A uniform magnetic field is restricted within a region of radius r. The magnetic field changes with time at a rate `(dB)/(dt)`. Loop 1 of radius R `gt` r encloses the region r and loop 2 of radius R is outside the region of magnetic field as shown in figure. Then, the emf generated is

Explanation

(a) To calculate the field magnitude E, we apply Faraday.s law in the form of Eq. 30-22. We use a circular path of integration with radius `rleR` because we want E for points within the magnetic field. We assume from the symmetry that `vecE` in Fig. 30-22b is tangent to the circular path at all points. The path vector `dvecs` is also always tangent to the circular path, so the dot product `vecE.dvecs` in Eq. 30-22 must have the magnitude E ds at all points on the path. We can also assume from the symmetry that E has the same value at all points along the circular path. Then the left side of Eq. 30-22 becomes
`intovecE.dvecs=intoEds=Eintods=E(2pir)` (30-25)
(The integral `into` ds is the circumference `2pir` of the circular path.)
Next, we need to evaluate the right side of Eq. 30-22. Because `vecB` is uniform over the area A encircled by the path of integration and is directed perpendicular to that area, the magnetic flux is given by Eq. 30-2:
`phi_(B)=BA=B(pir^(2))` (30-26)
Substituting this and Eq. 30-25 into Eq. 30-22 and dropping the minus sign, we find that
`E(2pir)=(pir^(2))(dB)/(dt)`
or `E=r/2(dB)/(dt)`
or `E=r/2(dB)/(dt)` (30-27)
Equation 30-27 gives the magnitude of the electric field at any point for which `rle R` (that is, within the magnetic field). Substituting given values yields, for the magnitude of `vecE` at r = 5.2 cm,
`E=((5.2xx10^(2)m))/2(0.13T//s)`
= 0.0034 V/m= 3.4mV/m.
(b) We can now write
`phi_(B)=BA=B(piR^(2))` (30-28)
Substituting this and Eq. 30-25 into Eq. 30-22 (without the minus sign) and solving for E yield
`E=(R^(2))/(2r)(dB)/(dt)`
Because E is not zero here, we know that an electric field is induced even at points that are outside the changing magnetic field, an important result that (as you will see in Section 31.9) makes transformers possible.
With the given data, Eq. 30-29 yields the magnitude of `vecE` at r = 12.5 cm:
`E=((8.5xx10^(-2)m)^(2))/((2)(12.5xx10^(-2)m))(0.13T//s)`
`=3.8xx10^(-3)V//m=3.8mV//m=3.8mV//m`
Because Eis not zero here, we know that an electric field is induced even at points that are outside the changing magnetic field, an important result that (as you will see in Section 31.9) makes transformers possible. With the given data, Eq. 30-29 yields the magnitude of E at r = 12.5 cm: