Understanding Capacitors: Principles, Calculations, and Applications

Chapters 24 and 26.4-26.5

Capacitor ++++++++++ ++++++++++++++ +q Potential difference=V - - - - - - - - - - - - - - - - - - - - - - - - - - -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

Units • 1 Farad=1 F= 1 C/V • Symbol: Indicates positive potential

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.

Simple Circuit Initially, H & L =0 After S is closed, H=+q L=-q -q L +q H S i -i

Recalling Displacement Current 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 -q +q i i id 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

If conductors had area, A • Then current density would be • Jd=id/A

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

Parallel plates of area A and distance, D, apart Area, A ++++++++++ ++++++++++++++ Distance=D - - - - - - - - - - - - - - - - - - - - - - - - - -

-q +q Coaxial Cable—Inner conductor of radius a and thin outer conductor radius b

Spherical Conductor—Inner conductor radius A and thin outer conductor of radius B

Isolated Sphere of radius A

Capacitors in Parallel i1 i2 i3 i E E C1 C2 q1 q2 C3 q3 i=i1+i2+i3 implies q=q1+q2+q3 i Ceq E q

Capacitors in Series i By the loop rule, E=V1+V2+V3 i C1 q E E Ceq C2 q C3 q

Energy Stored in Capacitors Technically, this is the potential to do work or potential energy, U U=1/2 CV2 or U=1/2 q2/C Recall Spring’s Potential Energy U=1/2 kx2

Energy Density, u Area, A • u=energy/volume • Assume parallel plates at right • Vol=AD • U=1/2 CV2 ++++++++++ ++++++++++++++ Distance=D - - - - - - - - - - - - - - - - - - - - - - - - - - Volume wherein energy resides

Dielectrics Area, A ++++++++++ ++++++++++++++ ++++++++++ ++++++++++++++ ++++++++++ ++++++++++++++ Insulator Distance=D - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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)

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 1.00059 • So we can replace, our old capacitance, Cair, by a capacitance based on the dielectric strength, k, which is • Cnew=k*Cair • An example is the white dielectric material in coaxial cable, typically polyethylene (k=2.25) or polyurethane (k=3.4) • Dielectric strength is dependent on the frequency of the electric field

Induced Charge and Polarization in Dielectrics ++++++++++++++++++++++ Note that the charges have separated or polarized • - - - - - - - - - - - - - - - - - - • +++++++++++++++++++ E0 Ei - - - - - - - - - - - - - - - - - - - - - - - ETotal=E0-Ei

Permittivity of the Dielectric • e=ke0 • For real materials, we define a “D-field” where • D=ke0E • For these same materials, there can be a magnetization based on the magnetic susceptibility, c, : • H= cm0B

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. move Va-Vb= -q/C Va-Vb= +q/C Vb Va i

R A S B V C 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

R A S B V C Switch to A • Start at S (loop clockwise) and use the loop rule

An Asatz—A guess of the solution

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?

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 R A S Circuit after a very long time B V ~V

R A S B V C Voltage across the resistor and capacitor Potential across resistor, VR Potential across capacitor, VC At t=0, VC=0 and VR=V At t=∞, VC=V and VR=0

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 • t=RC

Switch to B • The capacitor is fully charged to V or q=CV at t=0 R A S CV B V C

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

Three Connection Conventions For Schematic Drawings Connection Between Wires No Connection • A • B • C

Ground Connectors Equivalently

Household Wiring “hot” or black “return”/ “neutral” or white “ground” or green Single Phase Rated 20 A (NW-14) Max V 120 VAC Normally, the “return” should be at 0 V w.r.t. ground In THEORY, but sometimes no!

X X The Death of Little Johnny A short develops between the hot lead and the washer case Little Johnny hot Washer Uhoh! It leaks! neutral RG If RG=0, then Johnny is dead! If RG=∞, then Johnny is safe 120V RLittle Johnny RG

Saving Little Johnny A short develops between the hot lead and the washer case Little Johnny hot Washer Uhoh! It leaks! neutral RG No Path to Johnny! 120V RLittle Johnny RG

Understanding Capacitors: Principles, Calculations, and Applications