The external field then exerts a force F = I L × B on that current.
- Only segments inside the field feel a force. In a uniform field, forces on opposite segments that are both inside cancel.
- By Lenz’s law the force always opposes the relative motion (magnetic braking), whether the loop enters or leaves the field.
- No change in flux means no current and no magnetic force, even in a strong field.
- The forces can cause translational acceleration (net force) or rotational acceleration (torque).
- The force grows with speed, so the acceleration is not constant: use Newton’s second law as a differential equation.
Key relationships
| Quantity | Expression | Notes |
|---|---|---|
| Motional emf | ℰ = BLv | edge of length L, v ⊥ B |
| Induced current | I = BLv/R | R is the total resistance of the circuit |
| Magnetic force on the edge | F = ILB = B²L²v/R | opposes v |
| Power dissipated | I²R = B²L²v²/R = Fv | equals the power an agent supplies at constant v |
| Coasting (no other force) | v = v₀e^(−t/τ), τ = mR/(B²L²) | stopping distance v₀τ |
| Steady force F₀ (pull or weight) | v_T = F₀R/(B²L²); from rest v = v_T(1 − e^(−t/τ)) | falling loop: F₀ = mg |
Assumptions behind the results
- The field is uniform inside its region and zero outside; edges are perpendicular to B and v.
- Rails and connecting wires have negligible resistance unless stated; there is no friction.
- The loop’s own magnetic field is small compared with the external field (self-inductance comes in Topic 13.4).
Mistakes to avoid
- Force helping the motion. It never does; check with Lenz’s law.
- Counting segments outside the field. They feel no force.
- Constant-acceleration equations. F depends on v, so a changes as the rod slows.
- “More resistance, more braking.” Larger R gives a smaller current and weaker braking.
- Missing energy. Work against the magnetic force becomes thermal energy, I²R.
Quick self-check
- A rod is pulled at constant speed. You double the speed. By what factor does the pulling power change? (× 4, since P = B²L²v²/R)
- A square loop is wholly inside a uniform field, moving sideways. What is the magnetic force on it? (Zero: the flux is constant, so there is no current)
- A falling loop reaches terminal speed. You double B. What happens to v_T? (It falls to 1/4, since v_T = mgR/(B²L²))
Next: practice questions.