Understanding Magnetic Flux: Integrals vs. Simplified Equations

[Dan453234]

Homework Statement

So I understand that the magnetic flux is equal to the integral of (B dotted with dA) (new to the site don't know how to use math symbols). My question is, how come in some problems, is it ok to just say the magnetic flux is equal to the magnetic field times the area, while on other problems, you have to actually take the integral.

Homework Equations

The Attempt at a Solution

 

[axmls]

If it just so happens that the magnetic field doesn't change in strength anywhere on the surface and if the magnetic field's direction is always in the same direction as the normal vector to the surface, then the integral will just equal the product of the field with the area.

 

[drvrm]

My question is, how come in some problems, is it ok to just say the magnetic flux is equal to the magnetic field times the area, while on other problems, you have to actually take the integral.

the magnetic flux (often denoted Φ) through a surface is
the surface integral of the normal component of the magnetic field B
passing through that surface.

The vector representation of a surface element ds is a vector of magnitude IdsI in a direction perpendicular the surface.
In those cases where the B field is normal to the surface the Flux can be written equal to ( B. surface area) as the angle between B and ds is zero and the dot-product B.ds= Bds cos (0) =Bds
but in general cases it is surface integral of B.ds taken over the whole surface.