# Magnetic Flux and Magnetic Flux Linkage

Did you know fields are all around us? Yep, they exist in space and can be pretty useful to study. When it comes to electric and magnetic fields, we use something called "flux" to measure how much field is crossing a surface. This is different from the forces we learn about in basic physics, which only act on certain objects. Magnetic flux and magnetic flux linkage are two important concepts to understand in this area of study.

## Magnetic field and magnetic flux

Magnetic field is a fancy way of describing how magnets work in space. We use the letter B to represent it. Instead of looking at the whole field, we can just focus on how it affects a certain surface. This is where Faraday's law comes in. It talks about something called "magnetic flux," which is basically how much of the magnetic field crosses a surface. To calculate it, we use a formula with a dot and a vector, which might sound confusing but basically just means we're multiplying two things together. When the magnetic field isn't the same everywhere and the surface isn't flat, things get more complicated and we have to use integrals and other math stuff. But for now, we'll stick to flat surfaces and uniform magnetic fields. The angle between the field and the surface is what determines

Faraday's law is a rule that connects the electric field with magnetic flux. It's named after Michael Faraday, who discovered it through experiments. This law shows us how the energy needed to create an electric potential difference between two points (called the electromotive force or EMF) is related to how quickly the magnetic flux is changing. The equation for this might look complicated, but if we only look at the case of a uniform magnetic field and a fixed area, it becomes simpler. We can see this in an experiment where a moving surface and a uniform magnetic field create an electromotive force.

## What is magnetic flux linkage?

Maxwell's laws are a set of equations that describe how the electromagnetic field behaves. They are linear, which means that we can add different fields together to get a new field that also satisfies the equations. When we're trying to generate an electromotive force in an experiment, we can increase the output by using a concept called linkage. Linkage is a way of measuring how much of the magnetic field is "linked" to a certain circuit. The more linkage there is, the more electromotive force we can generate.

### Experimental setting of magnetic flux linkage

the previous scenario of a rotating coil in a magnetic field, we can increase the output of electromotive force by using multiple coils instead of just one. By doing this, we create N different surfaces that the magnetic field can interact with, and this increases the flux linkage. Flux linkage is a measure of how much of the magnetic field is linked to a certain circuit, and by increasing the number of coils, we increase the flux linkage and therefore the output of electromotive a

### Mathematical description of magnetic flux linkage

The graph shows the time evolution of the magnetic flux and the generated electromotive force in an experimental setup where a coil is rotating in a magnetic field with a value of 10 Teslas and an area of 1 m2. The coil is rotating at an angular velocity of 2 rad/s, and the magnetic field is directed in the x-axis while the normal vector evolves as given by the equation:

n(t) = (cos(2t), sin(2t), 0)

Using this equation, we can calculate the magnetic flux as:

ϕ(t) = B ⋅ A ⋅ n(t) = 10 ⋅ 1 ⋅ (cos(2t), sin(2t), 0)

This yields the following expression for the magnetic flux:

ϕ(t) = (10cos(2t), 10sin(2t), 0)

Using Faraday's law, we can compute the electromotive force as the time derivative of the magnetic flux:

ε(t) = -dϕ(t)/dt = (-20sin(2t), 20cos(2t), 0)

The graph shows that both the magnetic flux and the electromotive force are sinusoidal functions of time, with a period of π seconds. The amplitude of the magnetic flux is 10, while the amplitude of the electromotive force is 20. This is because we are using N identical coils that are synchronised and have the same three-dimensional orientation, which leads to an increase in the flux linkage by a factor of N. In this case, N is not specified, but we can assume that it is greater than 1 based on the discussion of flux linkage in the previous text.

Additionally, the number of coils used in a setup can greatly increase the output of the electromotive force by increasing the flux linkage. The use of multiple coils allows for easy manipulation of the magnetic field to generate an electromotive force. Faraday's law applies whenever there is a variation in either the intensity of the magnetic field, the area it goes through, or the orientation of the surface with respect to the field. Magnetic flux and Faraday's law are typically associated with a single rotating coil, while flux linkage is used to describe several static coils in the presence of a magnetic field. Overall, magnetic flux and magnetic flux linkage are important concepts in understanding how electromotive force is generated in different setups and how it can maximized

## Magnetic Flux and Magnetic Flux Linkage

Are magnetic flux linkage and density the same thing?

No, magnetic flux density designates the vector strength of the magnetic field, which we usually call B. The magnetic flux linkage is the growth of magnetic flux by having different surfaces crossed by a magnetic field.

What's the difference between flux and flux linkage?

The magnetic flux is a scalar quantity measuring the amount of magnetic field crossing a certain surface. The magnetic flux linkage is the growth of this magnetic flux by considering several surfaces.

What is the equation for magnetic flux linkage?

The equation for the magnetic flux linkage is: ΦL=N⋅Φ, where Φ is the magnetic flux and N is the number of coils.

What is the unit of magnetic flux linkage?

Magnetic flux linkage is measured in Webers (Wb).