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.

Step-by-step 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}*(0.08 * 0.30)\;\text{m}^(2) * 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)* 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 * 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((\pi)/(32)\right) = 90\;\text{V}* \sin\left(4\;\pi* (\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 = (\pi)/(2)`,

where

`sin(\omega \cdot t) = 1`,

and

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

`\displaystyle t = (\pi)/(2)/\omega = (1)/(8) = 0.125\;\text{s}`.