# Different formulas for magnetic flux

I am familiar with the formula $$\phi=BA\cos(\theta)$$ fo magnetic flux, where I know $$\theta$$ is the angle between $$B$$ and $$A$$. I came across another formula ; $$\phi=Ba\cos(\omega t)$$. Wanted to ask why is $$\theta$$ replaced by ωt here? are the two formulas used in different cases?

• Because in the second case the field is oscillating in strength (and perpendicular to the surface). – G. Smith Jul 24 at 21:25
• It's likely either what G. Smith has said, $B$ being the amplitude of the oscillation of the magnetic field, or the magnetic field is rotating with an angular frequency of $\omega$, relative to the loop through which the magnetic flux is being calculated. Rotating magnetic fields are commonly encountered in electric motors and generators. It's hard to say more unless you include a reference or describe the context. – Puk Jul 25 at 6:51

## 1 Answer

Maybe it's better to start from the "original" definition of magnetic flux which is the amount of the magnetic field passing perpendicularly through some surface area, or mathematically, is defined as the dot product of the magnetic field ($$\bf B$$) and that surface area ($$\bf A$$) (remember area is a vector which perpendicular to the surface):

$$\phi = {\bf B \cdot A} = B A \cos\left(\theta\right)$$

That's why you get $$\cos\left(\theta\right)$$. Now imagine, either the magnet or the surface is rotating, then the amount of the field passing perpendicularly through the surface will change as function of time and this variation (rotation) is expressed by the $$\theta$$ as a function of time ($$t$$), namely $$\theta(t) =\omega t$$. Therefore the flux can now be written as:

$$\phi = B A \cos\left( \theta(t) \right) = B A \cos\left( \omega t \right)$$