Alexander Fufaev
My name is Alexander FufaeV and here I will explain the following topic:

Magnetic Field, Magnetic Flux and Density Simply Explained

Important Formula

Formula: Magnetic Flux and Flux Density
What do the formula symbols mean?

Magnetic flux

Unit
Magnetic flux is a scalar quantity (i.e. not a vector) from electrodynamics, which can be used to describe the magnetic field.

Magnetic flux density (B-field)

Unit
Homogeneous magnetic flux density (short: magnetic field) describes how much magnetic flux penetrates a certain area. The flux density indicates the flux per area.

Area

Unit
Plane surface penetrated by the magnetic field (B-field).

Angle

Unit
Angle between the normal vector perpendicular to the \(A\) surface and the \(B\) field vector.
Magnetic Flux Through a Cross-Sectional Area

When you take two bar magnets #1 and #2 and play around with them a bit, you will make three observations:

  1. If you bring two specific sides of the bar magnets close to each other, they will repel.
  2. If you rotate bar magnet #1 and bring its other side close to bar magnet #2, they will attract.
  3. If you rotate bar magnet #2, the two bar magnets will repel each other again.

We can provide a theoretical explanation for this observation that works well: A bar magnet has two poles - a magnetic north pole N and a magnetic south pole S, which can attract or repel each other. With this theory, we can also formulate our three observations as follows:

  1. North pole of one magnet and north pole of another magnet repel each other.
  2. South pole of one magnet and north pole of another magnet attract each other.
  3. South pole of one magnet and south pole of another magnet repel each other.

We can summarize these three observations into one statement: Magnetic poles with the same name (S and S, N and N) repel and poles with different names (S and N) attract.

You will only understand WHY this is the case when you study quantum mechanics. In this lesson, we first want to understand the observations and be able to describe them physically.

A fundamental quantity involved in this repulsion and attraction of magnetic poles is the magnetic field that forms around the bar magnets. Bar magnet and its field lines

That this magnetic field actually exists and is not just a figment of physicists' imagination can be made visible using iron filings. To do this, scatter iron filings around the bar magnet and observe how the iron filings arrange themselves in line patterns around the magnet. For a bar magnet, the iron filings arrange themselves as shown in the image above.

If we take a pen and carefully trace the resulting lines of iron filings, we can then remove the iron filings and see what the magnetic field of a bar magnet looks like. The lines drawn around the magnet are called magnetic field lines (or simply field lines).

Looking at the image above with the magnetic field lines of the bar magnet, we can make three observations:

  1. The magnetic field lines start at one pole and end at the other pole. We say: The magnetic field lines are closed loops. We can define the direction of the magnetic field lines such that the lines originate from the north pole and end at the south pole.
  2. The magnetic field lines have different distances from each other. At the poles, they are closer together than around the middle of the bar magnet. We say: The magnetic field of a bar magnet is inhomogeneous. The »inhomogeneous« (non-uniform) means that the magnetic field lines do not have a uniform spacing between them.
  3. The magnetic field lines do not intersect.
trace

If we attach a bar magnet to the ground and walk around it with another bar magnet, we will notice that the magnetic field of the attached bar magnet interacts with the magnetic field of the other bar magnet to varying degrees. Directly at the poles, the attractive and repulsive force is greater than further away from them: The magnetic field strength decreases with distance from the magnet. This is typical for a inhomogeneous magnetic field. Additionally, we can state the following: The closer the magnetic field lines are to each other, the greater the attractive and repulsive force on the other bar magnet.

Unlike a bar magnet, the magnetic field inside a horseshoe magnet is homogeneous. If we scatter iron filings here, we see that the field lines run straight at a certain distance from each other. Magnetic field in horseshoe magnet

Magnetic Flux and Flux Density

If we look at the field lines of a bar magnet and their direction from the south to the north pole, we can imagine this as a kind of magnetic flux. Let's assign a large Greek letter \( \class{violet}{\mathit{\Phi}} \) (»Phi«) to this magnetic flux.

Let's take a ring enclosing an area \(A\) and place it into the homogeneous magnetic field lines of the horseshoe magnet, we can consider the quantity \( \class{violet}{\mathit{\Phi}} \) as a measure of how many field lines pierce the area \(A\).

If we take a rectangular ring enclosing a larger area \(A\), then this area will be pierced by more magnetic field lines. The larger the area \(A\), the greater the magnetic flux \( \class{violet}{\mathit{\Phi}} \).

Dividing the magnetic flux \( \class{violet}{\mathit{\Phi}} \) by the cross-sectional area \(A\), we obtain a new quantity, let's call it \( \class{violet}{B} \), which describes the »magnetic flux per area«: $$ \class{violet}{B} ~=~ \frac{ \class{violet}{\mathit{\Phi}} }{A} $$

We call the quantity \( \class{violet}{B} \) as magnetic flux density and assign it the unit Tesla (T).

As the name suggests, magnetic flux density describes how dense the magnetic field lines are. The larger \( \class{violet}{B} \) is, the denser the field lines. The advantage of the quantity \( \class{violet}{B} \) over \( \class{violet}{\mathit{\Phi}} \) is that \( \class{violet}{B} \) is independent of the chosen area \(A\) used to measure the magnetic field. No matter which area \(A\) we choose, the ratio \( \frac{\class{violet}{\mathit{\Phi}} }{A} \) remains the same for the considered magnetic field. Thus, magnetic flux density is a perfect quantity to characterize the magnetic field of a bar magnet, horseshoe magnet, or other geometric magnetic objects.

Let's rearrange the formula for \( \class{violet}{\mathit{\Phi}} \): $$ \class{violet}{\mathit{\Phi}} ~=~ \class{violet}{B} \, A $$

From this, we can read off the unit \( \text{T}\,\text{m}^2 \) (Tesla times square meter) of magnetic flux. Sometimes, magnetic flux is expressed in the unit \( \text{T}\,\text{m}^2 = \text{Wb} \) (Weber).

It may happen that we tilt the ring in the magnetic field, then the field lines do not pass through the area \(A\) exactly perpendicular. However, the above formula applies only when the magnetic field lines pass through the area perpendicular. Magnetic flux through a cross-sectional area

If the field lines enclose an angle \(\varphi\) with the area, then we can calculate the magnetic flux through an area \(A\) with the following formula: $$\class{violet}{\mathit{\Phi}} ~=~ \class{violet}{B} \, A \, \cos(\varphi)$$

The formula simplifies if the magnetic field lines penetrate the surface perpendicular to it. Then \( \varphi = 90^{\circ}\) and the cosine becomes \( \cos(90^{\circ}) = 1 \).

Example: Magnetic flux of a coil in a homogeneous magnetic field

A rectangular coil with a cross-sectional area of 0.2 m x 0.3 m is placed in a homogeneous magnetic field with a magnetic flux density of 0.5 T. The angle between the normal of the coil's surface area and the magnetic field is 60 degrees. What is the magnitude of the magnetic flux through the coil?

We need to substitute the given quantities into the formula: $$\begin{align}\class{violet}{\mathit{\Phi}} ~&=~ \class{violet}{B} \, A \, \cos(\varphi) \\\\ ~&=~ 0.5\, \mathrm{T} \cdot 0.2 \, \mathrm{m} \cdot 0.3 \, \mathrm{m} \cdot \cos(60^\circ) \\\\ ~&=~ 0.015 \, \mathrm{Tm^2}\end{align}$$

Therefore, the magnetic flux through the coil is 0.015 Weber.