Faraday's law — EMF = −dΦ - dt: Question bank

Recall the two tools we lean on the whole way through:

Flux $Φ_{B} = B A cos θ$ — see Magnetic Flux. Here $θ$ is the angle between $B$ and the loop's normal $\overset{n}{^}$ (the arrow sticking straight out of the loop's face), not the loop's flat surface.
The law $E = - \frac{d Φ _{B}}{d t}$ — EMF is the rate of change, and the minus sign is Lenz's Law doing bookkeeping for energy conservation.

True or false — justify

A constant magnetic field through a stationary loop induces an EMF.

False. EMF depends on

d Φ/ d t

; if

B

A

, and

θ

are all fixed then

Φ

is fixed and its rate of change is zero — no matter how enormous

B

is.

If the flux through a loop is momentarily zero, the induced EMF at that instant must also be zero.

False. EMF tracks the slope of

Φ (t)

, not its value. A sinusoidal flux

Φ = Φ_{0} sin ω t

passes through zero exactly when it is changing fastest, so the EMF is at its peak there.

Doubling the number of turns

N

doubles the induced EMF for the same flux change.

True. Each turn links the same flux, and their EMFs add in series:

E = - N d Φ/ d t

. This is exactly why real coils use many turns — see Electric Generators and AC.

A loop lying flat with

B

pointing along its plane (grazing the surface) has maximum flux.

False. "Along the plane" means

B ⊥ \overset{n}{^}

, so

θ = 9 0^{\circ}

and

cos 9 0^{\circ} = 0

— the flux is zero. Maximum flux is when

B ∥ \overset{n}{^}

(

θ = 0

The induced EMF can be nonzero even when neither

B

nor the loop is changing.

False.

Φ = B A cos θ

depends only on those three quantities. If none of

B

A

θ

changes with time,

d Φ/ d t = 0

. There is no fourth hidden knob.

Reversing the direction the loop is wound flips the sign of the induced EMF.

True (but it's a convention flip). Reversing the winding reverses your chosen

\overset{n}{^}

, which flips the sign of

Φ

and hence of

E

. The physical current direction relative to the loop is unchanged — you just relabelled "positive."

If two loops enclose the same changing flux, they get the same EMF regardless of their shapes.

True. Faraday's law cares only about the flux threading the loop, not the loop's geometry. Square, circle, or wiggly — same

d Φ/ d t

gives the same

E

Motional EMF and "changing-

B

" EMF are two different laws that happen to look similar.

False. They are two faces of the same law

E = - d Φ/ d t

. Motional EMF traces to the force

q v \times B

on charges (see Motional EMF); the flux picture unifies both.

Spot the error

"The rod slides faster, so the flux is bigger, so the EMF is bigger."

The error is "flux is bigger." Speed doesn't change the flux value — it changes the rate the area sweeps,

d A / d t = Lv

. EMF grows because

d Φ/ d t

grows, not because

Φ

is large.

θ

is the angle between the field and the surface of the loop."

θ

is measured from the normal

\overset{n}{^}

, not the surface. Field lying in the surface gives

θ = 9 0^{\circ}

and

Φ = 0

; this off-by-

9 0^{\circ}

error flips maxima and minima of every generator formula.

"Since we only want the magnitude, we can safely delete the minus sign forever."

Fine for a magnitude-only numeric answer, but the sign is Lenz's Law. Delete it and you lose the induced-current direction, hence the braking force and whether a generator resists or assists motion.

"The generator EMF

E = N B A ω sin ω t

peaks when the coil faces the field head-on."

Backwards. Head-on means

θ = 0

, where

Φ

is maximum but momentarily not changing — EMF is zero there. Peak EMF is when the coil is edge-on (

θ = 9 0^{\circ}

), where flux slews fastest.

"Induced current always flows so as to increase the flux through the loop."

It always opposes the change. If flux is rising, the current fights to reduce it; if flux is falling, the current tries to keep it up. Which way that is depends on the sign of

d Φ/ d t

, not a fixed direction.

"A superconducting loop with zero resistance gives infinite induced current, hence free energy."

The current isn't infinite — the loop's Inductance limits how fast flux can change, and Lenz's opposition means you do work to change the flux. Energy conservation is never violated.

Why questions

Why does the flux definition use a dot product

B \cdot d A

instead of just

B A

Only the field component perpendicular to the surface threads through it; the dot product's

cos θ

extracts exactly that perpendicular part and discards the sideways-sliding field.

Why must the induced current oppose the change (Lenz), rather than assist it?

If it assisted, the induced current would strengthen the very change that made it, feeding itself → runaway free energy. Opposition is the only option consistent with energy conservation.

Why does a generator produce alternating (sinusoidal) EMF rather than steady EMF?

Rotating the coil makes

θ = ω t

, so

Φ = B A cos ω t

. Differentiating brings out

sin ω t

— an oscillation whose sign flips every half turn as the coil face reverses relative to

B

Why can a purely time-varying magnetic field induce an EMF with no moving parts at all?

Faraday's law needs only

d Φ/ d t \neq = 0

. A changing

B

through a fixed loop changes the flux directly, no motion required — this is the seed of Maxwell's

\nabla \times E = - \partial B / \partial t

Why does more turns

N

multiply the EMF but not the flux

Φ_{B}

itself?

Φ_{B}

is the flux through one loop's area; adding turns doesn't change that. But the turns sit in series, so their individual EMFs stack: total

E = N \times (- d Φ/ d t)

Why is the minus sign the same physics as Lenz's law?

The minus sign says the EMF drives current in the direction whose own magnetic flux opposes the imposed change — which is precisely Lenz's statement, just written algebraically.

Edge cases

At the exact instant a rotating coil lies flat facing the field (

θ = 0

), what is the EMF?

Zero. Flux is at its maximum

Φ = B A

, so its slope

d Φ/ d t = 0

— the cosine is momentarily flat at its top. The coil is "coasting" through peak flux.

What happens to the EMF if a loop is folded into a figure-eight before the field changes?

The two lobes enclose opposite normals, so their fluxes subtract. If the lobes are equal, net

Φ \approx 0

and

d Φ/ d t \approx 0

— nearly no EMF, even in a strong changing field.

A loop moves at constant velocity through a uniform field (fully inside, no edges leaving or entering). What EMF appears?

Zero. The flux threading the loop doesn't change while it is entirely immersed — same

B

, same

A

, same

θ

. EMF appears only while an edge crosses the field boundary.

What is the EMF if

B

increases linearly with time,

B = k t

, through a fixed loop?

A constant EMF:

E = - A d B / d t = - A k

, steady in time. A linear ramp in flux gives a flat (DC) EMF — the only time-varying source that yields constant EMF.

If a loop's area shrinks to zero at some instant, is the EMF necessarily large there?

Not necessarily. What matters is

d Φ/ d t = B cos θ d A / d t

— the rate the area changes, not that

A

hits zero. A slow collapse gives small EMF even at

A = 0

Two coaxial coils: the inner carries a steady current. Is there EMF in the outer coil?

No, while the current is steady. Steady current → steady

B

→ steady flux → zero EMF. Only switching or varying the inner current (mutual Inductance) induces an outer EMF.

Recall One-line survival summary

EMF lives in the slope of the flux curve, never its height; the minus sign is Lenz guarding energy; and $θ$ is always measured from the normal $\overset{n}{^}$ . The three-word charm ::: "Change Makes EMF; Nature Says No."

Connections

Lenz's Law — the source of the minus sign and current direction
Magnetic Flux — the $Φ_{B} = B A cos θ$ these traps orbit
Motional EMF — the $q v \times B$ face of the same law
Maxwell's Equations — the field form for the "no moving parts" case
Electric Generators and AC — why edge-on gives peak EMF
Inductance — why zero resistance doesn't mean infinite current