Chapter revision

Revision summary

Key ideas, equations and common mistakes. Open any topic below for the full explanation.

Choose the area normal, identify what changes, then connect the linkage rate to e.m.f. Use the actual circuit to decide whether current flows and where energy is transferred.

Flux, linkage and induced e.m.f.

Φ = BA for perpendicular area
Φ = BA cos θ for angle θ to the normal
Λ = NΦ for equally linked turns
Mean E_ind = -ΔΛ/Δt
Instantaneous E_ind = -dΛ/dt

The normal is perpendicular to the plane. A plane parallel to B has zero flux; a plane perpendicular to B has maximum flux magnitude.
Match the positive normal to positive circuit traversal using the right-hand orientation. Reversing a reference changes the assigned sign, not the physical effect.
Lenz's law opposes the change producing the effect. An induced field can reinforce an existing field if that field is decreasing.
A finite graph chord gives an interval mean. A local tangent gives the instantaneous rate. For measured data, shorter intervals can increase sensitivity to reading noise.
Induced e.m.f. can exist with negligible current. A sustained loop current needs a complete conducting path; I = E_ind/R also requires the stated resistance-only model.

In the signed graph example, rising, constant and falling linkage produce -2.00, zero and +2.00 V. The final induced field remains outward as external B passes through zero, because the signed external field is still decreasing. Revisit the matched graph and current references when checking a direction.

Evidence needs controlled comparisons

The three central inferences are that changing linkage induces e.m.f., the induced direction opposes the producing change, and a greater linkage-change rate increases e.m.f. magnitude. Stationary/moving, approach/withdrawal and matched faster/slower comparisons address different parts of that evidence.

Keep the pole, path, coil, viewing side, leads and acquisition settings appropriate to the comparison. More turns can also change resistance, so a current ratio alone does not establish the e.m.f. ratio. A zero-looking record may have missed a pulse; a clipped record cannot give its true peak. The observation and inference table separates recorded responses from ideal predictions.

Applications and their conditions

Induction applications: cause, result and model limit
Application	Reasoning to retain
Rotating generator	Orientation changes linkage. Separate full slip rings preserve each coil connection. Maximum linkage gives zero e.m.f.; the maximum linkage-change rate gives peak output. Equating terminal voltage with induced e.m.f. requires the stated negligible winding/lead resistance and self-inductance. Mechanical input supplies loaded output.
Entering loop	Increasing outward flux gives clockwise current and a leftward force on the right edge. For steady entry, Fv = I²R in the supplied ideal model. A complete rigid loop wholly inside an unchanged uniform field has constant linkage and no sustained model current.
Iron-core transformer	Changing common flux links separate windings. Ideal ratios are N_s/N_p = V_s/V_p = I_p/I_s. Use rms values for the stated sinusoidal resistive-load power account. Steady d.c. gives no sustained secondary e.m.f.; real losses require a separate energy account.

Quantities and units

Quantities used in the flux, generator, braking and transformer models
Quantity and symbol	SI unit	Meaning or condition
Magnetic flux density B	T	The magnetic field at a location; use uniform B for the simple area product.
Flux Φ	Wb	1 Wb = 1 T m² = 1 V s.
Flux linkage Λ = NΦ	Wb-turn	Equal per-turn flux is assumed. A turn adds no dimension.
Area A; turn count N	m²; no unit	A is one turn's actual area; its perpendicular projection sets flux.
Induced e.m.f. E_ind; V_p, V_s	V	E_ind is a voltage, not field strength. Transformer voltages here are rms.
Current I, I_p, I_s	A	Specify reference direction or rms magnitude as appropriate.
Resistance R	Ω	The braking calculation uses total loop resistance.
Time t; period T	s	Period T is distinct from the unit symbol T for tesla.
Frequency f; angular frequency ω	Hz; rad/s	ω = 2πf; use radians in the supplied cosine and sine model.
Power P	W	Mechanical input, electrical transfer and heating must share the same account.

In base units, 1 Wb = 1 kg m² s^-2 A^-1. Dividing linkage change by time gives Wb/s = V. These checks help distinguish a flux, a voltage and a current even when their graphs share the same time axis.

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