Repeat Q: Ignore. Show what the magnitude of induced emf

[auk411]

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 -∫∫x²a³dydx.

The make the change to polar:

-aK³∫∫(rcosθ)²r dr dθ = -(K/4)a³R⁴∫cosθsinψt dθ.This doesn't get me anywhere.

 

[anathema]

To find the magnitude of the induced emf, we need to use Faraday's Law of Induction, which states that the induced emf is equal to the rate of change of magnetic flux through a surface.

In this case, the surface is a circle of radius R in the plane z = a, with its center at x = 0, y = 0, z = a. We can use the formula for magnetic flux, Φb = ∫B . dA, to find the magnetic flux through this circle.

Since the magnetic field B is given as B = K(x3z2, 0, -x2z3)sinωt, we can rewrite it as B = K(rsint, 0, -rcost)sinωt in polar coordinates, where r is the distance from the center of the circle and θ is the angle from the positive x-axis.

The magnetic flux through the circle is then given by:

Φb = ∫B . dA = ∫∫B . dS = ∫∫Bz dS = ∫∫-rcostsinωt r dr dθ = -ωR2sinωt ∫∫rcost r dr dθ

Since the circle has a radius R and is in the plane z = a, we can set the limits of integration as 0 ≤ r ≤ R and 0 ≤ θ ≤ 2∏. This gives us:

Φb = -ωR2sinωt ∫∫rcost r dr dθ = -ωR2sinωt ∫02∏∫0Rrcost r dr dθ

= -ωR2sinωt ∫02∏(R2/2)cost dθ = -ωR2sinωt (R2/2)2∏ = -πωR4sinωt

Finally, using Faraday's Law of Induction, we can say that the induced emf, ε, is equal to the rate of change of magnetic flux, or ε = -dΦb/dt. Taking the derivative with respect to time, we get:

ε = -dΦb/dt = πωR4cosωt

Therefore, the magnitude of the induced emf is given by:

|ε| = πωR4|cosωt| =