7/06 Electric Fields and Energy


 Bruno Pierce
 3 years ago
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1 Part ASome standard electric field and potential configurations About this lab: Electric fields are created by electric charges and exert force on charges. Electric potential gives an alternative description. You will study some simple cases illustrating how the field geometry is related to the potential geometry and to the charge distribution geometry.. Electric fields also store energy (as do magnetic fields). You will consider how and where the energy is stored, and how it can be dissipated. Some important electric field geometries: Besides the fundamental single point charge (monopole) field configuration (field directed radially out or i, depending on charge sign), there are some important special and simple electrostatic cases: plane parallel, coaxial cylindrical and dipole (oppositely charged points). > For infinite parallel conducting plates (in approximation, plates with separation << linear V dimensions), the field is uniform and perpendicular to the plates, of strength Separation. > For the dipole field (two opposite point charges) the field lines become circular close to the charges (<< separation distance) and form a distinctive pattern in general. (The simplest pattern for magnetic field lines is the dipole pattern there are no magnetic monopoles in the present cosmological epoch.)
2 Figure 1 Electric Potential and Field Mapping Arrangement From bottom up: Oppositely charged parallel electrodes with intermediate circular shielded region; Two oppositely charged points (dipole); Oppositely charged parallel electrodes The experimental arrangement for simulating the electric field between two point charges is shown below in Figure 2.. A DC power supply set just under 20 volts is connected across the two conductors. A digital multimeter set to read DC voltage on the 20V scale (if the applied voltage exceeds 20V the DMM will always read 1) has its negative lead fixed to the negative electrode; the positive meter lead may be touched to any point on the highresistance conducting sheet (centimeter grid) to measure the electric potential at that point.
3 Figure 2 Field map board, power supply and digital meter Electric charges flow from one electrode to the other through the conducting sheet; this situation simulates very closely that of two charged electrodes in electrostatic equilibrium. Procedure 1. Choose an electrode configuration and print outa graph page of Electric Maps Report.sxw, which are the graph grids for Dipole, Parallel Plates, Ring & Plates. Every time you identify an equipotential point (you will seek out the 4V points, then the 8V points, etc), mark it with pencil on the graph grid (not on the conducting paper). 2. For your electrode configuration (Dipole, Parallel Plates, Ring & Plates) locate on the conducting paper and plot on the graph four equipotential lines, at 4V, 8V, 12V, and 16V. (All points of one conductor are at 0V; all points of the other at (just under) +20V. Check it!) Indicate on the field plot the polarity of each electrode, (+ or ). (Put scale numbers on your plot axes.) The plot need not necessarily be exactly to scale; we are interested in shapes.) Take enough points so that you can draw fairly accurate equipotential lines, but not so many as to make it needlessly tiresome. For the parallel strip electrodes measure potential beyond and to the sides of the electrode region. For the circular electrode between parallel strips make measurements inside the ring to test shielding. For the two point electrode configuration, measure behind the points as well as between, to map the dipole (twopole) shape. In all cases be careful to take sufficient additional data to define the shape of the equipotential lines and corresponding E field lines close to the conducting electrodes. To simplify plotting it is a good idea to scan along grid lines of the conductor. If one scan direction gives slow variation, try the perpendicular direction. If time permits, do the entire conducting sheet; if time is short it will be sufficient to measure one side (left or right) and to sketch the other side symmetrically. Lift the probe to move it; don't gouge the black paper. Please do not write on or scratch the paper. Replacement is very costly.
4 3. Draw a smooth line among points at the same potential. Label all equipotential line with voltage. Then draw associated electricfield lines distinctively (use dashed lines for E. The E field lines should be everywhere perpendicular to the equipotential lines, from the defining relation between E and V, and should enter conductors at right angles (mobile charges very quickly readjust to make this so). The direction of E is toward decreasing potential V. Indicate on your plot the direction of the E field lines. Part B Electric energy storage and dissipation Figure 3 Capacitor chargingdischarging setup About this section: Capacitance ("C") is a numerical property of a particular electrode geometry. It specifies the relation between electrode charge and electrode potential difference. Resistance ("R") is a process involving electrical (organized) energy dissipation into (random) energy (heat).
5 Electrostatic energy stored in a capacitor arrangement is dissipated in circuit resistance when current is allowed to flow. The discharge voltage vs. time curve is characteristically exponential with time e  t R equivalent C equivalent. If the current is reversed with an EMF (battery or power supply), the charging voltage curve vs. time involves the same time constant [ 1 e  t R equivalent C equivalent ]. Figure 4 Charging/discharge curves with the same time constant R eq C eq = 15 seconds Recording and fitting such experimental curves determines the "time constant" product R equivalent C equivalent, where "equivalent" denotes the net electrical effect of combinations of C's or R's.
6 (Warning: The Graphical Analysis Analyze: Curve Fit function has standard forms like these. However, GA uses a parameter C which is the inverse of the time constant product.) The volume energy density (joules/cubic meter, in mks units) stored in an electric field is proportional to E 2. (A similar relation involving B 2 holds for magnetic energy storage.) For a fixed conductor geometry the E fields, corresponding charges +/ q and corresponding voltages V are all proportional. Thus the stored energy is proportional to the square of applied voltage V, to the square of charges q on conductors, and also to the space average of E 2 (space integral). The field strength representation of stored energy is the most fundamental, as traveling electromagnetic waves are kinetic, not electrostatic, and there is no applied voltage or charge to relate them to. Voltage is an energy concept, defined reciprocally (space derivative and space integral) to the corresponding electric field: E =  dv dl (dl = path element in direction of the max V increase), and V =  E dl, where the dot indicates the cosine between the two vectors. Apparatus: RC (ResistorCapacitor) circuit box, voltmeter, power supply, cables; additional resistors Capacitors are devices for storing electric field energy. They exist commercially in myriad forms and with varied properties. Depending on intended use, one property or another may be most desirable high voltage operation, compactness, low loss in AC operation, cost etc. They can serve to isolate one voltage level from another (an ideal capacitor does not pass DC current), as part of a timing circuit, as part of a voltage ripple filter, etc. Capacity occurs by virtue of electric field lines between the charge on two surfaces. It is sometimes undesired but unavoidable, as in electronic chips or other circuits, and designers must cope with the consequences of stray capacitance. Whether these are of major consequence or not may depend on the frequency of operation. (Similar considerations are involved in the presence of stray inductance, the analogous magnetic energy storage element.) The unit of capacitance is the farad, obviously named for Michael Faraday. A farad of capacitance is a very large amount; milli, micro, nano, or picofarads are much more common as discrete commercial devices. Stray capacitances may be even smaller.
7 Resistors are energy dissipating devices. As circuit elements, they involve a voltage drop ir, where i is the current. In series with a capacitor, they delay the charge or discharge when a switch is opened or closed., producing (for discharge) an exponential variation of the voltage across the capacitor. (See discussion below of charging and discharging.) But, air can be also considered to be resistive (dissipative) when it is ionized in an electrical discharge. Heat is produced and, when the air discharge is explosive, thunder also results. Getting the stored energy out of (or into) a capacitor Think of the capacitor as a cubical detention pond with a valved outflow pipe. When the valve is opened, the rate of fall of the water height (voltage) depends on the size of the pipe (resistance). And if two ponds are cross connected with a very large pipe (connected in parallel by a very low resistance), with a single small outflow pipe, the common level falls more slowly capacitors in parallel add. Discharging For discharge t / V =V 0 e where is the product RC. V = 0 as t > infinity. The voltage decreases by a factor 1/e every seconds. (This exponential decay is similar to that of a radioactive sample which is not being replenished.) (Note once again: Graphical Analysis has a Curve Fit function of the form: exp(cx). Here, the GA x is our time t, and the GA C is our inverse : C GA = 1/τ. (GA's C is obviously not a capacitance.) So, our = RC has dimensions of time, but GA's C has units of inverse time.) It is frequently easier to observe T 1 2 on a graph, the time for the voltage to reduce by half of starting value. The relation is T 1 2 = ln2 = ( ln 2 because we want the half time; if we wanted the 1/3 time, ln 3 would be involved, etc.) Charging For charging, the same time constant is involved
8 V = V 0 ( 1 e  t ) V approaches V 0 as t > infinity. The larger the time constant, the slower the charging or discharging. Combinations For various combinations of circuit elements, single equivalent values may be used by following these rules: > Resistors in series (same current through all) add directly: R series = R 1 + R 2 + R > Resistors in parallel (same voltage across all) add reciprocally: 1 R = 1 R R R (Your calculator 1/x function will handle this nicely. Don't forget the final inversion to get the equivalent R) > Capacitors in parallel (same voltage across all) add directly: C parallel = C 1 + C 2 +C > Capacitors in series (same charge (magnitude) on all plates) add reciprocally: 1 C series = 1 C C C The rules for equivalent resistors and capacitors thus interchange.
9 The schematics below represent charging/discharging circuits in which a capacitor and resistor are connected in series with a battery or power supply. Voltmeter +  A S + V(b) B C R Figure 5 Charge/discharge circuit When the switch S is placed in position A, the battery charges the circuit, i.e. charge flows from the battery thru the resistor into the capacitor, until the capacitor is fully charged. When the switch is placed in position B, the capacitor (which stores charge and energy) discharges through the resistor (which dissipates charge energy). You will investigate how quickly this charge enters and leaves the capacitor by measuring the voltage across it as a function of time and fitting the data. You will also connect capacitors in series (end to end) and parallel to measure the equivalent capacitance. Capacitance is the capacity to store charge, measured in farads, defined by: q = CV where q (Coulombs) is the charge on the capacitor and V (Volts) is the voltage across it. The electrical energy stored in the capacitor is given by stored energy = 1 2 CV 2 or, equivalently 1 q 2 2 C How to connect the RC circuit
10 A + B Two capacitors and one resistor are already wired into a single box with connection jacks. You need only connect the box to the power supply (which acts as the battery in the circuit) and the voltmeter. C E + D F The capacitors in the box are polarized and will only work if connected in one direction; Ground (black) on the power supply should only be connected to black on the box, red on the power supply should only be connected to red on the box. Figure 6 By connecting to different jacks (which are labeled A, B, C, etc) using the supplied cables, you can create various circuits as shown below: Power Supply Plug into +20V to Charge Capacitor Unplug from +20V to Discharge Capacitor +20V Ground Voltmeter +  A B F E C R Figure 7 Single capacitor discharge through a resistor:
11 Power Supply +20V Ground Plug into +20V to Charge Capacitor Unplug from +20V to Discharge Capacitor Voltmeter +  C D : A B F E C R Figure 8 Parallel Capacitors dischare through a resistor Figure 9 Single capacitor charging through a resistor: Procedure A. Single capacitor discharge Use your wristwatch to record voltage vs. time every five seconds, on scrap. Enter into Graphical
12 Analysis: Electric Fields and Energy, decide what theoretical function should describe the data and Analyze: Curve Fit to obtain the value of. Predict the time constant from the values of the resistances and capacitances in your circuit, using equivalent values as discussed above. Remember the warning that C GA is different than C cap. Save. B. Parallel capacitors discharge Proceed as above. Save. C. Series capacitors discharge Save. Note that there is no diagram for two capacitors in series. Design the experiment and hook up the cables according to what you think the circuit should be connected but have your instructor check the connection before performing the experiment. Then proceed as above. D. Single capacitor charge Save. Proceed as above. (Series and parallel resistors) Only if so directed, apply a known voltage to series and parallel resistor combinations and observe the current through the power supply. Sketch the circuit. Calculate and record the expected current from i calculated the circuit parameters and form the ratio. i measured Report Submit potential/field maps. Print composite charge/discharge graph. Show on each your calculation of the time constant R eq C eq, in terms of the given component values. Show ratio τ fit /τ calculated.
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