Lab 4

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.

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.


Experiment 1

You will measure the magnetic field strength near a current-carrying long straight wire, as a function of the perpendicular distance from the center of the wire.  You will verify that close to the wire and near its center Ampere's law can be used to make reasonable predictions, even if the wire is not infinitely long.  You will learn about giant magnetoresistive sensors.

Theory

imageOne of Maxwell's equations is Ampere's law.  Ampere's law applied to an infinitely-long wire predicts a magnetic field of strength B = μ0I/(2πr) a radial distance r from the wire.  The field B is tangential to a circle of radius r centered on the wire.
We therefore have (B/I) = (μ0/(2π))(1/r).
B/I is proportional to 1/r, and when plotted versus 1/r will yield a straight line with slope μ0/(2π).

Method

imageIn this experiment you will use a GMR sensor to measure magnetic field strength.  In a GMR sensor, the resistance of two thin ferromagnetic layers, separated by a thin, nonmagnetic conducting layer is altered by changing the magnetic moments of the ferromagnetic layers from parallel to anti-parallel or vice versa.

Layers with parallel magnetic moments have lower resistance than layers with anti-parallel magnetic moments.  The layers are typically less than 10 nm thick.  They are sputtered onto semiconductor wafers and they are patterned into narrow stripes.  A very small current flowing through the conducting layer across the stripes rotates the magnetic layers into anti-parallel, high-resistance alignment.  An external magnetic field applied perpendicular to the direction of current flow and parallel to the stripes can overcome the field produced by the current and rotate the magnetic moments of both layers parallel to the field.  The amount of current needed to destroy the alignment caused by the external field is a measure of the magnetic field strength.

Smart sensors with sensing elements and associated electronics on the same chip can be bought.  These sensors have a sensitive axis (along the stripes) and can directly detect the component of the magnetic field along this axis.  The diagram below shows two GMR sensors positioned to measure the magnetic field of a bar magnet.  The sensitive axes are indicated and the component of the field along the sensitive axes for the two sensors is plotted.

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Apparatus

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imageThe Ampere's Law apparatus is shown in the figure above.  It is used to measure B as a function of the radial distance r from a wire, for r much smaller than the length of the wire.  For this apparatus r is the distance between the center of the wire and the point at where the GMR sensor is located beneath the surface of its mounting package.

imageThe straight wire is  0.23 m long, and we measure the strength of the magnetic field near the middle of this wire as a function of the distance from the wire for r = 1.68 mm to 11.68 mm.

The distance r is determined with a digital scale to a precision of 0.1 mm.  Initially the top surface of the IC mounting package is positioned so that it touches the wire and the digital scale is zeroed.  At this position, r = r0 = 1.68 mm.  r0 is the sum of the radius of the wire (1.18 mm) and the sensor depth (0.5 mm).  r0 must be added to all subsequent readings of the digital scale.

To pass a current through the wire, it must become part of a larger circuit that includes a power supply.  The distance from the sensor to the connecting wires is much larger than r, and so the magnetic field produced by the connecting wires contributes very little to the measured magnetic field strength.

The magnetic field produced by the wire encircles the wire and its direction at the position of the sensor is perpendicular to the base plate.  The sensor is mounted so that its sensitive axis is also perpendicular to the base plate.  The output voltage of the sensor V is directly proportional to the magnetic field strength B to be measured if an appropriate offset voltage has been subtracted.  The offset voltage is partly due to the earth magnetic field and partly due to sensor electronic.  It must be determined before each measurement by reading the voltage when no current is flowing in the wire.  The current is turned on and off with a switch.

Procedure

Start the experiment by clicking the this link.

Open a new Excel spreadsheet and create a data table with columns as shown below.

r0 d V0 V I 1/r ΔV k B/I
0.00168                
0.00168                

The labels denote the following quantities.

r0 1.68 mm = 1.68*10-3 m
d reading of the digital scale  (meter)
V0 positive or negative offset Voltage with no current flowing  (Volt)
V output voltage with current flowing  (Volt)
I current flowing in wire when switch is closed  (Amp)
1/r 1/(r0 + d)
ΔV (V - V0)
k calibration constant (see apparatus label ~10-4 T/V)
B/I B/I = k*ΔV/I with I = 10 A.

Start taking data.  Click the switch to turn the power supply on or off.  As a function of r measure V0 (no current) and V (current) in quick succession, so that the offset voltage has very little time to change.

Record d (in units of m), V0, V, and I in your data table. 
Take data in 0.2 mm intervals for scale readings from 0 mm to 10 mm.

Data analysis


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.

(a)B and v(b)B and v

(c)B and v(d)B and v

Experiment 2

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 the PASCO 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.

imageIn the PASCO e/m apparatus 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.  A unique feature of the e/m tube is that the socket rotates, allowing the electron beam to be oriented at any angle (from 0-90 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.

You will analyze data obtained with this apparatus.


imageThe e/m Tube

The e/m tube is filled with helium at a pressure of 10-2 mm Hg.  It contains an electron gun and deflection plates.
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. 

imageThe electron gun is shown in the figure on the right.  The heater heats the cathode, which emits electrons.  The electrons are accelerated by a potential difference between the cathode and the anode.  A grid is held positive with respect to the cathode and negative with respect to the anode. It helps to focus the electron beam.

The Helmholtz Coils

The radius of the 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 e/m apparatus have a radius and a separation of 15 cm.   Each coil has 130 turns.  The magnetic field B produced by the coils is proportional to the current I through the coils times 7.80*10-4 T/A.  It is perpendicular to the plane of the coils.

B = (7.80*10-4 T/A) * I.

The Controls

The control panel of the e/m apparatus is straightforward.  All connections are labeled.

Cloth Hood

The hood can be placed over the top of the e/m apparatus so the experiment can be performed in a lighted room.

Mirrored Scale

A mirrored scale is attached to the back of the rear Helmholtz coil.  It is illuminated when the heater of the electron gun is powered.  By lining the electron beam up with its image in the mirrored scale, the radius of the beam path can be measured without parallax error.


Experiment:  Measuring e/m

Power supplies and meters are connected to the to the front panel of the e/m apparatus, as shown in the figure below.

image

The power supplies are adjusted to the following levels:

The current for the Helmholtz coils is adjusted with the current adjust knob.  The ammeter displays the current in units of Ampere (A).

After the cathode has heated up, the electron beam emerges from the electron gun.  Its path is curved by the field from the Helmholtz coils.  The electron beam's path is parallel to the Helmholtz coils.

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Table 1

Current
to coils
I (A)
Accelerating
voltage
(V)
Magnetic field B
=
(6.6*10-4T/A)*I
Radius of
circular path
r (m)
e/m
=
2V/(Br)2
         

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Laboratory 4 Report

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