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.

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

You have the following equipment at your disposal.

• An iOLab device and computer loaded with the software to use the iOLab to collect data.
• The purple wire wrapped around the body of the iOLab has 10 turns.

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.

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 post 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 post lab assignment discuss the following.

Your post lab assignment is due 48 hours after lab.