# why should we define "the magnetic flux is the dot product of the magnetic field and the plane area , instead of perimeter of the plane

When a uniform magnetic field $$\\mathbf B\$$ passes a plane at any angle, the magnetic flux $$\\phi_B\$$ is the dot product of the magnetic field $$\\mathbf B\$$ and the plane area $$\\mathbf{a}\$$,that is ,$$\\phi_B=\mathbf B \cdot \mathbf{a} \$$

Honestly,this explanation of magnetic flux is really abstract to me,i can write the formula above,but i don't really understand that,why should we define "the magnetic flux $$\\phi_B\$$ is the dot product of the magnetic field $$\\mathbf B\$$ and the plane area $$\\mathbf{a}\$$" instead of perimeter ,$$\p\$$ , of the plane ,that is ,$$\\phi_B=\mathbf B \cdot p\$$ ?

Can anyone explain this in detail?

• Why should we use the perimeter of the plane? – Hearth Apr 11 '20 at 15:48
• Total flux is flux density (B) x area! – Andy aka Apr 11 '20 at 16:33
• Because when we do the experiment with lots of different coils, we find the voltage is proportional to the area of the coil rather than the perimeter. That's what came first, the observation. Then we dreamt up mathematical formulae to describe what happened. Then we tried to add intuition as to why the formulae worked, and we invented 'lines of field' and 'flux' models to help visualise it. They're models, and all models are wrong, but they're useful, at non-relativistic speeds anyway. But always remember, the observations came first. – Neil_UK Apr 11 '20 at 19:02

"ϕB is the dot product of the magnetic field B and the plane area a"

No, it's: $$\Phi_A = \iint_A\vec{B}\cdot d\vec{A}$$ where B and dA are both vectors. You can't have a dot product between scalars.

B is a vector that points to the direction of the field and its magnitude is the field strength.

dA is the normal vector of the surface infinitesimal (an infinitely small part of the surface). That is, it points perpendicularly away from the surface and its magnitude is the area of the surface.

The dot product gives you the magnitude of the component of one vector that is parallel to the the other vector multiplied by the magnitude of the other vector.

Flux is the amount of magnetic field that is "flowing" through a given surface. So you only want to calculate with the component of B that is perpendicular to the surface. Hence the dot product.

If B is uniform in magnitude and is perpendicular to the surface at each point than the equation simplifies to just $$\Phi_A=B*A$$ where B is the magnitude of B and A is the total are of the surface. Because then B gives you the "flow rate" per unit area.

If B is uniform in magnitude but has a uniform angle with the surface at each point, than you have to take the part of B that is "flowing" through the surface, the perpendicular component that is. That's why you need the dot product. In this case the equation is $$\Phi_A=\vec{B}\cdot\vec{A}=B_\perp*A$$

"instead of perimeter ,p , of the plane"

Why would you possibly want to multiply by the perimeter? That is just simply not what flux is.

Given magnetic fields are just Efields with slight distortion by relativistic time shifts, the math we are taught is IMHO "because it works".

But I don't (yet) have a Masters in E&M so as to properly evaluate the history and evolution of the concepts and maths.

A friend of mine was delving into E&M, and noted the FORCE was orthogonal to the FLUX, "how convenient", he thought.

Ultimately its all about the electric fields.

Study on the A_VECTOR maths.