EE 3313 Electromagnetics Laboratory Experiment #1 Maxwell s Equations I: Gauss s Law for Electric Fields and Gauss s Law for Magnetic Fields 1. Objective The objective of Experiment #1 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 (q 1,q) 2 located in a medium characterized by permittivity ( ) is defined by Coulomb s law: 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. (1) 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 q yields the definition of the electric field (E). The electric field is defined as the vector force per unit charge on the (positive) test charge q 2. The electric flux density is obtained by multiplying the vector electric field by the permittivity. 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 2 (2) (3)
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 R as shown in Figure 1, the total charge on the electrode Q is given by o (4) (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 max defines the maximum radius where the electric flux density becomes negligible. The cylindrically symmetric electrode produces a cylindrically symmetric electric field (independent of ) 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 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 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 R o 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 (9) (10) (11) where max 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 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 200V. 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. 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. 2. 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.). 3. 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.). 4. 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 4 by placing the conductor geometry within a poorly conducting medium (distilled water, 10 /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. 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 12VAC 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. 6. 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.
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. 7. Reduce the separation distance between the two electrodes to one-half that of part (5). Repeat part (5.) for the new electrode geometry. 8. Repeat part (7.) for the new electrode geometry. Cylindrical Magnet Flux Density (Gauss s Law for Magnetic Fields) 9. 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 where =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( ) vs. in your report. 10. 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.