Day 4: Subtracting Background Fields to Reveal Desired Field
Yesterday, we mapped the magnetic field of a bar magnet and of the earth. We
said the Earth was an extra influence and something we wanted to remove from
the bar magnet map so we could get the true map of a dipole.
Today we will perform two different operations that remove the effect of the
earth to reveal the actual field of a dipole.
Activity 1: Visual Subtraction
Tape the Earth Magnet Map to a window or light table.
Carefully overlay the Bar Magnet Map over the Earth Map and tape in position.
Compare the measured Bar Magnet Map to the Earth Map. Note grid points
with the following characteristics:
Direction arrows are nearly parallel or anti-parallel?
Direction arrows are perpendicular?
Describe how smoothly the Bar Magnet arrows change direction in relation to
the Earth arrows.
Describe patterns to how the arrows change direction as you move around the
While we cannot give exact numbers for the strength of the bar magnet field at
any point, we can see from the arrows that it gets weaker fairly rapidly compared
to the relatively constant strength of the earth's contribution.
Activity 2 Mathematical Subtraction
In this activity, we will numerically compare observations against a reference
and try to understand the influence of the Earth on the actual observation.
Find the direction of the ambient magnetic field with the magnetometer. Check in
a few places to make sure that there aren't any local disturbances caused by
pieces of metal or currents. Tape down two pieces of 8.5x11 paper with their
long axis aligned with the north-south direction. Draw a line in dark ink on the
paper marking the North-South direction. You will make measurements along this
line. Tape a magnet to the paper at the end of the North-South line such that the
line forms a perpendicular bisector to the magnet. Every 5 centimeters, place a
point on this line. Use the magnetometer to observe the magnetic field direction
at each point. Draw a directed line segment representing the direction of the
magnetic field at that point.
- How do you know that the North-South line points north? Suppose the
magnets were not marked N and S; could you determine which direction was
north? Explain your answer.
- Each of the arrows that you draw at each point represent the direction
of the magnetic field. Why can't you also infer the strength of the magnetic
field from those lines as well?
- What contributes to the direction of the arrows you drew? That is, what
objects are contributing to what the magnetometer is measuring?
Make a table with 4 columns labeled distance, x, y, ratio (y/x). Distance is
the position of the observation point relative to the bar magnet.
Draw a right triangle on your paper using the arrow you drew as the hypotenuse
and the North-South line as one side. The third side is in the East-West direction
(of course!) Call the North-South side the x-direction and the East-West side the
y-direction. Measure the length of the sides x and y (in cm) for each triangle and
record your results in the table. Calculate and record the ratio (y/x) in the table.
We choose the geometry carefully to insure the magnetic field from just the bar
magnet to be entirely perpendicular (East-West) to the line. The line itself
represents the North-South field from the earth. Thus, the ratio (y/x) is the
ratio of the Bar magnet contribution divided by the Earth contribution.
Another way of thinking about the ratio is that it gives us the strength of the
magnet in units of the earth's field. The earth's field is nearly constant in
this observation, so we are simply dividing a measurement by a constant amount.
We haven't really changed the thing measured, just the idea of what one unit means.
We want to know how the magnitude of the bar magnet's field changes as you move
away from the bar magnet. The ratio (y/x) represents the magnitude of the bar
magnet field in terms of the constant strength of the earth's field. Thus, if
we make a graph of the ratio (y/x) as a function of the distance of the observation
point to the bar magnet, we can get a graphical representation of the changing
strength of the bar magnet field with position relative to the magnet.
In a computer spreadsheet program you can easily enter your data and calculate the
ratio (y/x). Then make a graph of the ratio (y/x) versus distance. Fit this graph
to a polynomial of the form B=Adn. We call this a power law as we are saying
the value of "B" is some power (exponential) of "d" times a constant. "A" is the
constant and "n" is the power of "d". We don't care what value you get for "A",
We are interested in the value of "n", which tells us how rapidly the magnetic field
strength of a dipole is reduced as you get farther from the dipole.
If you do not have access to a computer this analysis can still be done. You will
need to make this curve into a linear relationship you can find the power "n". You
assume a relationship exists in the data such that B=Adn. Take the log of both
sides of the equation.
I have included, in the 4th line, the general equation of a linear relationship on a
graph. The line above gives the 'linear' version of the power law form suggested
earlier. Notice how it matches the linear form (after having taken the logarithm
of both sides). You can see that y is equivalent to log(B), x is equivalent to
log(d), n is equivalent to the slope, and log(A) is equivalent to the y intercept.
Therefore if you graph log(B) vs. log d you will get a line where n is the slope and
log(A) is the y intercept.
- Why is it reasonable to expect the magnitude of the field to obey an equation of the form B=Adn?
- Would you expect n to be positive or negative? Give a logical argument for your answer.
- Why would this experiment be a great deal more complicated if you hadn't aligned your magnet perpendicular to the north-south direction?
- What assumption have you made about the magnetic field of the Earth throughout this experiment?
- How far from the magnet is the magnitude of the magnet's magnetic field and the Earth's magnetic field equal? Explain how you got your answer.
The suggested homework asks you to PREDICT the map you would observe for 5 different arrangements of 2 bar magnets. The subsequent activity to this one will involve testing the predictions.
Predict what the magnetic field map of two bar magnets would look like for
the following orientations.
a) | | |---------------------|
| | N | | S
| | |---------------------|
b) N | | S N | | S
c) N | | S S | | N
| | | |
d) | | | |
| | | |
| | | |
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e) | | | |
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Lesson Development/Writing: Ed Eckel
Web Design: Theresa Valentine
Last Updated: 8/24/2000