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.

 

[asteriatic]

I would approach this problem by first understanding the concept of induced emf. Induced emf is the electromotive force that is generated in a conductor when it is exposed to a changing magnetic field. This force is caused by the movement of free electrons in the conductor, which creates a current.

In this problem, we are given a magnetic field B that is varying with time and we need to find the magnitude of the induced emf in a circular loop at a specific location. To do this, we can use Faraday's law of induction, which states that the induced emf in a closed loop is equal to the negative rate of change of magnetic flux through the loop.

In mathematical terms, this can be written as:

ε = -dΦ/dt

Where ε is the induced emf, Φ is the magnetic flux, and t is time.

In order to calculate the magnetic flux, we need to first determine the magnetic field at the location of the circular loop. We are given the magnetic field B in terms of x, y, and z coordinates, so we can use the equation B = ∇ x A, where A is the vector potential.

In this case, A = (0, x3z2, -x2z3)sinωt, so B = (0, 3x2z2sinωt, -2xz3sinωt).

Next, we need to find the area vector for the circular loop. Since the loop is in the z = a plane, the area vector is in the k direction, and its magnitude is equal to the area of the circle, which is πR2. Therefore, the area vector is given by A = (0, 0, πR2).

Now, we can calculate the magnetic flux through the circular loop as:

Φ = ∫B . dA = ∫(0, 3x2z2sinωt, -2xz3sinωt) . (0, 0, πR2) dA

= ∫-2xz3sinωt dA

= -2sinωt ∫x(πR2)z3 dA

= -2sinωt ∫x(πR2)z3 rdrdθ

= -2sinωt (πR2) ∫∫xrcosθrdrdθ

=