Direction of Induced EMF (Faradays Law) Confusion

[jegues]

Homework Statement

See figure attached.

Homework Equations

The Attempt at a Solution

If we attempt to solve part a) in this question we encounter several confusions.

If we assume i(t) to be a positive current, the magnetic flux will flow through the core in a counter clockwise direction.

How are we supposed to deduce the direction of the induced voltage,

$$e_{ind}$$

?

If you tell me whether or not i(t) is increasing or decreasing or remaning constant I can then tell you how eind relates to v.

Are we just supposed to assume it is either increasing or decreasing? If so, the relation to v to eind will have a sign difference in each case.

Assume i(t) is increasing,

$$e_{ind} = -v = -\frac{d\psi}{dt}$$

Assume i(t) is decreasing,

$$e_{ind} = v = -\frac{d\psi}{dt}$$

As you can, depending on which case I assume, my expression of v will change by a negative sign.

This will affect the answer in part b).

How do we know which one to choose?

 

[Spinnor]

Looking at the picture, I take the up arrow to indicate positive current, an increasing current causes the polarity as indicated in the coil N2? So for part b increasing current leads to positive voltage? Any others?

 

[jegues]

an increasing current causes the polarity as indicated in the coil N2

Thus,

$$e_{ind} = v = -\frac{d\psi}{dt} = -\frac{250\mu_{0}N_{1}N_{2}d^{2}}{a}\frac{di(t)}{dt}$$

Why does my expression for v turn out to be negative while theirs is positive?

I was told that the expression for eind is always,

$$e_{ind} = -\frac{d\psi}{dt}$$

I agree that the positive induced current, if i(t) is increasing, will flow from the positive terminal of v to the negative terminal of v.