Concept of an area vector when finding magnetic flux

In summary, the textbook defines magnetic flux as the amount of field penetrating an area. The angle we measure it at is based off of the normal line to the plane of the area.
  • #1
RobSoko315
5
0
Concept of an "area vector" when finding magnetic flux

Hello,

I'm currently learning basic Electromagnetic Induction, specifically induced emf in a loop. According to my textbook, magnetic flux is defined as:
[tex]\Phi[/tex] = BA cos [tex]\theta[/tex]
Given that the field is uniform and is traveling through a constant area (the loop in this case).

The textbook then says [tex]\theta[/tex] is defined as the angle between the area vector (A) and magnetic field (B). The book's only explanation of an area vector is "Its direction is normal to the loop's plane, and its magnitude is equal to the area of the loop."

My question is, why do we measure the angle with respect to the area vector, and not the plane of the loop? In other words, why do we say magnetic flux is defined as the above equation instead of:
[tex]\Phi[/tex] = BA sin [tex]\theta[/tex] ?
Where [tex]\theta[/tex] is defined as the angle between the loop and the magnetic field.

Thanks in advance...

-Rob-
 
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  • #2


We use cosine because in regards to magnetic flux, we only concentrate on the component of the magnetic field that is perpendicular to the plane of the area. The normal of the area (the line perpendicular to the plane) is used because it essentially is the simplest method to follow in basic or intro physics.

If we use sine, where theta is defined as you say, and the given angle is to the normal, then theta is the complementary angle to the given angle.
In general, we can use your method, but the definitions for magnetic flux follows the use of the normal line to the plane of the area.
 
  • #3


Because flux through an area is the amount of field piercing the area. If the field were completely perpendicular to the area, then you would want the formula to be BA (you don't expect any correction for angle in this case, since the field is already piercing the area in the simplest way). The more general formula BAcos(theta) when the field is not perpendicular reduces to BAcos(0)=BA when the field is perpendicular, so that suggests that the cosine is indeed right.
 
  • #4


Simply because [tex]\Phi = \vec{A} \cdot \vec{B}[/tex]

And the magnitude of the dot product is defined as [tex]ABcos\theta[/tex], where [tex]\theta[/tex] is the angle between the two vectors.
 
  • #5


Thanks for all you help

-Rob-
 

1. What is an area vector?

An area vector is a mathematical concept used in physics and engineering to represent the direction and magnitude of an area. It is a vector that is perpendicular to the surface of an object and has a magnitude equal to the area of the surface.

2. How is an area vector used in finding magnetic flux?

When calculating magnetic flux, the area vector is used to determine the orientation of the surface with respect to the magnetic field. This helps to determine the amount of magnetic field passing through the surface, which is the definition of magnetic flux.

3. Is the area vector always perpendicular to the surface?

Yes, the area vector is always perpendicular to the surface. This is because it represents the direction of the surface itself, which is always perpendicular to itself.

4. Can the area vector change in value?

Yes, the area vector can change in value depending on the orientation of the surface. If the surface is rotated, the area vector will also change in direction and magnitude.

5. How is the area vector related to the magnetic field?

The area vector and the magnetic field are related through the dot product. The dot product is used to calculate the amount of magnetic field passing through a given surface, and the area vector helps to determine the orientation of the surface in relation to the magnetic field.

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