Question

In a model AC generator, a 505 turn rectangular coil 8.0 cm by 30 cm rotates at 120 rev/min in a uniform magnetic field of 0.59 T.
(a) What is the maximum emf induced in the coil?

(b) What is the instantaneous value of the emf in the coil at t = (π/32) s? Assume that the emf is zero at t = 0.

(c) What is the smallest value of t for which the emf will have its maximum value? s

Answer 1

• (a) Maximum emf: 90 V (2 sig. fig.)
• (b) Emf at π/32 s: 85 V.
• (c) t = 0.125 s.

Explanation

(a)

The maximum emf in the coil depends on

• the maximum flux linkage through the coil, and
• the angular velocity of the coil.

Maximum flux linkage in the coil:

$\phi_\text{max} = B\cdot A\cdot N = 0.59\;\text{T}\times(0.08 \times 0.30)\;\text{m}^{2} \times 505 = 7.2\;\text{Wb}$.

Frequency of the rotation:

$f = 120\;\text{rev}\cdot\text{min}^{-1} = 2 \;\text{rev}\cdot\text{s}^{-1}$.

Angular velocity of the coil:

$\omega = 2\;\pi\;\text{rev}^{-1}\times 2\;\text{rev}\cdot\text{s}^{-1} = 4 \pi \;\text{s}^{-1}$.

Maximum emf in the coil:

$\epsilon_\text{max} = \omega\cdot\phi_\text{max} = 4\;\pi \times 7.2\;\text{Wb} = 90\;\text{V}$.

(b)

Emf varies over time. The trend of change in emf over time resembles the shape of either a sine wave or a cosine wave since the coil rotates at a constant angular speed. The question states that emf is "zero at t = 0." As a result, a sine wave will be the most appropriate here since $\sin{0} = 0$.

$\displaystyle \epsilon(t) = \epsilon_\text{max}\cdot \sin{(\omega\cdot t)}$.

Make sure that your calculator is in the radian mode.

$\displaystyle \epsilon\left(\frac{\pi}{32}\right) = 90\;\text{V}\times \sin\left(4\;\pi\times \frac{\pi}{32}\right) = 85\;\text{V}$.

(c)

Consider the shape of a sine wave. The value of $\displaystyle \sin\left(\omega \cdot t\right)$ varies between -1 and 1 as the value of $t$ changes. The value of $\epsilon$ at time $t$ depends on the value of $\sin(\omega \cdot t)$.

$\sin(\omega \cdot t)$ reaches its first maximum for $t\ge 0$ when what's inside the sine function is equal to $\pi/2$.

In other words, the first maximum emf occurs when

$\omega \cdot t = \dfrac{\pi}{2}$,

where

$\sin{\omega \cdot t} = 1$,

and

$\epsilon = \epsilon_\text{max}$.

$\displaystyle t = \frac{\pi}{2}/\omega = \frac{1}{8} = 0.125\;\text{s}$.