Is Magnetic Flux A Vector: Detailed Insight And Facts

The total number of magnetic field lines passing through a given area is magnetic flux. Is magnetic flux a vector? Let’s find out. 

Magnetic flux, which tells us about the number of field lines that cross the surface, is a scalar. It is the dot product of two vectors. So, Is magnetic flux a vector? The answer is simply no, but let’s get detailed insight. 

Magnetic field lines are imaginary lines that determine the space around the magnet where its effect is exerted. Whatever may be the type of magnet is, they will always consist of two poles, a north and south. 

The magnetic field lines outside the magnet are from north-south, while inside, the direction gets reversed. The region where lines are clustered is the region of strong magnetic effect. As field lines move apart, magnetic effects become weak. 

The magnetic flux does let us know about the field lines that pass through any plane. It is an important concept that lets us know about the effect of any magnet.

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Image Credit: Wikipedia

Why is magnetic flux a vector?

It is well known that a magnetic field has direction and thus is a vector, but this does not make magnetic flux also a vector. The magnetic flux is the scalar product of magnetic field lines and surface area.

is magnetic flux a vector
Image Credit: Wikipedia

Thus we have the formula as:

Φ= B.A

Φ = B A cos θ


Φ is the magnetic flux

B denotes the magnetic field 

A is the surface area. 

θ is the angle made by the field lines with a closed surface. 

The fundamental unit of magnetic flux is Volt-second, and the standard unit is weber (Wb).

The angle theta plays a vital role in determining the magnetic flux over a given surface. In case the magnetic field lines are normally falling to the surface, then the magnetic flux will be zero. Let us understand this. 

Φ = B A cos θ

Substituting the value of theta as 90°, we get

Φ = B A cos 90

We know cos 90 is equal to 0; thus, magnetic flux becomes zero. 

Is magnetic flux density vector?

Apart from magnetic flux, the magnetic flux density is also used to describe the effect of a magnet. Many get confused between these two magnetic concepts and use them to describe the same thing. But magnetic flux and magnetic flux density are quite different.

If we talk in simple language, then magnetic flux density tells us about the density of the field. The high value of magnetic flux indicates that the magnetic effect is strong, and a small value means a low magnetic effect. 

Magnetic Flux density is dependent on the area. The area is vector and changes with direction. This brings us to the conclusion that magnetic flux density is also a vector. 

As the name suggests, magnetic flux density determines the flux per given area, which brings us to the formula: 

B = Φ/A

Here, B is the magnetic flux density

Φ is the magnetic flux

A is the given surface area. 

The standard unit of magnetic flux density is Tesla. It is a vector quantity as it is in a way similar to the electric field by the relation B = εE. Here since ε is the constant, magnetic flux density is very much proportional to the electric field. As we know, electric fields have both magnitude and direction so is the magnetic flux density. 

Is magnetic flux linkage a vector?

Magnetic flux linkage is a value that represents the linking of a magnetic field with the coil. We can simply say that the magnetic flux linkage is the flux times the number of turns in the coil. 

It is generally used for solenoids. For example, a solenoid has 25 turns. Suppose the magnetic flux through the surface is 5 weber. Then magnetic flux linkage would be a product of magnetic flux and number of turns, i.e., 125. So, it is nothing but the total flux. 

The emf is induced in case the magnetic flux changes. This magnetic flux is termed magnetic flux linkage. And thus, it is the vector quantity as it is proportional to the current, which is also a vector quantity. So here, it is clear that magnetic flux is scalar, but flux linkage is a vector. 

How can magnetic flux be a scalar, but magnetic flux density is a vector?

Flux, in general in all the cases, is a scalar as it represents the total number. The number of anything is never associated with the direction. For instance, let’s count the number of birds flying over your roof. It doesn’t matter in which they fly; the total number will be a scalar. 

Let us look at a more proper explanation; we know that area and magnetic field are both vectors. Now in the figure above, we have given a surface with area A and magnetic field passing making angle theta with the surface. 

We know magnetic flux will be a product of magnetic field and area that is:.’

Φ = BA

From the figure, we can see that on splitting B into its component, we get B cos θ . Therefore:

Φ = B cos θA

Φ = B A cos θ

Φ = B . A

Which is a scalar dot product, and hence magnetic flux is a vector. On the other hand, the magnetic flux density is dependent on the surface area; it will vary in different areas. Since the area is a vector quantity, so is the magnetic flux density. Now we have got the answer to is magnetic flux a vector and why magnetic flux density is a vector. 

Frequently Asked Questions (FAQs)

What is magnetic flux?

For studying the magnetic field, magnetic flux is a vital concept. 

The magnetic field lines that cross a particular area, their total number, are said to be the magnetic flux. Its unit is weber and Tesla.

Is magnetic flux a vector quantity?

Though the quantities involved to find magnetic flux are vector, it is a scalar.  

How is magnetic flux different from magnetic flux density?

Magnetic flux and flux density have a minute but significant differences. 

Magnetic flux is used to describe the number of magnetic field lines, whereas magnetic flux density tells us about the density of the field lines—both in the given area. 

Is the magnetic field a vector?

The magnetic field has a significant direction and therefore is a vector. 

The magnetic field lines start from the north pole and enter the south pole. Whereas inside the magnet, the direction is opposite; it moves from the south pole to the north pole. 

What is magnetic flux linkage?

The magnetic flux linkage is usually the concept of solenoids. 

To understand it in an easy way, consider a solenoid has ‘n’ number of turns, and magnetic flux through one turn is Φ. Then flux linkage will be nΦ, which is basically the total flux through a solenoid. 

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