How Is EMF Induced in Different Magnetic Fields?

[popo902]

Homework Statement

Figure below shows two circular regions R1 and R2 with radii r1 = 21.7 cm and r2 = 30.5 cm. In R1 there is a uniform magnetic field of magnitude B1 = 51.3 mT directed into the page, and in R2 there is a uniform magnetic field of magnitude B2 = 75.7 mT directed out of the page (ignore fringing). Both fields are decreasing at the rate of 11.6 mT/s. Calculate intergral of E dot ds, (in mV) for (a) path 1, (b) path 2, and (c) path 3.

here's the diagram i took from the problem
http://i36.photobucket.com/albums/e47/jo860/pict_30_36.gif

Homework Equations

i know that integral is V, but not potential
its the EMF
EMF also equals d[tex]\phi[/tex]/dt
EMF = 2[tex]\pi[/tex]rE

The Attempt at a Solution

i basically have no idea how to go about starting this.
i know i have dB/dt and some radii.
i understand that since they are both decreasing and pointing in opposite ways,
the induced emf will make a "current" to counteract the change
so for for each B field, the emf would make a B field in that same direction:
R1 emf is down and R2 emf is up
i don't know where to begin with the calculations though :(
can someone point me in the right direction?

 

[anathema]

Thank you for posting your question. This is an interesting problem that involves calculating the induced EMF along different paths in a changing magnetic field.

To start, let's review some relevant equations. As you mentioned, the induced EMF can be calculated using Faraday's law: EMF = -N(dΦ/dt), where N is the number of turns in the coil and Φ is the magnetic flux. In this case, we have a uniform magnetic field, so we can simplify this equation to EMF = -NBAcosθ, where B is the magnetic field strength, A is the area of the coil, and θ is the angle between the magnetic field and the normal vector to the coil.

Now, let's consider each path individually. For path 1, the coil is located entirely within R1, so we only need to consider the magnetic field in R1. Using the equation above, we can calculate the induced EMF as:

EMF1 = -N1B1Acosθ1

Where N1 is the number of turns in the coil, B1 is the magnetic field strength in R1, A is the area of the coil, and θ1 is the angle between the magnetic field and the normal vector to the coil. In this case, the magnetic field is directed into the page, so θ1 = 90°. We can also calculate the area of the coil using the formula for the area of a circle: A = πr^2, where r is the radius of the coil.

For path 2, the coil is located within both R1 and R2, so we need to consider the magnetic field in both regions. We can calculate the induced EMF as:

EMF2 = -N1B1Acosθ1 - N2B2Acosθ2

Where N1 and N2 are the number of turns in the coil in R1 and R2 respectively, B1 and B2 are the magnetic field strengths in R1 and R2 respectively, A is the area of the coil, and θ1 and θ2 are the angles between the magnetic fields and the normal vector to the coil. In this case, θ1 = 90° and θ2 = 270°, since the magnetic field in R2 is directed out of the page.

For path 3, the coil is located entirely within R2