Question

You are shipwrecked on a deserted tropical island. You have some electrical devices that you could operate using a generator but you have no magnets. The earth's magnetic field at your location is horizontal and equal to 8.0×10−5T, and you decide to try to use this field for a generator by rotating a large circular coil of wire at a high rate. You need to produce ϵmax = 9.0V and estimate that you can rotate the coil at 30rpm by turning a crank handle. You also decide that to have an acceptable coil resistance, the maximum number of turns the coil can have is 2000.

A. What area must the coil have?
B. If the coil is circular, what is the maximum translational speed of a point on the coil as it rotates?

Answer 1

(A) The area of coil must be = 17.9
`m^(2)`

(B) Maximum translation speed of a point is = 7.5
`(m)/(s)`

Step-by-step explanation:

Given :

Magnetic field
`B = 8 * 10^(-5)` T

Induced emf = 9 V

Angular frequency
`\omega =`
`2\pi f`

No. of turns
`N = 2000`

Where
`f = (30)/(60) = (1)/(2)` so
`\omega = \pi`

(A)

From the laws of electromagnetic induction,

Induced emf =
`- N(d \phi)/(dt)`

Where
`\phi =` magnetic flux, in our case magnetic flux
`\phi = BA \cos \omega t`

Max. induced emf =
`N \omega BA`

Therefore area of loop is given by,

`A = (9 )/(2000 * \pi * 8 * 10^(-5) )`

`A = 17.9`
`m^(2)`

(B)

Coil is circular so maximum translation speed is given by,

`v = r \omega`

Where
`r =` radius of circle
`r = \sqrt{(A)/(\pi ) }`

`v = \sqrt{(A)/(\pi ) } \omega`

`v = \sqrt{(17.9)/(\pi ) } \pi`

`v = 7.5`
`(m)/(s)`