2 Capacitor q -q Any two conductors separated by either an insulator or vacuum for a capacitor The “charge of a capacitor” is the absolute value of the charge on one of conductors. This constant is called the “capacitance” and is geometry dependent. It is the “capacity” for holding charge at a constant voltage Potential difference=V
4 Interesting Fact When a capacitor has reached full charge, q, then it is often useful to think of the capacitor as a battery which supplies EMF to the circuit.
5 Simple Circuit S H L Initially, H & L =0 +q i -i -q After S is closed, H=+q L=-q
6 Recalling Displacement Current i +q -q i idid This plate induces a negative charge here Which means the positive charge carriers are moving here and thus a positive current moving to the right Maxwell thought of the capacitor as a flow device, like a resistor so a “displacement current” would flow between the plates of the capacitor like this
7 If conductors had area, A Then current density would be J d =i d /A
8 Calculating Capacitance Calculate the E-field in terms of charge and geometrical conditions Calculate the voltage by integrating the E-field. You now have V=q*something and since q=CV then 1/something=capacitance
9 Parallel plates of area A and distance, D, apart Distance=D Area, A
10 Coaxial Cable—Inner conductor of radius a and thin outer conductor radius b +q -q
11 Spherical Conductor—Inner conductor radius A and thin outer conductor of radius B
12 Isolated Sphere of radius A
13 Capacitors in Parallel E C1C1 C2C2 C3C3 C eq E E i1i1 i2i2 i3i3 i i i=i 1 +i 2 +i 3 implies q=q 1 +q 2 +q 3 q q1q1 q2q2 q3q3
14 Capacitors in Series E C1C1 i i q q qC2C2 C3C3 By the loop rule, E=V 1 +V 2 +V 3 E C eq
15 Energy Stored in Capacitors Technically, this is the potential to do work or potential energy, U U=1/2 CV 2 or U=1/2 q 2 /C Recall Spring’s Potential Energy U=1/2 kx 2
16 Energy Density, u u=energy/volume Assume parallel plates at right Vol=AD U=1/2 CV Distance=D Area, A Volume wherein energy resides
17 Dielectrics Distance=D Area, A Insulator Voltage at which the insulating material allows current flow (“break down”) is called the breakdown voltage 1 cm of dry air has a breakdown voltage of 30 kV (wet air less)
18 The capacitance is said to increase because we can put more voltage (or charge) on the capacitor before breakdown. The “dielectric strength” of vacuum is 1 Dry air is So we can replace, our old capacitance, C air, by a capacitance based on the dielectric strength, , which is C new = *C air An example is the white dielectric material in coaxial cable, typically polyethylene ( =2.25) or polyurethane ( =3.4) Dielectric strength is dependent on the frequency of the electric field
19 Induced Charge and Polarization in Dielectrics E0E0 EiEi E Total =E 0 -E i Note that the charges have separated or polarized
20 Permittivity of the Dielectric 0 For real materials, we define a “D-field” where D= 0 E For these same materials, there can be a magnetization based on the magnetic susceptibility, , : H= B
21 Capacitor Rule For a move through a capacitor in the direction of current, the change in potential is –q/C If the move opposes the current then the change in potential is +q/C. i move VaVa VbVb V a -V b = -q/C V a -V b = +q/C
22 RC Circuits Initially, S is open so at t=0, i=0 in the resistor, and the charge on the capacitor is 0. Recall that i=dq/dt B A V S R C
23 Switch to A Start at S (loop clockwise) and use the loop rule B A V S R C
24 An Asatz—A guess of the solution
25 Ramifications of Charge At t=0, q(0)=CV-CV=0 At t=∞, q=CV (indicating fully charged) What is the current between t=0 and the time when the capacitor is fully charged?
26 Ramifications of Current At t=0, i(0)=V/R (indicates full current) At t=∞, i=0 which indicates that the current has stopped flowing. Another interpretation is that the capacitor has an EMF =V and thus B A V S R ~V Circuit after a very long time
27 Voltage across the resistor and capacitor B A V S R C Potential across resistor, V R Potential across capacitor, V C At t=0, V C =0 and V R = V At t=∞, V C =V and V R =0
28 RC—Not just a cola RC is called the “time constant” of the circuit RC has units of time (seconds) and represents the time it takes for the charge in the capacitor to reach 63% of its maximum value When RC=t, then the exponent is -1 or e -1 =RC
29 Switch to B The capacitor is fully charged to V or q=CV at t=0 B A V S R C CV
30 Ramifications At t=0, q=CV and i=-V/R At t=∞, q=0 and i=0 (fully discharging) Where does the charge go? The charge is lost through the resistor
31 Three Connection Conventions For Schematic Drawings A B C Connection Between Wires No Connection
32 Ground Connectors Equivalently
33 Household Wiring “hot” or black “return”/ “neutral” or white “ground” or green Normally, the “return” should be at 0 V w.r.t. ground Single Phase Rated 20 A (NW-14) Max V 120 VAC In THEORY, but sometimes no!
34 The Death of Little Johnny Washer hot neutral Little Johnny A short develops between the hot lead and the washer case RGRG 120V R Little Johnny RGRG If R G =∞, then Johnny is safe Uhoh! It leaks! If R G =0, then Johnny is dead! X
35 Saving Little Johnny Washer hot neutral Little Johnny A short develops between the hot lead and the washer case RGRG 120V R Little Johnny RGRG Uhoh! It leaks! No Path to Johnny!