Question

Do you think it is possible to measure the earth's magnetic field in a given direction by looking at the induced emf in a coil? If so, how would you construct the experiement and determine its value?

Answer 1

Yes, it is possible.

You can make an square fit coil with N spires and connect it to an osciloscope (or better, to some kind of data aqquisition system).

If you point the coil axis in North - South direction, and suddenly spin it 180 degrees (so the axis now will point South - North), you will change the effective area of the coil facing the Magnetic field, the magnetic flux will change and will produce a EMF, you will record that EMF in your osciloscope.

What now?

I will use $\Phi$ for the magnetic flux, and $\varepsilon$ for the EMF

At the osciloscope you will see

$\varepsilon = -\frac{d \Phi}{dt} \Rightarrow \int_{0}^{\Delta t} \varepsilon dt= \Phi_{NS} - \Phi_{SN}$

Where indexs SN and NS denote the coil direction

now, if you know the time taken by the coil to spin 180 degrees, the left side of the equation will be simply the area under the osciloscope curve and you can obtain it by any technique you like (that's why it could be usefull a data aqquisition system)

The right side of the equation

$\Phi_{NS} - \Phi_{SN} = NBS \cos(0) \ - \ NBS \cos(180) = 2NBS$

where B is the magnetic field and S is the coil area (remember N is the number of spires in the coil)

Finally:

$B = \frac{\Phi_{NS} - \Phi_{SN}}{2NS} = \frac{\int_{0}^{\Delta t} \varepsilon dt}{2NS}$

An easy way to obtain a fixed wave form at the osciloscope is joining the coil to a electric motor of known angular speed, so you can know the time taken by the coil to accomplish a half turn. Equation will be the same.

If you align your coil N - S, you will be measuring the horizontal component of the earth magnetic field, if you want to know the magnitude, you have to know your actual lattitude.

Enjoy