====== Faraday's Law and Measuring Earth's Magnetic Field ======
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 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 //experimental physics//.
====== Induced Current In A Loop ======
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 seemingly simple phenomena can be found in a wide range of applications ranging from production and detection of magnetic fields, to the wireless chargers now available for charging you phone.
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 $\Phi = B \cdot A$.
====== Apparatus ======
This photo shows what the apparatus at your station looks like and what your station should look like at the end of lab.
{{ phylabs:lab_courses:phys-120_130-wiki-home:winter-experiments:pxl_20220228_182225475.jpg?400 |}}
You have the following equipment at your disposal.
* A function generator.
* An oscilloscope.
* A Heath Coil.
Each Heath coil has 3400 windings with a total resistance of 62.5Ω.
* An iOLab device and computer loaded with the software to use the iOLab to collect data.
* An approximately 20cm to 30cm length of wire.
===== IOlab Magnetic Sensor Location =====
It may be useful to know the precise location of the magnetometer in the iOLab device. The image below gives that information.
{{ phylabs:lab_courses:phys-120_130-wiki-home:winter-experiments:iolab-mag.png?400 |}}
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.
Here is the link to the Google Doc for this lab.
[[https://docs.google.com/document/d/1ulWcHjSl1T59lXb_cwegBf7TIIRfkdKPyResTQKp82Q/copy]]
===== Test Case =====
An important part of developing any experimental technique is testing your assumptions and apparatus by making measurements where you know what the result should be. For this lab you will use an iOLab device to directly measure the induced $\epsilon$ in a loop of wire, or possibly $N$ loops of wire. Once $\epsilon$ has been measured and the parameters of the loop(s) are known you can calculate $\vec B$.
To verify that your assumptions are correct, and that you know how to properly use your apparatus to make the measurements, setup and perform the following experiment.
What you want to do is the following.
- Generate a time varying magnetic field, of known magnitude and frequency, along a known axis.
- Place a single coil of wire in this magnetic field so that the time varying flux will induce an $\epsilon$ in it.
- Measure $\epsilon$ and calculate B to verify everything is working as you expect it to.
The function generator, Heath coil, scope and iOLab can be used to accomplish #1. The wire and iOLab can be used for #2.
====== Two Ways To Measure B ======
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 $\Phi = B \cdot A$. 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 $\Phi = B A sin(\omega t)$. Differentiating Faraday's Law with respect to time then yields $emf = −ωNBA cos(ωt)$. The iOLab and coil can be setup to simultaneously record the rotational velocity of the coil (about the proper axis) and the induced $\epsilon$ at the same time.
Based on the above, one way to measure a constant magnetic field would be to rotate the wire loop while simultaneously measuring the induced $\epsilon$ and angular velocity $\omega$.
Another way to approach the problem is to rotate the iOLab in the $\vec B$ field between two known angles $\theta_{1}$ and $\theta_{2}$. 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 = |^{\theta(t2)}_{\theta(t1)} NBA cos(\theta) = 2NBA $
Using the above if you can record $\epsilon$ while rotating the coil between two known angular positions and then integrate $\epsilon$ you can get the magnitude of $\vec B$.
Now work out a plan for how you intend to measure both the horizontal and vertical components of the net magnetic field in the lab. 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 you need to test your planed technique using a magnetic field of known strength. Use the Heath coil to produce a constant magnetic field of known strength. Use your planned technique(s) to measure the known field in order to verify that your plan will work, compare your measured field to the known value.
====== Measurement of Earth's Magnetic Field ======
Now that you are familiar with how magnetic fields can be created and detected by current carrying loops of wire, use this knowledge to come up with a technique for measuring the vertical and horizontal components of the Earth's magnetic field using one or more loops of wire connected to the G- and G+ inputs on the IOlab.
In order to estimate the uncertainty in your measured values try to make use of more than one measurement technique for each component of the Earth's magnetic field as well as averaging multiple measurements.
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.
{{phylabs:lab_courses:phys-120_130-wiki-home:winter-experiments:chicago_mag.png?800}}