How much field threads the loop

Magnetic flux measures how much field passes through a surface. For a uniform field over a flat loop,

$$\Phi_B = \vec{B}\cdot\vec{A} = BA\cos\theta,$$

where $\vec{A}$ is the area vector: magnitude equal to the loop's area, direction along the loop's normal. The SI unit is the weber, $1\ \text{Wb} = 1\ \text{T}\cdot\text{m}^2$.

Three separate quantities sit on that line, and everything in Unit 13 depends on keeping them apart.

1. The field $B$ is what a probe reads at a point. It knows nothing about your loop.
2. The flux $\Phi_B$ belongs to the field and a chosen surface. Tilt the loop, shrink it, or slide it out of the field region and the flux changes while $B$ does not.
3. The flux rate $d\Phi_B/dt$ is the only one of the three that induces an EMF, and it needs a clock.

A superconducting magnet bore at $1.5$ T is an enormous field, and a coil held still inside it carries enormous flux and induces nothing, because nothing on that list is changing. Rank problems by the third rung, never the first.

Every wrong cosine in this unit comes from measuring $\theta$ to the wrong reference. It is the angle between $\vec{B}$ and the loop's normal, not between $\vec{B}$ and the loop's surface.

$$\theta = 0^\circ: \ \Phi = BA \ \ \text{(normal along the field)}; \qquad \theta = 90^\circ: \ \Phi = 0 \ \ \text{(the field lies in the loop's plane)}.$$

Read that second case slowly, because intuition fights it. A loop held so the field runs along its surface catches nothing: the field lines skim past instead of poking through. That is the maximum-looking case that is actually zero.

The reliable procedure is one extra pen stroke:

1. Draw the normal arrow sticking out of the loop, perpendicular to its surface.
2. Measure the angle from that arrow to $\vec{B}$.
3. Take the cosine of that angle. If you had to take a sine, you measured from the plane.

Choosing the normal to point the other way flips the sign of $\Phi$. That is allowed: flux is signed relative to a choice you make, and Lenz's law only ever asks about the change, so a consistent choice is all that is required.

A coil of $N$ turns is $N$ loops stacked in the same place, and each one is threaded by the same $\Phi_B$. The quantity that matters is the flux linkage $N\Phi_B$, and Faraday's law carries the $N$ with it:

$$\varepsilon = -N\frac{d\Phi_B}{dt}.$$

A 500-turn coil produces 500 times the EMF of a single loop in the same changing field. Dropping the $N$ is not a small slip; it is a factor of several hundred, and it is the single most common arithmetic error on induction free-response.

Flux linkage is also how inductance is defined, which is why it is worth being fluent in now:

$$L = \frac{N\Phi_B}{I}.$$

Watch the geometry when you count turns. Doubling the turns on a fixed-length solenoid also doubles the field inside it, so the linkage goes as $N^2$. Stacking two separate coils in series does not.

$BA\cos\theta$ assumes one value of $B$ over the whole surface. When the field varies across the loop, the general definition takes over:

$$\Phi_B = \int \vec{B}\cdot d\vec{A}.$$

The standard case is a rectangular loop lying in the plane of a long straight wire. The field $B = \mu_0 I/(2\pi x)$ is strong at the near side and weak at the far side, so slice the loop into strips along which $B$ is constant. For a loop of height $h$ running from $x = a$ to $x = b$:

$$\Phi_B = \int_a^b \frac{\mu_0 I}{2\pi x}\, h\, dx = \frac{\mu_0 I h}{2\pi}\ln\!\left(\frac{b}{a}\right).$$

Two things to notice. The strips run parallel to the wire, because that is the direction along which $B$ does not change, and the answer is a logarithm rather than a power, so doubling the loop's width does not double the flux. Substituting the field at one edge, or at the midpoint, and multiplying by the area is the shortcut that fails here.

A closed surface is a special case worth stating once: $\oint \vec{B}\cdot d\vec{A} = 0$ always, because magnetic field lines close on themselves and there is no magnetic charge for them to start on.