Electromagnetism Lessons

A Quantitative Study of Helmholtz Coils

These coils come in pairs with the same number of windings of wire on each of the two coils. In "true Helmholtz" configuration: (1) the coils are wired in series with identical currents in the same direction in each coil, and (2) the coils are placed a distance apart that is equal to the radius of each coil. When in this configuration, they produce a very uniform magnetic field that is directed along their common central axis. One of the most common uses for such coils in physics education is in determining the charge to mass ratio of electrons, accomplished by immersing an electron tube in the central region of the coils and measuring the resultant curvature of the electron beam.

The two photos directly below show a self-contained e/m apparatus manufactured by Daedalon, with a light blue electron beam that is visible due to a small quantity of helium in the otherwise evacuated bulb. The accelerating potential of the electron beam and coil current are shown by digital displays at the bottom left and right of the apparatus, respectively.

e/m Apparatus

Circular Electron Beam

The purpose of this experiment, however, is not to determine the e/m ratio for electrons. Rather, the purpose is (1) to investigate the magnitude of the magnetic field produced at varying distances from the center point of the two coils along the axis and (2) to see how this field is affected by coil separation distance. While Helmholtz coils can cost thousands of dollars, there is a nice pair including a base track available from Sargent-Welch for only $99.90 (Cat# WLS1804-47 and price as of October 24, 2016). The photo below shows these coils, with PocketLab's magnetometer centered on the common central axis of the coils. The X on the top PocketLab shows the location of the magnetometer within PocketLab.

PocketLab Centered in Coils 

The complete setup for this experiment is shown in the photo below. A power supply provides a constant current, shown as 0.35 amp. The two coils, on the provided track, are separated by a distance equal to their radius with current in the same direction for each coil. PocketLab is taped to a 5/8" diameter wood dowel rod, with PocketLab's magnetometer at the center point on the axis of the coils.The dowel is in turn attached to a burette clamp, which is then attached to a ring stand. The ring stand can then be easily moved back and forth along the meter stick to provide known distances from the center point along the axis of the coils.

Experiment Setup

With the power supply off, the magnetometer is zeroed. By moving the ring stand, PocketLab is then moved to a distance 10 cm to the left (negative position) of the zero position. The power supply is turned on and the PocketLab app is set to begin recording data. Data is recorded for 10 seconds, and the the ring stand is moved 1 cm to the right. Data is again recorded for ten seconds. This process continues until the PocketLab magnetometer is 10 to the right of the center point. The result is a graph similar to that below, which shows the X magnetometer readings collected by the PocketLab app for the 210 seconds of data collection. The numbers above each of the "steps" is the magnetic field in microTesla. These values can be obtained easily by examination of the data file, or by using Excel to find averages for each of the steps. Finding averages is not really necessary, however, as the readings are quite stable as long as PocketLab is kept stationary.

x Magnetometer

The experiment is repeated a second time with the coils closer together than the radius of the coils, and a third time with the spacing of the coils greater than the radius of the coils. The graph below summarizes the results of the three experiments. Red represents data in which the coils are too close together, green for data in which the coils are spaced "just right" at coil radius, and blue for data in which the coils are too far apart. The vertical colored lines on the graph are at the location of the coils for each of the three experiments. The graph clearly demonstrates that when the coils are separated by a distance equal to the coil radius, then the magnetic field is most uniform within the coils, particularly in the central region equal to roughly 1/2 the radius of the coils.

(Note that the range for the PocketLab magnetometer is plus or minus 2000 microTesla.  If you find that you are exceeding this range, simply lower the current in the coils until you are within range.)

Magnetic Field vs. Axial Distance

For anyone who might be interested, the video below shows the author quickly moving PocketLab from the far left of the coils to the far right of the coils a distance of 20 cm, with coil separation equal to the coil radius. The superimposed X magnetometer graph clearly shows the uniformity of the magnetic field near the center of the coils.

Optional Investigations

1. Investigate off-axis magnetic field strength, in an effort to determine the uniformity of the magnetic field on a plane perpendicular to the axis and centered between the coils.


2. Compare the experimental results with theoretical equations that predict the magnetic field strength along the axis of a pair of Helmholtz coils. This can be accomplished by using the equation for the axial magnetic field strength B at a distance x from a single coil of radius R with N turns of wire. Many calculus-based physics textbooks derive this equation (multiply the right side by N for N turns of wire):

 Equation

Using this equation, a spreadsheet can be developed to produce theoretical graphs of magnetic field along the axis of the coils. The formulas become a bit tedious, but do show that experiment and theory are in good agreement.  See the graph below for a comparison of theory and experiment for the case in which the coils are spaced a "true Helmholtz" distance (i.e., separation equals radius).

Theory vs. Experiment

3. Set up the coils so that the currents are in opposite directions. Such coils are sometimes referred to as being "reverse Helmholtz coils". The result will be a "magnetic quadrupole", with zero magnetic field strength at the axial center point between the coils, with polarity changing either side of the center point.


4. Have your students do a Web exercise investigating graphite levitation, an application of quadrupoles produced by strong neodymium magnets. Images Scientific Instruments (www.imagesco.com) sells a Pyrolytic Graphite Levitation Kit for $49.95 (as of October 25, 2016).

 

 

Magnetic Field on the Axis of a Current Loop

In this lesson students will find that a current-carrying loop can be regarded as a magnetic dipole, as it generates a magnetic field for points on its axis.  The figure below shows a diagram and the equation for the magnetic field B.  Derivation of this equation requries knowledge of the Biot-Savart Law, calculus and trigonometry.  But in this lesson we are interested only in comparing experimental results from PocketLab's magnetometer to the theoretical equation in the figure below.  More advanced students can consider derivation of the equation, if they wish.
 
Diagram and Theory
 
There are many ways that you can make a current loop.  The author used a plastic ribbon spool approximately 3" in diameter and 3/4" wide, and then wrapped 10 turns of insulated wire around the spool.  The ends of the wire were connected to a DC power supply that supplied constant current for the current loops.  The photo below shows PocketLab with its magnetometer centered in the middle of the spool on the axis of the spool.  PocketLab's magnetic sensor is located about 0.5 cm in from the its edge, shown by the black X drawn on PocketLab.  PocketLab is set to provide magnetic field magnitude data and is zeroed when there is no current in the wire loops.
 
PocketLab at x=0 Position
 
The two photos below provide two more views of the apparatus setup.  It is important to keep the magnetic sensor on the loop axis, as it is moved to known distances from the center of the loop.  The author used some small blocks of wood for this purpose.  A meter stick with its zero point at the center of the loops allows moving PocketLab gradually outward, increasing the value of x by one cm for each move of PocketLab.  The author's setup used a current of i = 5.12 amp, R = 0.0361 m, with x varying from 0.00 m to 0.10 meters by steps of 0.01 m each.  The number of loops N = 10.
 
Pocket Lab along Axis
 
Axis View
 
The graph shown below, constructed in Excel, was obtained from data in the magnetometer.csv file produced by the PocketLab app.  The horizontal plateaus are labeled with the average value of the magnetic field for each plateau.  These averages could be obtained using Excel, but are much easier and quicker to obtain using Logger Pro, an exceptional educational data analysis software package produced by Vernier Software & Technology (vernier.com).  The highest plateau is where PocketLab was at x=0.  The next plateau is for x=0.01 m, the next for x=0.02 m, and so on, through x=0.10 m.
 
Magnetic Field vs. Time
 
The graph below, produced in Excel, summarizes the experimental results for magnetic field vs. distance along the axis, as they compare to the expected results from the theoretical equation.  A good discussion would be for students to suggest reasons for discrepancies between theory and experiment.
 
Experiment vs. Theory

Investigating Ampere's Law for a Long Current Carrying Wire

One of the classes of problems dealing with magnetic fields concerns the production of a magnetic field by a current-carrying conductor or by moving charges.  It was Oersted who discovered back in the early 1800's that currents produce magnetic effects. The quantitative relationship between the magnetic field strength and the current was later embodied in Ampere's Law, an extension of which made by Maxwell is one of the four basic equations of electromagnetism.
 
PocketLab, used in conjunction with a long, straight current carrying wire, offers a great opportunity for students to quantitatively study the relationship between magnetic field strength B and (1) the current i in the wire and (2) the distance r from the wire's center.  Students will be able to confirm the experimental results that
bformula1
The picture below shows the experimental setup used by the author.  A long, straight wire (red) is strung from a ring stand to the floor, and the loose ends of the wire are attached to a DC power supply that allows varying the current as desired.  The current value is shown in amperes in the right-most digital display on the power supply.  The table allowed pulling the sections apart so that the wire could be in the center of the table.  The wire could just as well have been placed along the outer edge of the table.  A ruler is placed on the table zeroed at the center of the wire.  PocketLab can then be placed at the desired distance from the wire.
Experiment%20Setup2
The figure below shows a close up of PocketLab, the NSTA ruler, and the wire.  The ruler is zeroed on the center of the wire, and PocketLab is shown with its left edge at the 3 cm mark on the ruler.  Since PocketLab's magnetic sensor is located about 0.5 cm in from the left edge, shown by the black X drawn on PocketLab, the distance r from the wire in this photo would be 3.5 cm.  PocketLab is set to provide magnetic field magnitude data and is zeroed when there is no current in the wire.
Measuring%20Distance%20from%20Wire3
VARYING CURRENT WHILE KEEPING DISTANCE CONSTANT
 
The video below contains data for magnetic field magnitude while varying the current, but keeping the distance r constant at 1.5 cm throughout.  Data from this video or the attached magnetometer file can be used in Excel to obtain a chart of B vs. i.
The author's Excel chart below clearly shows that the magnetic field B is directly proportional to the current i in amps.  The linear trend/regression fit provides an R-squared value of 0.9999.  It is seen from the linear regression equation that the magnetic field increases by about 7.3 microT for each ampere increase in current.
constantdistance5
VARYING DISTANCE WHILE KEEPING CURRENT CONSTANT
The video below contains data for magnetic field magnitude while varying the distance, but keeping the current constant at about 6 amps throughout.  Data from this video or the attached magnetometer file can be used in Excel to obtain a chart of B vs. r.
The author's Excel chart below clearly shows that the magnetic field B is inversely proportional to the distance r in cm.  The power trend/regression fit provides an R-squared value of 0.9981.  It is seen from the power regression equation that the power is -1.072, very close to the -1 expected for an inverse first-power proportionality.
constantcurrent7

The Inverse Cube Law for a Neodymium Dipole Magnet

PocketLab makes is quite easy to investigate and verify the inverse cube law for the magnetic field of a neodymium magnet as a function of distance from the magnet.  All that is needed in addition to The PocketLab is a centimeter ruler, small neodymium magnet, a small block of wood and a little double stick tape.  The photo below shows how the neodymium magnet is taped to the block of wood with the magnet located at the 10 cm mark on the NSTA ruler.  The height of the center of the magnet is at about the height of the circuit board inside of PocketLab.  The X on the front face of PocketLab is very close to the location of the magnetic field sensor inside of PocketLab, 0.5 cm from the left edge of Pocket Lab, in line with the Y-axis of PocketLab.

Setup%20View%202

The photo below shows the set up from above with the left edge of Pocket lab at the 15 cm mark.  The distance between the dipole and the sensor is therefore about 5.5 cm in this photo.

Setup%20View%201

In preparation for data collection, PocketLab is set to display magnetic field magnitude.  It is then moved far from the neodymium magnet and zeroed.  It is then placed at the 12 cm mark on the ruler, making the distance between magnet and sensor 2.5 cm.  After a few seconds, PocketLab is moved to the 13 cm mark, thus increasing the distance by 1 cm to 3.5 cm.  This process is continued through a distance of 8.5 cm.  The magnetic field magnitude can be read directly from the movie, shown below, at each of the known distances.

(Distance, magnetic field magnitude) data pairs are then entered into an Excel spreadsheet, and a chart of Magnetic Field vs. Distance is created.  The chart, shown below, appears to show some sort of an inverse relationship between magnetic field and distance.  The Excel "Add trendline" feature is then used and the "Power" regression fit is applied.  It is found that the power is -2.832, very close to the -3 expected for an inverse cube relationship.

excel%20data%20and%20chart

A copy of the Excel spreadsheet is included in the attachments for anyone interested in viewing it.

PocketLab Investigation of Fuel Efficiency

With gas prices as high as they are and having a growing concern for the environment, Americans today are becoming conscious about things they can do to improve fuel efficiency.  Many realize that driving at the posted speed limits promotes both safety and reduces the rate at which fuel is consumed.  With these things in mind, some have purchased hybrid vehicles including the Toyota Prius, all-electric vehicles such as the Nissan LEAF, or range-extending vehicles such as the Chevy Volt.  Those with EV's soon realize that they get more miles per charge if they avoid driving at excessively high speeds on the open road.
 
With this background as a base, it would be very instructive if students could perform a laboratory investigation that would provide a quantitative relationship between distance per unit of fuel consumed (a measure of efficiency) and speed.  One way to accomplish this is by investigating these factors using an N-scale electric train set, such as those sold by Bachman for around $100, and commonly available in toy and hobby stores.  In this case, fuel is the unit of electrical energy, i.e. the kilowatt-hour (kWh), or on the scale of a hobby train, more appropriately the watt-second (W-s).  The picture below shows a train set, with a power supply providing the electrical power to run the train,  The power supply provides readings for voltage (V) in volts and current (I) in amps, from which power can be calculated by the product VI.  A meter stick is used to measure the diameter of the circular train track, from which the radius is found to be 0.285 m.  PocketLab is mounted to the top of the engine using tabs that come with the PocketLab Maker Kit.
Setup (2)
Starting with a low voltage, the train was allowed to run for about one lap.  This process was repeated for a sequence of voltages to slightly above the train manufacture's recommended limit of 16 volts.  Voltages can be read from the video.  PocketLab was set to provide angular acceleration.  With PocketLab mounted on the engine with the train moving in the XY plane, Z angular acceleration (shown in green in the video) is the variable of interest, telling us the number of degrees per second that the train revolves on its circular track.  It is important to make sure that angular acceleration is zeroed prior to capturing data and video with the PocketLab app.
The image shown below contains a graph of Z angular velocity vs. time from data produced by the PocketLab app.  The graph was obtained by importing data from the PocketLab app into Logger Pro, an exceptional educational scientific analysis software from Vernier Software & Technology (vernier.com).  The statistical analysis capability of Logger Pro is used to obtain mean angular velocity for each of the voltage steps during the experiment.  This is an extremely useful capability as the angular acceleration does vary a fair amount at each step level.  For example, the graph shows that the mean Z angular velocity for the second voltage step was 32.59 degrees/s. The step nature of the Logger Pro chart is a result of the voltage being increased to a new level after a few laps have been made by the train at a given voltage.
logger
The Microsoft Excel table below summarizes all of the raw data as well as come calculated columns containing power, period, speed, energy per lap, and efficiency.  A copy of the Microsoft Excel file is attached so that you can look at the formulas used in these calcualted columns.  Power is voltage times current (P= VI).  Period is 360/angular velocity.  Speed is 2*Pi*R / period.  The energy per lap in W-s is power multiplied by period.  Efficiency in Laps per W-s is the reciprocal of Energy per lap.  Therefore, efficiency for our N-scale train is measured in laps per W-s.  This is analogous to mpg for a traditional gas consuming car, or to miles per charge or miles per kWh for an EV.
datasheetexcel
The Microsoft Excel table below summarizes all of the raw data as well as come calculated columns containing power, period, speed, energy per lap, and efficiency.  A copy of the Microsoft Excel file is attached so that you can look at the formulas used in these calcualted columns.  Power is voltage times current (P= VI).  Period is 360/angular velocity.  Speed is 2*Pi*R / period.  The energy per lap in W-s is power multiplied by period.  Efficiency in Laps per W-s is the reciprocal of Energy per lap.  Therefore, efficiency for our N-scale train is measured in laps per W-s.  This is analogous to mpg for a traditional gas consuming car, or to miles per charge or miles per kWh for an EV.
graph (1)
Questions for Students:
 
1.  Why is the efficiency of the train so low at low speeds?
 
2.  What are some possible explanations for the train's efficiency dropping after reaching the speed of peak efficiency?
 
3.  Considering a moving automobile as an analogy, (1) What is the efficiency of a car when it is at rest in a traffic jam and why?  (2) What factors reduce the efficiency of a typical car at very high speeds?
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