Describe how magnetic flux can be sensed by the changing alignment of a magnet on a compass needle and quantitatively analyse the concept of magnetic flux density B and flux Phi = B A cos theta in a magnetic field

What this dot point is asking

NESA wants you to distinguish magnetic flux density $B$ (the field at a point, in tesla) from magnetic flux $\Phi$ (the total field through a surface, in weber), apply $\Phi = B A \cos \theta$ correctly, and connect this concept to the qualitative idea of a compass needle responding to field direction. Flux is the bridge to Faraday's law in the next dot point.

The answer

Magnetic flux density B

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 ($F = BIL \sin \theta$).

SI unit: the tesla (T). Equivalent forms:

Typical magnitudes:

• Earth's surface field: about $5 \times 10^{-5}$ T.
• Bar magnet near a pole: 0.01 to 0.1 T.
• MRI scanner: 1.5 to 3 T (some research machines reach 7 T).
• Strong laboratory electromagnet: up to 10 T.
• Neutron star: $10^8$ T (and rising).

Magnetic flux

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

where $\theta$ is the angle between $\vec{B}$ and the normal to the surface (the vector perpendicular to the surface). SI unit: the weber (Wb), where 1 Wb = 1 T m$^2$.

Two ways to picture it:

1. Flux is the "amount of field passing through" the surface. More field, more area, or more alignment with the surface normal all increase flux.
2. Flux is the dot product $\Phi = \vec{B} \cdot \vec{A}$, where $\vec{A}$ is the area vector (magnitude $A$, direction along the normal).

Special angles:

• IMATH_19 (field along the normal): $\Phi = BA$, maximum flux.
• IMATH_21 (field in the plane of the surface): $\Phi = 0$, no flux through the surface.
• IMATH_23 : $\Phi = -BA$, maximum negative flux (the field passes through the surface in the opposite sense).

The angle convention (watch this)

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.

"Plane at 30° to the field" $\Rightarrow$ normal at 60° to the field $\Rightarrow$ $\cos 60° = 0.5$.

"Normal at 30° to the field" $\Rightarrow$ $\cos 30° = 0.866$.

Flux through multiple turns

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

Faraday's law uses flux linkage: $\varepsilon = - N \, d\Phi / dt$, not just $d\Phi / dt$. We treat this in the induction dot point.

Compass needles and flux qualitatively

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.

If you tilt a small loop of wire in a uniform field while watching the field lines, the number of lines threading the loop changes as $\cos \theta$. That is the geometric content of $\Phi = BA \cos \theta$.

Worked example: rotating coil

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.

Area: $A = 0.20^2 = 0.040$ m$^2$.

Maximum flux linkage (normal aligned with field, $\theta = 0°$):

$\Psi_{\max} = N B A = 50 \times 0.30 \times 0.040 = 0.60$ Wb.

At $\theta = 45°$:

$\Psi = N B A \cos 45° = 0.60 \times 0.707 = 0.42$ Wb.

As the coil rotates, the flux linkage oscillates between $+0.60$ Wb and $-0.60$ Wb, with the rate of change driving the induced EMF in a generator (Faraday's law).

Common traps

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

Using the angle to the surface instead of to the normal. Questions love to phrase the geometry as "the coil is at 60 degrees to the field," meaning the plane of the coil is at 60 degrees, so the normal is at 30 degrees. Always check what $\theta$ refers to before substituting into $\cos$.

Forgetting the $N$ for a multi-turn coil. 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.

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

Quoting flux as the dot product without specifying the area direction. Always state which way the area normal points; a flipped normal flips the sign of flux but does not change physics.

In one sentence

Magnetic flux density $B$ is the local field strength (tesla, T = Wb/m$^2$), and magnetic flux $\Phi = B A \cos \theta$ (weber, Wb = T m$^2$) is the total field threading a surface of area $A$ whose normal lies at angle $\theta$ to $\vec{B}$.