Quick questions on Magnetic flux and flux density: HSC Physics Module 6

The magnetic flux density (often just called the magnetic field) $\vec{B}$ at a point is a vector describing the strength and direction of the magnetic field there. It is what a compass needle aligns with, and it determines the force on a moving charge ($F = qvB$) or on a current ($F = BIL \sin \theta$).

For a flat surface of area $A$ placed in a uniform field $\vec{B}$, the magnetic flux through the surface is:

The $\theta$ in $\Phi = B A \cos \theta$ is the angle between $\vec{B}$ and the normal to the surface, not between $\vec{B}$ and the surface itself. Questions sometimes give the angle between the field and the plane of a coil; you must take the complement.

A coil of $N$ turns links flux $N$ times (each turn intercepts the same flux, in series). The flux linkage is:

A compass needle is a small magnetic dipole. It aligns with the local field direction so that its north pole points along $\vec{B}$. By placing compasses (or sprinkling iron filings) over a region you can map the direction of $\vec{B}$ at every point, hence the field line pattern. The density of the lines (lines per unit area perpendicular to them) is proportional to the flux density $B$, hence the name.

A square coil of side $0.20$ m and $50$ turns is rotated in a uniform field of $0.30$ T. Find the maximum flux linkage and the flux linkage when the coil normal is at $45°$ to the field.

$B$ is a field strength per unit area; $\Phi$ is a total field through an area. They have different units (T vs Wb).

A 100-turn coil with 1 Wb through each turn has 100 Wb of flux linkage, not 1 Wb. For Faraday's law, you must use $N \Phi$ or apply the $N$ outside the derivative.

Flux $\Phi$ is a scalar. The direction information is buried in the sign through $\cos \theta$ (positive or negative depending on orientation).