A rate of change, and what resists it

$$\Phi = BA\cos\theta, \qquad \theta \ \text{between} \ \vec{B} \ \text{and the area vector}.$$

The area vector points perpendicular to the plane of the loop. So the procedure is to draw that arrow first, then measure $\theta$ from the arrow to $\vec{B}$.

The case that exposes the error: a loop lying edge-on to the field, with its plane parallel to $\vec{B}$. Measuring from the plane gives $\theta = 0$ and the maximum $\Phi = BA$. Measuring correctly from the normal gives $\theta = 90^\circ$, $\cos 90^\circ = 0$, and no flux threads the loop at all, which is right: nothing passes through a hoop held edge-on to the flow.

The other detail that goes missing is the turn count. An $N$-turn coil links the flux $N$ times, so $N$ multiplies before any emf is quoted.

$$\varepsilon = -N\frac{\Delta\Phi}{\Delta t}.$$

Two consequences worth stating separately, because both get missed:

1. A magnet held motionless inside a coil induces nothing. $\Delta\Phi = 0$ over any interval, however strong the magnet.
2. On a graph of flux against time, read the slope. At the flux maximum the graph is flat, so the emf there is zero, and the emf peaks on the steepest segments.

Reporting the largest emf at the largest flux puts the answer exactly where the correct answer is zero, which makes it a good self-check: if your peak emf lands at the peak of the flux curve, you read the height.

Since $\Phi = BA\cos\theta$, an emf appears when any of the three moves. Checking only $B$ misses two standard setups.

1. $B$ changes. A magnet moved toward a coil, or a nearby current switched on.
2. $A$ changes. A bar sliding along conducting rails grows the enclosed area at rate $Lv$, so $\varepsilon = BLv$, in a perfectly constant field.
3. $\theta$ changes. A coil rotating in a steady field induces continuously. That is a generator.

So "the field is constant, therefore nothing is induced" is wrong in two of the three cases. What genuinely induces nothing is a loop that slides around inside a uniform field with its area and orientation unchanged, since then all three factors hold still.

The induced current makes a magnetic field that opposes the change in flux. That is not the same as opposing the external field, and the difference shows up every time the flux is decreasing.

1. Into-page flux increasing: the loop drives current counterclockwise, making an out-of-page field to resist the increase.
2. Into-page flux decreasing: the loop drives current clockwise, making its own into-page field to prop the flux up.

The blanket rule "always oppose the external field" is right in the first case and backward in the second, so it fails for magnets being pulled away, loops being shrunk, and currents being switched off.

The mechanical check is worth having, because it never depends on remembering a direction: the force on the induced current always resists the motion that produced it. Push a magnet toward a loop and the loop pushes back; pull it away and the loop pulls after it. That is also the energy argument, since anything else would let you get current for free.