Chabot College Physical Science Lab

Determining the Earth’s Magnetic Field

Scott Hildreth

 

 

Goal:

  • Determine the value of the horizontal component of Earth’s local magnetic field in the laboratory using a solenoid – a current-carrying loop -  as well as using oscillations of a magnet suspended in an artificial magnetic field.

 

 

Background:

You can determine the Earth’s local magnetic field (which we’ll denote as Be  for this lab) field using just a compass and a current-carrying solenoid, in at least three neat ways. Two easier methods use the predictable decrease in strength of the magnetic field with distance, and with careful observation you can get within 10% of the correct value.  A third method uses harmonic oscillation created by torques – turning forces -  on a small magnet to determine the local magnetic field; if timed correctly, you can achieve accuracy to within 5-10%.

 

 

Method 1: Bracketing the value for the Earth’s field using the Biot-Savart Law.

 

  1. Using a compass, determine the direction of Be in our lab at your station by observing the needle pointing towards “North”.  Take readings around the table, and note any positions where the compass needle deflects noticeably.  (Can you hypothesize about why such deflections might occur?  And can you test these hypotheses?  Try it!)  Make a brief sketch of the lab table and the direction of “North” as indicated by your team’s compass.

  2. Set up a 3400-turn coil, Ammeter (a digital multimeter), and power supply in series.  Turn on the power supply so that the current reads 0.20 Amps.  Align the resulting Bs field of the coil so that it points “South” opposite the direction you found in step (1).  Use the compass to verify that this direction indeed points “South” by holding it close to the coil along its centerline.

  3. Position a ruler extending from the center line of the coil towards the “South” direction.  Put the compass on the ruler, and slide it slowly away from the coil.  At some point, the needle of the compass will begin to shift from pointing “South” in the room (because the coil’s field Bs dominates) to pointing “North” in the room (because the Earth’s Be field dominates.)
    RECORD in your lab notes the three points on the ruler:

    • xa, where the compass still points directly towards “South” just before it begins to shift;

    • xb, where the compass points 90 degrees from “North”/”South”; and,

    • xc, where the compass has resumed pointing directly towards “North” in the room.

 

  1. We can calculate the field at any distance x along the centerline of a solenoid of average radius a and N turns, created by a current I, from an interesting relationship called the “Biot-Savart Law”:

Bs(x) =

 

Here, the value m0 defines the strength of magnetic fields in space.  Its typical value is 4p x 10-7 Tesla-meters/Amp, where a “Tesla” is the unit of measurement for a magnetic field.  You might have heard of a “Gauss” as another measure of magnetic fields.  One Tesla is 10,000 Gauss. 

The Earth’s field is typically about ½ of a Gauss (or 0.5 x 10-4 Teslas)  

MEASURE the outer and inner radii of the coil, determine the average radius “a”. 

 

DETERMINE three values of the net B field at your three measured distances  xa, xb, and xc using the known values for N, I, and a,    The calculation will be set up for you on the computer; you’ll need to enter the data and evaluate the answer.

ANSWER the following questions:

4a) Which point represents an upper limit to the value of the Earth’s field, i.e. where Be < Bs? 
4b) Which point represents a lower limit to the value of Earth’s field, i.e. where Be > Bs? 
4c) Approximately what is the value of the Earth’s field, and your uncertainty from this experiment?

 

  1. QUESTIONS: Where are the sources of error in this experiment?  What could you do to minimize the error?

 

Method 2: Determining the value for Be using angles and the Biot-Savart Law.

 

  1. Now rotate the coil so that it points 90 degrees from the North-South line you established earlier.  Again locate a ruler extending from the centerline of the coil.  Slide the compass along the ruler and estimate the angle of deflection for the compass needle at various distances from the coil.  MEASURE and RECORD values for at least 4 angles qi, and the corresponding distances from the coil xi.   

  2. You can determine the value of Be from the angle by noting that the direction of the compass needle is the vector sum of the fields from the Earth and the solenoid.  

      so        
             Use the computer or a calculator to determine Be for each angle.



  1. DETERMINE your average value for Be from the measurements of qi, and compare your result with the value from Method 1.  Discuss your errors in this experiment, and analyze which experiment should produce a more accurate value of  Be.


Method 3: Measuring Harmonic Oscillation caused by Magnetic Moments

 

If you suspend a bar magnet with a small thread tied about its middle, the magnet will act like a compass, and move to align itself with the Earth’s field, but its momentum will cause the magnet to “overshoot” the North direction, and the magnet will oscillate.  We can time this oscillation and use the period to determine the value of Be, if we use the magnetic field of the solenoid as a “driver.”

The net field B “felt” by the magnet causing the restoring torque arises from both the solenoid and the Earth (if the solenoid is aligned parallel to the Earth’s field) = Be + Bs. 



  1. Set up the experiment with a small cylindrical magnet suspended at the middle from a thread within the core of the solenoid. Connect the solenoid in series with the Ammeter (multimeter) and power source, and vary the current from 0.10 A through 0.50 A, measuring the resulting elapsed time of oscillation of the magnet for each value of the current. 

    MEASURE and RECORD both the values of the current and the total time for a fixed number of  oscillations.  Enter these into the computer.

    A good experimental technique that improves your precision is to time 20 measurements and divide your elapsed time by 20 to get the period. A better technique is to repeat this measurement at least once to verify that the elapsed times are relatively constant at the same currents.  An even better technique is to have two people count the number of oscillations each time the experiment is performed at a particular value of the driving current.

 

  1. I’ll help you use Excel to plot T-2 vs. Is  (or, frequency2 vs. Is), determine the slope and y-intercept, and find the value of  Be.  COMPARE this value with those you obtained in the first two parts of the experiment.

  2. DETERMINE at least two sources of random error in this experiment (things you cannot control), and two sources of systematic error (resulting from errors in the derivation or factors not taken into account in the derivation of the harmonic motion.)