6/06 E Field Energy Storage

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Herbert Kelley
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Capacitor charging-discharging setup About this lab: The volume energy density (joules/cubic meter, in mks units) stored in an electric field is proportional to E 2.

1 Capacitor charging-discharging setup About this lab: 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:

2 E = - dv (dl = path element in direction of the max V increase), and dl V = - Edl. Apparatus: RC (Resistor-Capacitor) circuit box, voltmeter, power supply, cables; additional resistors Some important field geometries Besides the fundamental single point charge (monopole) field configuration (field directed radially in out 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 dimensions), the field is uniform and perpendicular to the plates, of strength V/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.) 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.)

3 The unit of capacitance is the farad, obviously named for Michael Faraday. A farad of capacitance is a very large amount; mill, micro, nano, or pico farads are much more common as discrete commercial devices. Stray capacitances may be even smaller. 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.) 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 V = V 0 e t / 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: 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 has dimensions of time, but GA's C has units of inverse time.) It is frequently easier to observe T 1 2, the time for the voltage to reduce by half of starting value. The relation is T 1 2 = ln2 = ( ln2 because we want the half

4 time; if we wanted the 1/3 time, ln3 would be involved, etc.) Charging For charging, the same time constant is involved 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

5 > Capacitors in series (same charge (magnitude) on all plates) add reciprocally: 1 C series = 1 C C C The rules for resistors and capacitors thus interchange. The schematics below represent charging/discharging circuits in which a capacitor and resistor are connected in series with a battery. Voltmeter + - A S + V(b) B C R When the switch S is placed in position A, the battery charges the circuit charge flows from the battery 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 analyzing 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

6 How to connect the RC circuit Two capacitors and one resistor are already wired into a single box with connection jacks. You need only connect the box to A + B 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. By connecting to different jacks (which are labeled A, B, C, etc) using the supplied cables, you can create various circuits as shown below: Single capacitor discharge through a resistor: Power Supply Plug into +20V to Charge Capacitor Unplug from +20V to Discharge Capacitor +20V Ground Voltmeter + - A B F E C R Parallel Capacitors discharging through a resistor:

7 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 Single capacitor charging through a resistor:

8 Procedure A. Single capacitor discharge Record voltage vs. time on scrap. Enter into Graphical Analysis, 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. Save. B. Parallel capacitors discharge Proceed as above. C. Series capacitors discharge 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 Proceed as above. (Series and parallel resistors) 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 the circuit parameters and form the ratio i calculated i measured. Report Print graphs as directed. Sketch on the graph your circuit (show parameter values) and show your calculation of the time constant. Show ratio τ fit /τ calculated.

Exercises on Voltage, Capacitance and Circuits. A d = (8.85 10 12 ) π(0.05)2 = 6.95 10 11 F

Exercises on Voltage, Capacitance and Circuits Exercise 1.1 Instead of buying a capacitor, you decide to make one. Your capacitor consists of two circular metal plates, each with a radius of 5 cm. The