This lab is devoted to understanding how to use the concept of magnetic induction to design and test a method of measuring the ambient magnetic field in the lab (which will be pretty close to but not exactly the same as the value of the Earth's magnetic field in Chicago).

You will use Faraday's Law to measure the induced emf ($\epsilon$) in a coil of wire. Part of the lab is ensuring you know how to use the apparatus at your disposal to create a magnetic field of known value and then measure that known field using Faraday's Law. This exercise will give you the knowledge you need to devise a way of using a coil of wire to measure the horizontal and vertical components of the ambient field in the lab. Once you have a plan for measuring the ambient field, you will verify your technique by again creating a magnetic field of known value and measuring it in the same manner as you plan to use for the ambient field measurement. Once you have established that your technique is sound and produces the results you expect, then you will proceed to make the final measurement of the ambient field.

Pedagogically speaking, measuring the ambient field is not the point of the lab. It is the end point of the experimental task you have been given. But what we are teaching you is how to figure out for yourself how to use your physics knowledge (i.e. Faraday's Law) and common lab apparatus to design, test and execute an experiment without being told explicitly what to do every step of the way. Said another way, we are teaching how to do physics.

Ampere's Law can be used to show that passing an electrical current ($I$) through a loop of wire with radius $R$ produces a magnetic field ($\vec B $) given by,

$\vec B = \frac{\mu_{o} I}{2R}$

where $\mu_{o} = 4\pi \times 10^{-7} Tm/A$.

This relatively straight forward phenomena can be found in a wide range of applications ranging from production and detection of magnetic fields, to wireless chargers.

Faraday's Law shows how a time varying magnetic flux ($\Phi$) induces an electro magnetic force ($\epsilon$) in $N$ loops of wire as,

$\epsilon = -N \frac{d\Phi}{dt}$

where $\Phi = \int B \cdot dA$.

This photo shows what the apparatus at your station looks like and what your station should look like at the end of lab.

You have the following equipment at your disposal.

• A DC power supply.
• A Heath Coil.
• An iOLab device and computer loaded with the software to use the iOLab to collect data.
• An approximately 20cm to 30cm length of wire.

It may be useful to know the precise location of the magnetometer in the iOLab device. The image below gives that information.

In addition there are various ring stands, rods, clamps, spools of wire and string, tape, rulers, etc in the room which you can make use of.

Let us now consider how to use magnetic induction in a wire loop as part of an experiment to measure the strength of the horizontal and vertical components of the Earth's magnetic field here in Chicago.

The Earth's $\vec B$ field is constant, at least on time scales relevant to this lab. According to Faraday's Law the induced $\epsilon$ is proportional to $\frac{d\Phi}{dt}$ where $\Phi = \int B \cdot dA$. So if $\vec B$ is constant we need to find a way to vary the area $A$ of our loop in order to induce an $\epsilon$.

Here are some tips to get you started.

Not only does the iOLab have the differential input amplifier (inputs G- and G+) for reading the small induced $\epsilon$ from a wire loop, it also has built in three gyroscopes on three axes. Using these two features along with the fact that the device uses wireless transmission to the computer opens up some interesting possibilities. The built in gyroscopes can be used to measure the rotational motion of the body of the iOLab. If a coil of wire is wrapped around the body of the device, which is then spun with angular velocity $\omega$ around an axis orthogonal to the axis of the magnetic field component you want to measure, the time dependent dot product of the magnetic field and area vectors becomes $\Phi = B A cos(\omega t)$. Differentiating Faraday's Law with respect to time then yields $\epsilon = ωNBA sin(ωt)$. The peak emf $\epsilon_{peak}$ occurs when $sin(\omega t)$ = 1. Therefor if one can record the induced emf and the angular velocity of the coil about an axis orthogonal to the normal vector of the area $\vec{A}$, $B$ can be found given Knowledge of $N$ and $A$.

Another approach is to make use of the integral form of Faraday's Law. If we rotate the coil from $\theta = 0^\circ$ to $theta = 180^\circ$ integrating Faraday's law gives $ \int^{t_{2}}_{t_{1}} \epsilon dt = - \int{ \frac{d \phi}{dt}dt} = NBA( cos(\theta(t1)) - cos(\theta(t2))$. If we rotate the iOLab from $\theta (t1) = 0^\circ$ to $\theta (t2) = 180^\circ$ we get $ \int^{t_{2}}_{t_{1}} \epsilon dt = 2NAB$.

Work out the details of a plan for how you intend to measure both the horizontal and vertical components of the net magnetic field in the lab using both. Confirm with your TA that your plans are reasonable before proceeding.

The field in the lab should be close to that of the Earth's magnetic field in Chicago, whose approximate value is given later in this wiki. You may have to use different techniques for each component of the field. In addition to the apparatus at your station, you should feel free to use other items in the lab such as ring stands, clamps, additional wire, etc.

Once you have a plan in place for both measurements it is always a good idea to test that your basic idea is workable in terms of the apparatus you are using. This also gives you an opportunity to make sure you are doing all of the necessary calculations correctly.

Your station has a pair of Heath coils and a function generator. By setting up the function generator to output a sine wave and sending that signal through a pair of coils you can generate a time varying magnetic field between them. By placing your sensor (i.e. the loop of wire attached to the iOLab device) in between the coils you should be able to measure the induced emf.

Ideally we would make use of a pair of Helmholz coils which are designed to produce a uniform magnetic field which can easily be calculated from knowledge of the current and the coil dimensions. This would make for a nice, first principles test. Unfortunately all of the Helmholz coils are being used in other lab courses at this time, so we will have to make due with the coils provided. It is not easy to calculate the field you get from these coils, but the iOLab does have a magnetic field sensor which can be used to directly measure the field produced by the coils when driven by a sine wave.

Use the apparatus provided to perform a test measurement of both of your planned techniques.

The Earth's magnetic field in Chicago can be estimated using the NOAA Magnetic Field Calculator https://www.ngdc.noaa.gov/geomag/calculators/magcalc.shtml?#igrfwmm as shown in the screen shot below.

Some context for your out of lab assignment. During the lab you performed two measurements to determine the value of an unknown quantity. There is no known value for the magnetic field in KPTC-216 to which you can compare your results to see if you “got the right answer”. This is the context for pretty much all physics research, both experimental and theoretical.

The purpose of this out of lab assignment is to communicate to another physicist (your TA) what you did, how you did it, what was the final result, why should they have confidence in it, and to do it in a manner which is clear and concise.

For your individual summary discuss the following.

Your individual summary is due 48 hours after lab.