LAB 10
Magnetic Fields due to Currents
Formal Lab Report
OBJECTIVES
1. Observe different magnetic fields associated with various current distributions.
2. Predict and measure the magnetic fields using Biot-Savart’s Law for three different
current distributions.
3. Plot the magnetic field versus distance and verify mathematically the shape of the
magnetic field functions.
EQUIPMENT
DataStudio (Magnetic Field Sensor), Helmholtz Coils setup, Power Supply, blocks, and
meter stick.
THEORY
Magnetic Field of an N-turn Coil
When current flows through a single N-turn circular coil of radius R , the magnetic
field of the coil a distance z along the central axis above the center of the coil can be
determined by the Biot-Savart Law and is given by
N0iR2
1
2
2 (z  R2 )3/2
coil
Magnetic Field of a Helmholtz Coil
For the Helmholtz configuration, the distance between the coils is set equal to their
mean radii. According to the Biot-Savart Law, if the currents are in the same
direction through the two coils, the magnetic field is nearly uniform at the
midpoint between the coils. The resultant magnetic field at the midpoint is a
superposition of two N-turn coils:
N0iR2
N0iR2
1
1
BHelmholtz 

3/ 2
2
2 3/ 2
2
2
(z  R / 2)  R 
(z  R / 2)2  R2 




where z is the distance from the midpoint between the two coils. Near the midpoint
between the coils (z << R), one can show that the maximal field between the coils is
equal to its value at the center
N0i
BHelmholtz (z  0) 
(5/4)3 / 2 R
On the other hand, if the currents are opposite in direction instead, the magnetic
field is close to zero at the midpoint between the coils. To determine the maximal
(and minimal) value of the field one must find the critical points (there are two) using
dBHelmholtz/dz  0. Substitute the critical points back into BHelmholtz to determine the
maximal and minimal values of the field.
BN-turn 
PROCEDURE
Use a Dynamics track to measure the position of a Magnetic Field Sensor (MFS) as it is
moved along the central axis of the coil. Compare the measured and predicted magnetic
field strength of a single N-turn coil and a Helmholtz coil configuration.
a. Connect the MFS [1] (select the axial setting and set the Range Select to 10X) to
DataStudio in a Digits display mode and set the Magnetic Field Strength to 10X.
Setup an Excel spreadsheet to record both the magnetic field and position of the
MFS.
b. Construct a “track” for the MFS as shown below using a meter stick and lab jacks.
Adjust the position of the lab jacks so they are at the ends of the meter stick.
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
Place the meter stick through the coil and center the 50-cm mark at the center of
the coil. Now adjust the height of the track so the MFS can move along the
centerline. Make sure the meter stick is level.
c. Connect the coil and an ammeter in series with a Power Supply. Set the upper
current limit of the power supply to 1.0 A before starting the experiment! Adjust the
voltage of the Power Supply such that the current is about 0.5 A.
d. Before recording any data, make sure that you do several practice runs. The general
procedure is
 Zero the MFS at the center of the coil when there is no current running through
the coil.
 Place the MFS on top of the meter stick with the end of the sensor rod pointing
towards the coil and aligned with the 30 cm mark.
 Turn on the power supply, record the current value and start measuring data.
 Slowly and steadily move the MFS [2] along the Dynamics track from the 30 cm to
the 70 cm mark. Measure the magnetic field strength (from the Digits display)
and position at every 5-cm increment and record your data in a table.
Part 1: The Magnetic Field of a Single N-turn Circular Coil
Connect an N-turn circular coil to a DC power supply and measure the magnetic field as
stated in the Procedure above. Measure the radius R of a coil.
a. Measure the magnetic field Bexpt as a function of position (Bexpt vs. z) for current i =
0.5A. Remember to perform a coordinate shift in order to make sense of the data.
b. Calculate the theoretical maximal magnetic field Bthy for the coil using the measured
current, radius, and number of turns.
c. Compare the maximal magnetic field for Bthy and Bexpt using a percent difference?
How do they compare?
d. Plot and explain Bexpt vs. z both physically and mathematically. Does the plot agree
with the Biot-Savart law predictions?
Part 2: The Magnetic Field of Helmholtz Coils
Connect two coils in a Helmholtz configuration and readjust the current to be 0.5 A.
a. Measure the magnetic field BH,expt as a function of position (BH,expt vs. z) for currents
in the same direction (make sure that the coils are wired in series).
b. Calculate the theoretical maximal magnetic field BH,thy for the coil using the
measured current, radius, and number of turns.
c. Compare the maximal magnetic field for BH,expt and BH,thy using a percent difference?
How do they compare?
d. Plot and explain BH,expt vs. z both physically and mathematically. Does the plot
agree with the Biot-Savart law predictions?
e. Repeat parts (2a) – (2d) for currents in the opposite directions. Note that there
are two maximal “humps” in this case.
f. Plot three magnetic fields on a single plot: single coil, Helmholtz coils with same and
opposite current directions. Discuss physically the similarities and difference of the
three plots. Furthermore, compare the maximal field of a single coil and the
Helmholtz coils (same and opposite currents) – how do they compare? Which coil
produced the most uniform magnetic field? Did the theoretical values show that?
10-2
[1] The Magnetic Field Sensor produces a voltage that is directly proportional to the
magnetic field strength as follows: 10 mV = 10 Gauss (G) (where 1000 G = 0.1 T).
The sensor’s range is ±2000 G. Depending on the strength of the magnetic field
around or inside the coil you may want to increase the sensitivity of the Magnetic
Field Sensor. For this laboratory, set the Range Select switch on the 10X setting.
(Adjust if needed either 1X or 100X.)
[2] Warning: point the field sensor with the same orientation throughout the experiment.
The sensor will give two different readings at the same location if the sensor is
pointing in different directions.
10-3