Concept of an area vector when finding magnetic flux

[RobSoko315]

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:
$$\Phi$$ = BA cos $$\theta$$
Given that the field is uniform and is traveling through a constant area (the loop in this case).

The textbook then says $$\theta$$ 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:
$$\Phi$$ = BA sin $$\theta$$ ?
Where $$\theta$$ is defined as the angle between the loop and the magnetic field.

Thanks in advance...

-Rob-

 

[Nivlac2425]

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.

 

[atyy]

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.

 

[Nabeshin]

Simply because $$\Phi = \vec{A} \cdot \vec{B}$$

And the magnitude of the dot product is defined as $$ABcos\theta$$, where $$\theta$$ is the angle between the two vectors.

 

[RobSoko315]

Thanks for all you help

-Rob-