KAHIBARO
Table of Contents
Seeing how much magnetic field passes through a surface
Magnetic flux is a way to measure how much magnetic field goes through a surface. It is not just about how strong the magnetic field is, but also about how large the surface is and how the surface is oriented.
If a magnetic field passes straight through a surface, the flux is large. If the field only grazes the surface, the flux is smaller. If the field runs exactly parallel to the surface, no magnetic field passes through it, so the magnetic flux is zero.
Magnetic flux is important because changing flux is what produces induced effects in circuits, which is studied in later topics such as Faraday's law and Lenz's law.
Definition of magnetic flux
For a uniform magnetic field passing through a flat surface, magnetic flux is defined as
$$
\Phi_B = B A \cos\theta
$$
where $B$ is the magnetic field magnitude, $A$ is the area of the surface, and $\theta$ is the angle between the magnetic field and the surface's normal direction.
The normal direction is an imaginary line perpendicular to the surface. This direction matters because the magnetic field component that passes through the surface is the component along the normal.
Important formula:
$$
\Phi_B = B A \cos\theta
$$
Here, $\theta$ is the angle between $\vec B$ and the normal to the surface, not the angle between $\vec B$ and the surface itself.
If the angle is $0^\circ$, then $\cos 0^\circ = 1$, so the flux is maximum:
$$
\Phi_B = BA
$$
If the angle is $90^\circ$, then $\cos 90^\circ = 0$, so the flux is zero:
$$
\Phi_B = 0
$$
Why the normal direction is used
A surface can be tilted in many ways, so we need a clear way to describe its orientation. The normal vector gives that description. It points straight out from the surface.
This means magnetic flux depends on the part of the field that goes through the surface, not the part that slides along it. If the field is mostly sideways, little flux passes through.
Flux as a dot product
Magnetic flux can also be written in vector form. If $\vec A$ is the area vector, then
$$
\Phi_B = \vec B \cdot \vec A
$$
The area vector $\vec A$ has magnitude equal to the area $A$, and direction along the normal to the surface. Because this is a dot product,
$$
\Phi_B = BA\cos\theta
$$
for a uniform field over a flat surface.
This form is useful because it shows that flux is a scalar quantity. The result is a number, not a vector.
Vector form of flux:
$$
\Phi_B = \vec B \cdot \vec A
$$
Magnetic flux is a scalar, even though it is built from vectors.
Units of magnetic flux
The SI unit of magnetic flux is the weber, abbreviated as Wb.
Since
$$
\Phi_B = BA
$$
and magnetic field has unit tesla, $\mathrm{T}$, while area has unit $\mathrm{m}^2$, we get
$$
1\ \mathrm{Wb} = 1\ \mathrm{T \cdot m^2}
$$
The unit can be summarized as follows.
| Quantity | Symbol | SI unit |
|---|---|---|
| Magnetic flux | $\Phi_B$ | weber, $\mathrm{Wb}$ |
| Magnetic field | $B$ | tesla, $\mathrm{T}$ |
| Area | $A$ | $\mathrm{m^2}$ |
Positive and negative flux
Magnetic flux can be positive or negative, depending on the chosen direction of the surface normal.
If the magnetic field points in the same general direction as the normal, then $\cos\theta$ is positive and the flux is positive. If the field points opposite the normal, the flux is negative.
This sign becomes especially important for closed surfaces and for induction, where the direction of the induced effect matters.
Flux through curved or nonuniform surfaces
If the magnetic field changes from place to place, or if the surface is curved, then the simple formula $BA\cos\theta$ is not enough for the whole surface at once. In that case, we imagine the surface broken into many tiny pieces, each with its own small area and direction. Then the total flux is
$$
\Phi_B = \int \vec B \cdot d\vec A
$$
You do not need advanced calculus to understand the idea. The formula just means, add up the tiny amounts of magnetic field passing through each tiny part of the surface.
Open surfaces and closed surfaces
A surface can be open, like a loop of wire stretched across an imaginary film. It can also be closed, like the surface of a sphere.
For an open surface, flux tells how much magnetic field passes through that chosen surface.
For a closed surface, the total magnetic flux has a special result in physics. That result belongs to Gauss's law for magnetism, which is covered elsewhere. Here, the key idea is simply that flux can be defined for both open and closed surfaces.
Physical interpretation
Magnetic flux does not mean that magnetic field is a substance flowing like water. It is a geometric measure of how much field crosses a surface.
A good mental picture is to imagine magnetic field lines crossing a surface. More lines crossing means larger flux. Fewer lines crossing means smaller flux. This picture is only a visualization, but it helps build intuition.
If you keep the field strength fixed and make the surface bigger, the flux increases. If you keep the area fixed and increase the field strength, the flux also increases. If you tilt the surface so that less field goes through it, the flux decreases.
Simple examples
Suppose a magnetic field of magnitude $0.50\ \mathrm{T}$ passes perpendicularly through a flat area of $0.20\ \mathrm{m^2}$. Then
$$
\Phi_B = BA = (0.50)(0.20) = 0.10\ \mathrm{Wb}
$$
Now suppose the same field and area are present, but the angle between the field and the normal is $60^\circ$. Then
$$
\Phi_B = BA\cos 60^\circ = (0.50)(0.20)(0.5) = 0.050\ \mathrm{Wb}
$$
If the field is parallel to the surface, then the angle with the normal is $90^\circ$, so
$$
\Phi_B = BA\cos 90^\circ = 0
$$
Common angle mistake
A very common mistake is using the angle between the magnetic field and the surface itself. The formula uses the angle with the normal, not with the plane.
If $\alpha$ is the angle between the field and the surface, then the angle with the normal is
$$
\theta = 90^\circ - \alpha
$$
so the flux becomes
$$
\Phi_B = BA\sin\alpha
$$
Both forms are correct if the angle is defined carefully.
Be careful with angles.
If the given angle is with the surface normal:
$$
\Phi_B = BA\cos\theta
$$
If the given angle is with the surface itself:
$$
\Phi_B = BA\sin\alpha
$$
Summary idea
Magnetic flux tells how much magnetic field passes through a surface. It depends on field strength, area, and orientation. For a flat surface in a uniform field,
$$
\Phi_B = BA\cos\theta
$$
and in general,
$$
\Phi_B = \int \vec B \cdot d\vec A
$$
This idea is the foundation for electromagnetic induction, where changes in magnetic flux lead to important physical effects.