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

Apparatus

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.

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.

If you took the PHYS141 labs last quarter you are already familiar with the iOLab device and software. If you need a refresher here is a link to the iOLab device introduction page.

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.

Group Notebook For Faradays Law

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

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 = - \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$.

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.

Assignment submission and grading

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!

Your individual summary and conclusions are due 48 hours after the end of the lab.