Induced EMF Problem 6.6 Solution Help

[izelkay]

Homework Statement

It's Problem 6.6 on this page:
http://web.ece.ucdavis.edu/~lxgliu/eec130a/2012winter/hw9s.pdf

It has the solution and everything. I don't understand part (b) though.

Homework Equations

The Attempt at a Solution

[/B]
For part (b), I thought the direction of the induced current flow would flow as to oppose the change in the magnetic flux. The change in the magnetic flux here is dΦ/dt = -6.9sin(2πx10^4t) (V).

They chose t = 0, but on an interval t =0 to say t = π/2, the sine function in the magnetic flux is greater than 0, which means dΦ/dt is negative and less than 0 right? Which in turn means the flux is decreasing. However, the flux is going INTO the page (-x direction) so shouldn't a decreasing flux mean it's getting less negative? So then the induced current I think should flow CLOCKWISE to oppose this change. This isn't what the solution says though, and I can't understand the way they did it. Can someone explain the fault in my reasoning?

 

[gleem]

The solution is correct. the flux initially into the page is decreasing so the induced current will be in a direction so as to oppose this change i.e. to increase the flux. by producing a field coming out of the page, counter clockwise. Your derivative is negative but it is deceasing into the page so the induced current mus be increasing the flux out of the page.

 

[izelkay]

The solution is correct. the flux initially into the page is decreasing so the induced current will be in a direction so as to oppose this change i.e. to increase the flux. by producing a field coming out of the page, counter clockwise. Your derivative is negative but it is deceasing into the page so the induced current mus be increasing the flux out of the page.

Okay so like they said the initial flux is in the -x direction into the page and is maximum at t = 0. It varies as a cosine initially, so it is decreasing. Here's where my confusion is:

in the -x direction,
cos(0) = -1, maximum
cos(pi/4) = -1/√2

Is it not getting more positive (i.e, closer to 0) and starting to try to point in the +x-direction?

 

[izelkay]

bump

 

[gleem]

The induced current (CCW) is producing a field out of the page.