Show what the magnitude of induced emf

[auk411]

Someone who knows what they are talking about: Show what the magnitude of induced emf

Homework Statement

Consider a magnetic field B = K(x³z²,0, -x²z³)sinωt in the region of interest, where K and ω are positive constants and t is variable time. Show that the magnitude of the induced emf around a circle R in the plane z = a with its center at x = 0, y = 0, z = a is:
ε = (K/4)∏a³R⁴ωcosωt

Homework Equations

Flux_b = ∫B . dA

The Attempt at a Solution

Since the normal vector points in the k direction, we only have to worry about B_z.

∫B_zdydx. So -∫∫(sinwt)x²a³dydx.

The make the change to polar:

-aK³∫∫(sinwt)(rcosθ)²r dr dθ = -(K/4)a³R⁴∫cosθsin(wt) dθ.

This doesn't get me anywhere. I'm not really sure what I'm supposed to be integrating over, which is probably why I'm stuck.

 

[I like Serena]

Hi auk411!

Let's start with Bz.
Can you write it down separately?
It seems you did not copy it correctly.

Secondly you did not bring the constant out of the integral properly.

Furthermore in polar coordinates you would integrate r from 0 to R, and theta from 0 to 2pi.

And for the tip: rewrite (cosθ)² using cos2θ.

 

[auk411]

Hi auk411!

Let's start with Bz.
Can you write it down separately?
It seems you did not copy it correctly.

Secondly you did not bring the constant out of the integral properly.

Furthermore in polar coordinates you would integrate r from 0 to R, and theta from 0 to 2pi.

And for the tip: rewrite (cosθ)² using cos2θ.

(cosθ)² using cos2θ.[/QUOTE] ... huh, what trig identity are you using.

this still doesn't answer the most pressing question. t varies, theta varies and r varies. we have 3 varying variables in a DOUBLE integral. I see no way to reduce them to two. How do I get around this.

 

[I like Serena]

First things first.
You appear to have skipped my question, so I'll answer it myself:

B_z = -Kx²a³sinωt

This is not what you used.

(cosθ)² using cos2θ. ... huh, what trig identity are you using.

cos(2θ) = 2 cos²θ - 1

this still doesn't answer the most pressing question. t varies, theta varies and r varies. we have 3 varying variables in a DOUBLE integral. I see no way to reduce them to two. How do I get around this.

No, you have 2 variables.
You appear to be thinking spherical coordinates, but you should be thinking in cylindrical coordinates.

z is constant at z=a.
Only the other 2 vary.