Ampere's law and the e/m experiment

Moving charged particles produce magnetic fields and are acted on by magnetic fields.  Currents are moving charges and therefore they produce and are acted on by magnetic fields.  Permanent magnets are the result of "magnetization currents" flowing inside the material.

In experiment 1 of this lab you will simulate using a giant magnetoresistive (GMR) sensor to measure the strength of the magnetic field B produced by a current flowing in a circuit.  One section of the circuit is a long, straight wire.  You will measure the strength of the magnetic field near the middle of this wire as a function of the distance from the wire, for distances much smaller than the length of the wire.  You will also use Ampere's law to calculate the magnetic field strength B produced by a current flowing in an infinitely-long, straight wire (with the rest of the circuit at infinity).  You will compare the results of your measurements with the results of your calculations.

In experiment 2 of this lab you will observe the deflection of electrons in a magnetic field and use this deflection to determine the electron's charge to mass ratio.

Equipment Needed:

• e/m Apparatus
• Pasco Low Voltage Power Supply
• Patch cords
• Magnet
• Compass needle

Open a Microsoft Word document to keep a log of your experimental procedures, results and discussions.  This log will become your lab report.  Address the points highlighted in blue.  Answer all questions.

Currents produce magnetic fields and current-carrying wires are deflected in magnetic field.

Experiment 1

Use the "Low Voltage" power supply.  Before you turn it on, turn the voltage knob all the way up (clockwise) and the current knob all the way down (couterclockwise).  After you have connected a load, you can then control the current passing through the load by slowly turning the current knob clockwise.

(a)  Let the wire cross the needle at right angle as shown.
    
Slowly turn up the current to ~ 3 A.  Do you see any deflection of the needle? 
Turn the current back down before the wire gets hot.

Record your observation.  Is the needle deflected?  Do you expect a deflection?  Why or why not?
Switch the direction of the current flow by switching the red and black lead at the power supply.
Record your observation.  Is the needle deflected?  Do you expect a deflection?  Why or why not?

(b)  Now align the wire with the needle. 
    
Let the current pass over the needle from North to South.
Slowly turn up the current to ~ 3 A.  Do you see any deflection of the needle?  T
urn the current back down before the wire gets hot.

Record your observation. Is the needle deflected?  Do you expect a deflection?  Why or why not?
Let the current pass over the needle from South to North and repeat.
What do you see?  Can you explain your observation?

Near Knoxville, TN, the strength of the Earth magnetic field is ~ 53 microT.  It has a declination D (deviation from North) of about 3^o and an inclination I (downward tilt) of ~ 65^o, so its horizontal component has a magnitude of approximately 53 microT*cos(65^o) = 22 microT.


The magnetic field of the wire and the Earth magnetic field are vectors and add vectorially.  If the wire is aligned with the compass needle when no current flows through the wire, and the magnitude of average field due to the wire at the compass needle is approximately equal to magnitude of the horizontal component of the Earth field, you should see a deflection of ~45^o.

Can you deflect the needle by ~ 45^o with a current of less than 4 A?  Would you expect to be able to? 
What current do you need to produce a magnetic field of magnitude 22 microT at a distance of the needle from the wire?

(Review:  The magnetic field produced by a steady current flowing in a  long straight wire)

Activity 1

Magnetic fields exert forces on other moving charge.  The force a magnetic field exerts on a charge q, moving with velocity v, is called the Lorentz force.
It is given by F = qv × B. 

Assume a charged particle is moving with velocity v through a region with magnetic field B.  Predict the direction of the magnetic force for each situation below.   Assume that the particle is positively charged.  Record your predictions in your log.

Experiment 2

Part 1:  Measuring e/m

Mass spectrometry has become an important measurement tool in clinical chemistry, microbiology, toxicology and in the pharmaceutical world.  A mass spectrometer deflects ionized and accelerated molecular fragments using a magnetic field and sorts them according to their charge to mass ratio.  For this exercise you will use an e/m apparatus like a mass spectrometer to determine the electron's charge to mass ratio, e/m, by measuring the radius of curvature of an electron's path in a uniform magnetic field of known strength.

You will use the "Fine Beam Tube System" show below.  All components are labeled.

Inside a glass tube a beam of electrons is accelerated through a known potential difference, so the kinetic energy and therefore the velocity of the electrons is known.  A pair of Helmholtz coils can produce a uniform magnetic field at right angles to the electron beam.  This magnetic field then deflects the electron beam in a circular path.  The diameter of this circular path can be measured using the calibration marks in the center of the tube, which are equally spaced by 2 cm. 

The glass tube is filled with helium at a low pressure of 13 Pa.  It contains an electron gun and an electrostatic lens (Wehnelt) system.  A heater heats the cathode, which emits electrons.  The electrons are accelerated by a potential difference between the cathode and the anode.  The electron beam leaves a visible trail in the tube, because some of the electrons collide with helium atoms.  The atoms are excited and then radiate visible light.

The tube socket rotates, allowing the electron beam to be oriented at any angle (from -10 to 270 degrees) with respect to the magnetic field produced by the Helmholtz coils.  The vector nature of the magnetic force on moving charged particles can therefore be explored.  A small permanent magnet can also be used to deflect the electron beam.

In order to rotate the tube, loosen the screw, but do not screw it all the way out.  Do not turn the tube itself, instead rotate the turntable.  then tighten the screw again.

The radius of the Helmholtz coils is equal to their separation.  This geometry provides a highly uniform magnetic field near the center of the coils.  The Helmholtz coils of the apparatus have a radius and a separation of ~15 cm.  Each coil has 124 turns.  The magnetic field B produced by the coils is proportional to the current I through the coils times 7.56*10^-4 T/A.   It is perpendicular to the plane of the coils.
B = (7.56*10^-4 T/A)*I.

Analysis of an e/m measurement

The magnetic force F_m acting on a charged particle of charge q moving with velocity v in a magnetic field B is given by the equation

F = qv × B.

If the electron beam velocity is perpendicular to the magnetic field, we have the following equation relating the magnitudes F_m, q, v, and B.

F_m = qvB.

The electron is moving in a circular path of radius r, with the magnetic force being equal to the centripetal force mv²/r.  We therefore have

qvB = mv²/r   or   q/m = v/Br.

We denote the magnitude of the charge q_e of the electron by e and therefore have e/m = v/Br.
The electrons are accelerated by the accelerating potential V, gaining kinetic energy equal to their charge times the accelerating potential.
Therefore eV = ½mv².
The velocity of the electrons is v = (2eV/m)^½.  Inserting this expression for v in the equation above and squaring both sides we obtain

e/m = 2V/(Br)²  or  2V = (e/m) (Br)².

The slope of a plot of 2V versus (Br)² is equal to electron charge to mass ration e/m.

Procedure

• Align the tube as shown in the picture above, with the electron gun perpendicular to the with the magnetic field produced by the Helmholtz coils.  Then turn the power supply on and work in a darkened room.
• Set the adjustment knob for the heater voltage to a position in the middle (~6V).
• Turn the knob for the coil current all the way to the left (0 A).
• Wait about a minute for the heater temperature to stabilize.  Then slowly increase the anode voltage to 300 V.  The electron beam path will become visible.
• Select the Wehnelt voltage so that the electron beam is narrow and clear.  Optimize the focus and brightness of the beam by varying the heater voltage.
• Increase the coil current until the electrons move in a circle of radius r = 4 cm or diameter 8 cm.  (Take some time to adjust the current carefully. Remember that the calibration marks in the center of the tube are equally spaced by 2 cm.  Place your eye so that the front and back calibration marks line up and the electron beam passes in between.)
• Paste the table below into an Excel spreadsheet and enter your measured value for the coil current into the table.
• Repeat for the other accelerating voltages in the table.  (Fill the table from the bottom up.)
• Let Excel calculate the remaining columns in the table.
• Produce a graph of 2V versus(Br)² and label the axes.
• Add a trendline, set the intercept to zero, and display the equation on the chart.  Format the trendline label Scientific, 2 decimal places.

• Insert your table and your graph into your Word document.

• What is the slope of your trendline?  (Note: The slope will have units (J/C)/(Tm)² = C/kg.)

• What is your measured value of e/m?

• Calculate e/m from the accepted values of the electron's charge and mass.  Does the value of e/m from your experiment agree with the calculated value?  Given your experimental procedure, how close do you expect them to agree?

Convert your log into a lab report.  See the grading scheme for all lab reports.

Name:
E-mail address:

Laboratory 4 Report

• In one or two sentences, state the goal of this lab.
• Make sure you completed the entire lab and answered all parts.  Make sure you show your work and inserted and properly labeled relevant tables and plots.
• Add a reflection at the end of your report in a short essay format.

Save your Word document (your name_lab4.docx), go to Canvas, Assignments, Lab 4, and submit your document.