# How can the area be a vector in the equation $\Phi=BA\cosθ$

I'm a high school student. We were taking Magnetic flux in school and the formula is $$\Phi=BA\cosθ$$.

Please Bear with me. My issue lies in the cosθ. I'm assuming here we treated the area as a vector but how can we treat the area as a vector? and how can the area decrease due to change in orientation?

I really want to understand the math behind it because if it is treated as a vector won't the area have two components? Again I'm not even sure if the area is even a vector. I'm not even sure what am I asking I'm just fed up with understanding physics it's too taxing

I'm assuming here we treated the area as a vector but how can we treat the area as a vector?

Yes, area is considered as a vector in physics. You will come across another law called Gauss Law which also considers area as vector.

and how can the area decrease due to change in orientation?

The area does not change. I am assuming that you are not familiar with the concept of flux. Basically the number of magnetic field lines(in your case) passing through the given area changes when the orientation is changed (second diagram)

I really want to understand the math behind it because if it is treated as a vector won't the area have two component?

Since area is a vector, it can be resolved into two directions. While finding flux, we take the components of the area vector which are directed parallel to the electric/magnetic field lines. $$|\overrightarrow{A}cos\theta|$$ gives the effective area in the direction of the field lines.

EDIT: I forgot to add vector signs in the picture.

• So.... If I have a Square of Area A whos normal makes an angle x with the field lines can I say it's as if there are two smaller areas in the x and y-direction? and the area where the normal is perpendicular to the field lines is neglected since field lines won't pass?
– HEL
Nov 16, 2020 at 14:00
• That is correct. I was just about to upload a picture showing that. Nov 16, 2020 at 14:04
– HEL
Nov 16, 2020 at 14:06
• Thank you so much I got it!
– HEL
Nov 16, 2020 at 14:34

I am pretty sure your book explains how the vector associated with the area is defined. The direction of the vector is perpendicular to the surface of the given area. Changing the angle between the area and the magnetic field does not change the magnitude of the area. Same as when you rotate any other vector, you don't change the magnitude of the vector. The flux through the surface changes with angle and you can, intuitively, see this looking at the field lines for uniform field (to make it simpler). If the surface is perpendicular to the magnetic field, many field lines go through the surface. If the surface is inclined relative to the magnetic field, fewer field lines go through. And if the surface shows its edge to the field lines there are no lines going across the surface. In all these cases the area of the surface is the same.

First of all keep this thing in mind that Dot product and cross products are a tool. They are defined so as to ease the calculations.

Now when you go for calculation of flux you define it as a number of field lines crossing an area perpendicularly. Now for slanted field lines, we take the area perpendicular to the field lines and we find that we have to take a factor of $$\cos\theta$$ Now magnetic field intensity is directly proportional to the number of lines crossing the surface. So $$\Phi=BAcos\theta$$

In vectors, we have developed some tools that are useful so to use that tool of Dot product we define a direction to the Area. $$\mathrm{Dot \ product}=\overrightarrow{A}.\overrightarrow{B}$$

So we take $$\Phi=\overrightarrow{B}.\overrightarrow{A}$$ where B is magnetic field intensity and A is area vector. We have defined this area vector a direction perpendicular to the surface.. See image so that angle between these two vectors becomes $$\theta$$.

AB cos¶ is a product of two vectors a and b, where a and b are vectors and A, B their magnitudes

but both a and b are just representations, not the actual quantities they represent.

Area can't be an ordinary arrow vector nor can the magnetic field be an arrow vector.

They can both be the products of 2 arrow vectors. For example a magnetic field is a certain product of the current vector and the position vector.

In 3d this product can be represented by yet another vector b but don't be swayed by this since it's only a representation.