Question 15.3: Suppose that we have a toroidal core with μr = 5000, R = 10 ...

First, the permeability of the core material is
μ = μ_rμ_0 = 5000 × 4π × 10^{−7}
Using Equation 15.18, we compute the flux:

Φ = BA = \frac{μNI}{2πR} πr^2 = \frac{μNIr^2} {2R}      (15.18)
Φ = \frac{μNIr^2}{ 2R} = \frac{5000 × 4π × 10^{−7} × 100 × 2 \sin(200πt) × (2 × 10^{−2})^2}{ 2 × 10 × 10^{−2}}
= (2.513 × 10^{−3}) sin(200πt) Wb
The flux linkages are
λ = NΦ
= 100 × (2.513 × 10^{−3}) sin(200πt)
= 0.2513 sin(200πt) weber turns
Finally, using Faraday’s law (Equation 15.8), we can find the voltage induced in the coil by the changing field:

e = \frac{dλ} {dt}        (15.8)
e = \frac{dλ}{dt} = 0.2513 × 200π cos(200πt)
= 157.9 cos(200πt) V