Day 4: Subtracting Background Fields to Reveal Desired Field
Student Pages

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 combined images.

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.

1. 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.
2. 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?
3. 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.

1. Why is it reasonable to expect the magnitude of the field to obey an equation of the form B=Adn?
2. Would you expect n to be positive or negative? Give a logical argument for your answer.
3. Why would this experiment be a great deal more complicated if you hadn't aligned your magnet perpendicular to the north-south direction?
4. What assumption have you made about the magnetic field of the Earth throughout this experiment?
5. 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.

Homework:

Predict what the magnetic field map of two bar magnets would look like for the following orientations.

```

S

-----

|   |

a)  |   |        |---------------------|

|   |      N |                     | S

|   |        |---------------------|

-----

N

|---------------------|        |---------------------|

b)   N |                     | S    N |                     | S

|---------------------|        |---------------------|

|---------------------|        |---------------------|

c)   N |                     | S    S |                     | N

|---------------------|        |---------------------|

N            S

-----        -----

|   |        |   |

d)  |   |        |   |

|   |        |   |

|   |        |   |

-----        -----

S            N

N            N

-----        -----

|   |        |   |

e)  |   |        |   |

|   |        |   |

|   |        |   |

-----        -----

S            S

```

Lesson Development/Writing: Ed Eckel
Web Design: Theresa Valentine
Last Updated: 8/24/2000