Wize High School Grade 12 Physics Textbook > Magnetism

Magnetic Flux

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Magnetic Flux

Magnetic flux measures how many magnetic field lines pass through a surface. It is key to understanding electromagnetic induction. 
Consider a constant magnetic field B⃗\vec{ B}Bpassing through a flat area A⃗\vec{A}A.
The magnetic flux through the surface is given by the following equation, where θ\thetaθ is the angle between the magnetic field and area vectors:
Φm=B⃗⋅A⃗=∣B∣⃗∣A∣⃗cos⁡(θ)\boxed{\Phi_m=\vec{B}\cdot\vec{A}=\vec{\left|B\right|}\vec{\left|A\right|}\cos(\theta)}Φm​=B⋅A=∣B∣​∣A∣​cos(θ)​
The SI unit for magnetic flux is the Weber. 1 Wb=1 Tm2\boxed{1~Wb=1~Tm^2}1 Wb=1 Tm2​ 
Wize Tip
For the vector A⃗\vec AA, the magnitude is equal to the surface area, and the direction is perpendicular to the surface of the area. If you have a closed 3D shape (e.g. a sphere), the vector points to the outside of the shape.

Wize Concept
For any closed shape, the net magnetic flux must be zero. This is because magnetic monopoles do not exist, so all magnetic field lines that enter a 3D volume must also leave the 3D volume.

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Example: Magnetic Flux

A box is oriented as shown in the diagram below. There is a uniform magnetic field of B=0.3 TB=0.3\ TB=0.3 Tpointing parallel to the positive z-axis.
a) Find the magnetic flux across the surface ABDC.
b) Find the magnetic flux across the surface BDEF.
c) Find the total magnetic flux through the box.
 

Part a)

The surface ABDC has an area vector that points in the positive z-direction, which is the same direction as the magnetic field. Therefore, the dot product reduces to a simple multiplication.
Φm=BAcos⁡θΦm=(0.3 T)(5m×2m)cos⁡(0o)Φm=3.0 Wb\begin{aligned}
\Phi_m&=BA\cos\theta \\
\Phi_m&=(0.3~T)(5m\times2m)\cos(0^o) \\
\Phi_m&= 3.0 ~Wb\\
\end{aligned}Φm​Φm​Φm​​=BAcosθ=(0.3 T)(5m×2m)cos(0o)=3.0 Wb​
Part b)

The surface BDEF has an area vector that points in the positive y-direction, which is perpendicular to the magnetic field. Therefore, the magnetic flux is zero (cos⁡90o=0\cos90^o=0cos90o=0).

Part c) 

For all closed surfaces (such as this box!), the total magnetic flux passing through it is zero!

If we wanted to be more detailed, we could see that the flux through four of the surfaces is zero (the area vector is perpendicular to the field direction), and the flux passing through the top surface would be equal and opposite to our result from Part (a) (cos⁡180o=−1\cos180^o=-1cos180o=−1). So the flux would be given as: +3 + 0 + 0 + 0 + 0 - 3 = 0 Wb.