Retain every term when a coil and its field change

Flux linkage is not just field strength

A coil has N turns only if the winding geometry actually supplies that many linked surfaces. Here ten identical turns share area 0.2 m² and the same uniform field. The normal rotates with theta(t) = omega t while B(t) = B0(1 + t/(2 s)). Flux linkage is N A B(t) cos(theta). Changing field, area or orientation can change it; a large steady flux alone need not induce an emf.

Apply the product rule with signs

Differentiation gives emf = −N A [Bdot cos(theta) − B omega sin(theta)]. The first term comes from changing field, the second from changing projected area. They may reinforce or cancel. At t = 0 the normal is aligned with the field, so the rotational term vanishes even though the coil is moving; the field-change term need not vanish.

Compare limiting cases before a mixed case

Set Bdot = 0 in a separate uniform-field thought packet. Then emf = N A B omega sin(theta): at theta = 0 the flux is maximal and emf zero; at theta = pi/2 the flux is zero and emf magnitude maximal. If the loop instead changes area at fixed normal and field, emf = −N B dA/dt. For N = 5, B = 0.2 T and area increasing at 0.03 m²/s, the emf is −0.03 V. These are limits of the same linked-flux derivative, not unrelated formulas.

Do not infer energy without defining the driver

The prescribed rotation is not a free solution of Newton’s laws. For rotational generation in a fixed magnetic field with an ideal resistive load, induced current exerts a torque opposing the driven motion and the mechanical driver supplies energy. When B also changes externally, the field-change contribution can instead produce an instantaneous torque that assists rotation. Lenz’s law concerns opposition to the total flux change, not a universal opposition to every motion. Rotational mechanical exchange and the external field-source energy must then be accounted for separately. This widget does not model that torque, circuit inductance or field-source work. A voltage plot alone is not a complete generator efficiency measurement.

Worked original case

B0 = 0.2 T and omega = 1 rad/s give at t = 1 s a field-change term −0.2 cos(1) ≈ −0.108060 V and a rotation term 0.6 sin(1) ≈ +0.504883 V. The total is +0.396822 V. Dropping either term changes the result materially.

Investigation

Predict the two terms separately at zero time and at a later orientation. Retain their algebraic sum, identify a sign-change or cancellation interval and test the fixed-field and fixed-orientation limits. Explain what additional force and source-work records would be needed for an energy claim.

Changed-case practice

Sources and boundaries

Original teaching checked 31 October 2026 against current official Unit 13 and the relevant OpenStax discussion. No source exercises or diagrams are reproduced. All records and apparatus are virtual, supplied or generated; no real wiring, switching, magnetic braking or electrical measurement is requested. Public feedback and retained work do not establish independent mastery, supervised practical participation or AP authorization.