Aloop of wire of area 71 cm^2 is placed with its plane parallel to a 16 mt magnetic field. the loop is then rotated so that its plane is perpendicular to the field, in a time of 0.7 s. what is the average emf generated in the coil while the loop is rotating, in units of volts?

By Faraday's Law of Induction, the EMF [tex]\epsilon[/tex] induced in a coil (one loop) is equal to the rate of change in the magnetic flux [tex]\Phi[/tex] through the coil.

[tex]\displaystyle \epsilon(t) = \frac{d}{dt}(\Phi(t))[/tex].

Finding the average EMF in the coil is similar to finding the average velocity.

[tex]\displaystyle \text{Average}\; \epsilon = \frac{1}{t}\int_0^t \epsilon(t)\cdot dt[/tex].

However, by the Fundamental Theorem of Calculus, integration reverts the action of differentiation. That is:

[tex]\displaystyle \int_0^{t} \epsilon(t)\cdot dt = \int_0^{t} \frac{d}{dt}\Phi(t)\cdot dt = \Phi(t) - \Phi(0)[/tex].

Hence the equation

[tex]\displaystyle \text{Average}\; \epsilon = \frac{1}{t}\int_0^t \epsilon(t)\cdot dt = \frac{\Phi(t)- \Phi(0)}{t}[/tex].

Note that information about the constant term in the original function will be lost. However, since this integral is a definite one, the constant term in [tex]\Phi(t)[/tex] won't matter.

Apply this formula to this question. Note that [tex]\Phi[/tex], the magnetic flux through the coil, can be calculated with the equation

[tex]\Phi = B \cdot A \cdot N \; \sin{\theta}[/tex].

For this question,

[tex]B = \rm 16\; mT = 16\times 10^{-3}\; T[/tex] is the strength of the magnetic field.
[tex]A = \rm 71\; cm^{2} = 71\times \left(10^{-2}\right)^2 \; m^{2}[/tex] is the area of the coil.
[tex]N = 1[/tex] is the number of loops in the coil.
[tex]\theta[/tex] is the angle between the field lines and the coil.
At [tex]\rm 0\;s[/tex], the field lines are parallel to the coil, [tex]\theta = 0^{\circ}[/tex].
At [tex]\rm 0.7\; s[/tex], the field lines are perpendicular to the coil, [tex]\displaystyle \theta = 90^{\circ}[/tex].

Initial flux: [tex]\Phi(0)= 0[/tex].

Final flux: [tex]\Phi(0.7) = \rm 1.1136\times 10^{-4}\; Wb[/tex].

Average EMF, which is the same as the average rate of change in flux:

[tex]\displaystyle \frac{\Phi(0.7) - \Phi(0)}{0.7} \approx\rm 1.62\times 10^{-4}\; V[/tex].