EE 3324 Electromagnetics Laboratory
Experiment #4
Maxwell’s Equations I: Gauss’s Law for Electric Fields
and Gauss’s Law for Magnetic Fields
1. Objective
The objective of Experiment #4 is to investigate two of Maxwell’s equations: Gauss’s law for
electric fields and Gauss’s law for magnetic fields. The relationship between the surface charge
distribution on a conductor and the vector electric field produced by that charge is illustrated through
experimental measurements. The basic characteristics of the magnetic field produced by a permanent
magnet is also illustrated through experimental measurement.
2. Introduction
The vector force (F) between two point charges (q1 ,q2) located in a medium characterized
by permittivity (,) is defined by Coulomb’s law:
(1)
where r is the separation distance between the charges and â is a unit vector pointing from the charge
exerting the force to the charge on which the force is applied. From Coulomb’s law, we find that the
force between two point charges is directly proportional to the product of the charges and inversely
proportional to the square of the separation distance. Coulomb’s law is not explicitly included in
Maxwell’s equations (the fundamental equations describing any electromagnetic field). However,
Coulomb’s law is implicitly defined in Gauss’ law for electric fields as shown below.
Starting with Coulomb’s law, the vector electric field and vector electric flux density can be defined.
Dividing both sides of Coulomb’s law in (1) by q2 yields the definition of the electric field (E).
(2)
The electric field is defined as the vector force per unit charge on the (positive) test charge q2. The
electric flux density is obtained by multiplying the vector electric field by the permittivity.
(3)
Given the electric flux density, Gauss’ law relates the total electric flux through a surface S to the
charge enclosed by the surface. Gauss’s law for electric fields in integral form is
(4)
where ŝ defines the outward-pointing unit normal to the surface S. Gauss’s law is valid for any charge
distribution and any surface S that encloses the charge.
Given an electrode over a ground plane, a simple electric field measurement can be used to
determine the charge on the conductor by way of Gauss’s law. Consider a cylindrically symmetric
electrode (such as a sphere or cylinder) over a ground plane. Assuming the Gaussian surface S
enclosing the charged electrode is defined as the ground plane plus the upper hemispherical surface
of radius Ro as shown in Figure 1, the total charge on the electrode Q is given by
(5)
Figure 1. Electrode over a ground plane.
If the radius of the hemispherical surface is large in comparison to the size of the electrode, then the
electric flux through the hemispherical surface is negligible. Thus, the total flux through the closed
surface S is the flux through the ground plane surface only such that Equation (5) may be written as
(6)
where Dmax defines the maximum radius where the electric flux density becomes negligible. The
cylindrically symmetric electrode produces a cylindrically symmetric electric field (independent of N)
which gives
(7)
The integral in Equation (7) may be evaluated numerically to obtain the total charge on the electrode.
The electrode charge can also be found using the definition of capacitance given the applied
voltage and the capacitance. The analytic formula for the capacitance of a conducting sphere over
a ground plane (using image theory) is
(8)
where a is the radius of the sphere and d is the distance from the center of the sphere to the point on
the plane directly under the sphere. An accurate approximation to the capacitance can be obtained
by truncating the infinite series in Equation (8) at a sufficient number of terms. The total charge on
the electrode at a voltage V is then
(9)
Gauss’s law for magnetic fields has the same form as Gauss’s law for electric fields with the
electric flux density replaced by the magnetic flux density. The total electric charge on the right hand
side of Gauss’s law for electric fields [Equation (4)] must be replaced by zero in Gauss’s law for
magnetic fields since magnetic charge does not exist. The integral form of Gauss’s law for magnetic
fields is
(10)
From Equation (10), we see that, for any magnetic field, the total magnetic flux into the volume
bounded by the surface S must equal the total magnetic flux out of the volume.
Given the cylindrical permanent magnet shown in Figure 2, a simple magnetic field
measurement can be used to verify Gauss’s law for magnetic fields. Just as in the case of the
electrode over ground, the Gaussian surface S is defined as the x-y plane plus the upper hemispherical
surface of radius Ro as shown in Figure 2. If the radius of the hemispherical surface is large in
comparison to the size of the solenoid, then the magnetic flux through the hemispherical surface is
negligible. The total flux through the closed surface S is then the total flux through the x-y plane.
For the given orientation of the magnet, the flux through the x-y plane just above the magnet is in
positive z-direction while the flux further away from the magnet is in the negative z- direction.
Gauss’s law for magnetic fields applied to the air-core solenoid may be written as
(11)
where Dmax defines the maximum radius where the magnetic flux density becomes negligible. The
magnetic flux density produced by the cylindrical magnet lying on the z-axis is independent of N based
on symmetry. Equation (11) then reduces to
(12)
Equation (12) can also be evaluated numerically to verify Gauss’s law for magnetic fields.
Figure 2. Magnetic Flux of Cylindrical Magnet.
3. Equipment List
Electrodes Over a Conducting Ground Plane (Gauss’s Law for Electric Fields)
Electrostatic field meter
High voltage power supply
LCR meter
Multimeter
Electrodes mounted on a adjustable scale (spherical electrode, cylindrical electrode)
Ground plane
Mapping Equipotential Contours (Gauss’s Law for Electric Fields)
Electrolytic tank
AC Voltage source
Multimeter
Steel needle mounted on an insulated stand
Rectangular electrodes, grid paper
Cylindrical Magnet Flux Density (Gauss’s Law for Magnetic Fields)
Gaussmeter with transverse probe mounted on adjustable scale
Small cylindrical magnet
4. Procedure
Measurement of the Electrostatic Field
The Monroe Electronics Model 257 Electrostatic Field
Meter measures the electrostatic field in units of kV/cm. The
Model 257 provides two scales (0 to 2 kV/cm and 0 to 20
KV/cm) which are activated using the 3-position switch on the
front of the instrument. A grounding cable is provided at the
rear of the instrument and must always be connected to a reliable
ground reference (bench ground terminal) for accurate readings.
The Model 257 uses a Model 257C-1 electrostatic field probe
which must be oriented as shown in Figure 1 for accurate
measurements. Proper orientation of the probe is essential to
ensure the accuracy of the field measurement.
The probe must be “zeroed” before making any
measurement. Before zeroing the meter with the ZERO knob,
be sure to ground yourself and any conductors in the vicinity
which may be charged as these represent a significant source of
electrostatic field which may contaminate the measurements.
Figure 1. E-field Probe
Alignment.
Note: You should ground yourself periodically during the execution of electrostatic field
measurements to prevent static charge buildup on your body from corrupting the measurements.
Electrodes Over a Conducting Ground Plane (Gauss’s Law for Electric Fields)
1.
Mount the spherical electrode on the adjustable scale and adjust the height of the spherical
conductor so that the minimum distance between the ground plane and the surface of the
spherical electrode is 6 mm. Use the spacer provided to accurately position the electrode at
the proper height. Carefully center the electrode directly above the electric field probe which
is mounted within the conducting ground plane. Position the probe mount scale to some
convenient starting value so that the position of the electrode can be accurately recorded as
it is moved away from its initial position. Measure the capacitance of the sphere-plane
conductor system using the LCR meter. Connect the ground plane and the ground cable of
the electrostatic field meter to the bench ground. With the high voltage power supply off,
connect the positive side of the supply to the electrode and connect the negative side of the
supply to the ground plane. Use a voltmeter to monitor the high voltage power supply
output. Turn on the electrostatic field meter using the 2 kV/cm scale and zero the output
using the zero adjust knob. Adjust the voltage level of the high voltage power supply to its
minimum value. Turn on the high voltage power supply and increase the applied voltage to
200 V. Measure the resulting electric field as you move the spherical electrode away from its
original position. You will need to accurately measure the variation in the electric field in
order to calculate the charge on the electrode by evaluating the integral in Equation (8)
numerically. Thus, you may need to take more data points in regions where the electric field
varies rapidly. Take measurements out to a distance where the electric field approaches zero.
2.
3.
4.
Perform the numerical integration on your measured data to estimate the charge on the
spherical electrode. Compare your Gauss’ law results with that obtained from the definition
of capacitance in Equation (10). Include the details of your computations in your report.
Move the spherical electrode to a position where the minimum separation between the
electrode and the ground plane is 12 mm and repeat the measurements and calculations of
part (1.).
Replace the spherical electrode with the cylindrical electrode and position the electrode such
that the minimum spacing between the electrode and the ground plane is 6 mm. Repeat the
measurements and calculations of part (1.).
Move the cylindrical electrode to a position where the minimum separation between the
electrode and the ground plane is 12 mm and repeat the measurements and calculations of
part (1.).
Mapping Equipotential Contours (Gauss’s Law for Electric Fields)
A plot of the electric field between a two-dimensional conductor geometry can be measured
by placing the conductor geometry within a poorly conducting medium (distilled water, F .10!4 ®/m,
,r = 81), applying a potential difference between the conductors and measuring the resulting potential
distribution. The resulting potential distribution in the conductor should match the potential
distribution that would exist given an insulating medium between the conductors. An AC voltage is
used with distilled water to prevent potential drops at the electrodes due to electrolysis.
5.
6.
Place one sheet of the grid paper underneath the electrolytic tank. Note that the grid paper
is divided into square millimeters (small divisions) and square centimeters (large divisions).
With the 12V AC source off, connect the source to the two rectangular electrodes and
carefully position the electrodes as designated on the grid paper. Fill the electrolytic tank with
400 ml of distilled water. Place the steel needle in the insulated mount and connect the
voltmeter so as to measure the voltage of the needle probe relative to the grounded electrode.
Turn on the AC source and record the source voltage. Starting at the center of the electrode,
map the 2V, 4V, 6V, 8V and 10V points on each vertical centimeter line (using symmetry,
you can reduce the number of points to be measured) . Carefully draw the 2V, 4V, 6V, 8V
and 10V equipotential contours for this electrode configuration and include this plot in your
report.
Carefully map the potential completely around the positive electrode at points 1mm away
from the electrode. Use enough points to achieve an accurate plot in the variation of the
potential. At the corners of the electrode, average the potential obtained in the vertical and
horizontal directions. You can approximate the electric field normal to the electrode at the
points you have mapped using the following approximation:
(13)
where )V defines the change in potential over the distance )x. Determine the electric field
on the surface of the electrode and plot your electric field verses the distance around the
electrode l [plot E (l ) vs. l] . Use the center point on the upper surface of the electrode as
your reference (l = 0) where l increases in the clockwise direction around the electrode.
7.
8.
Assuming the electric field you determined on the electrode surface represents that on an
equivalent three dimensional parallel plate conductor system of length 1m immersed in air, use
Gauss’s law to determine the total charge on the charged conductor.
Reduce the separation distance between the two electrodes to one-half that of part (5).
Repeat part (5.) for the new electrode geometry.
Repeat part (7.) for the new electrode geometry.
Cylindrical Magnet Flux Density (Gauss’s Law for Magnetic Fields)
9.
10.
Install the transverse probe on the gaussmeter and place the probe on the probe mount. Be
sure to zero the probe before making any measurements. Place the small cylindrical magnet
on a flat, non-magnetizable surface. Set the probe position to a convenient starting point and
carefully move the probe stand to position the probe directly over the axis of the magnet
where the maximum flux density is measured (the height of the probe above the magnet is
unimportant as long as a large enough flux density to obtain an accurate measurement is
achieved). Accurately measure the vertical magnetic flux density as a function of D where
D = 0 represents the axis of the magnet. You will need many more points in regions where the
flux density varies rapidly. Include a plot of your measured B(D) vs. D in your report.
Using your measured results, evaluate the Gauss’s law for magnetic fields integral of Equation
(12). Determine the total magnetic flux into your Gaussian surface and the total magnetic
flux out of your Gaussian surface. Compare these values to the net magnetic flux that you
determined by integration in Equation (12).
5. Additional Question
1.
The electric field on the surface of a conducting sphere is measured and found to be:
Using Gauss’ law, determine how much charge is located on the sphere.