If you have a rectangular loop of area A placed in a region where the magnetic field is perpendicular to the plane of the loop, than would the emf be zero because they are perpendicular?

First the field through the loop must be changing with time to generate voltage around the loop.

If the B field vector lies in the plane of the loop, there is no flux through the loop. However if the field is changing or equivalently the loop is rotating, there is a rate of change of magnetic field flux through the loop and an emf appears.

No, the electromotive force (emf) induced in a rectangular loop by a perpendicular magnetic field is not always zero. The emf is actually determined by the rate of change of magnetic flux through the loop.

To understand this, let's consider Faraday's law of electromagnetic induction, which states that the emf induced in a closed loop is equal to the rate of change of magnetic flux through the loop. Mathematically, this is expressed as:

emf = -dΦ/dt

where emf is the electromotive force, dΦ/dt is the rate of change of magnetic flux, and the negative sign indicates the direction of the induced current.

Now, let's look at a rectangular loop placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnetic flux (Φ) through the loop is given by the product of the magnetic field (B) and the area (A) of the loop, i.e., Φ = B * A.

If the magnetic field remains constant, the rate of change of magnetic flux (dΦ/dt) is zero, and hence, no emf is induced. However, if there is a change in either the magnetic field or the area of the loop, the rate of change of magnetic flux becomes non-zero, and an emf is induced.

For example, if the magnetic field is constant but the area of the loop is changing (e.g., if the loop is being stretched or compressed), then the rate of change of flux is non-zero, resulting in an induced emf.

Therefore, the emf induced in a rectangular loop placed in a region with a perpendicular magnetic field is not determined solely by their perpendicularity, but rather by the rate of change of magnetic flux through the loop.