Topic 13.1 Magnetic Flux: define magnetic flux as the surface integral of the field and compute it for uniform and changing configurations.

What this topic is asking

The College Board (Topic 13.1) wants you to define magnetic flux as the surface integral of the field through a loop, compute it for uniform and angled configurations, and recognize the three ways it can change. Magnetic flux is the quantity whose rate of change drives induction in the rest of the unit.

Defining magnetic flux

The dot product means only the component of $\vec{B}$ along the surface normal threads the surface. A field skimming parallel to the surface produces no flux; a field perpendicular to it produces the maximum, $BA$. When the field is non-uniform or the surface is curved, the flux is the genuine surface integral $\displaystyle\int\vec{B}\cdot d\vec{A}$, summing $\vec{B}\cdot d\vec{A}$ over every patch. For the loops in this course the field is usually uniform across a flat area, so the simple $BA\cos\theta$ form applies, but the integral definition is what makes magnetic flux the same kind of quantity as the electric flux of Unit 8.

The angle and the area vector

The angle $\theta$ is measured between $\vec{B}$ and the surface normal, not the surface itself. A common slip is to use the field's angle with the plane of the loop; the two differ by $90^\circ$. When the loop's plane faces the field (normal parallel to $\vec{B}$), $\theta = 0$ and $\Phi = BA$; when the plane lies along the field (normal perpendicular to $\vec{B}$), $\theta = 90^\circ$ and $\Phi = 0$.

Flux linkage and the three ways flux changes

For a coil of $N$ identical turns, the total flux linkage is $N\Phi_B$, which is what matters for induction: each turn contributes the same flux, so a coil of many turns multiplies the effect, which is why generators and transformers use coils of hundreds of turns. The flux can change in exactly three ways:

• the field $B$ changes (for example a magnet moves closer),
• the area $A$ enclosed changes (a loop is stretched or a bar slides),
• the orientation $\theta$ changes (the loop rotates, as in a generator).

Try this

Q1. A $0.50$ T field passes perpendicularly through a $0.10$ m squared loop. Find the flux. [1 point]

• Cue. $\Phi = BA\cos 0 = (0.50)(0.10) = 0.050$ Wb.

Q2. List the three ways the flux through a loop can change. [1 point]

• Cue. Change the field $B$, the area $A$, or the orientation angle $\theta$.