# Chapter 16 Capacitors Batteries Parallel Circuits Series Circuits.

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• Chapter 16CapacitorsBatteriesParallel CircuitsSeries Circuits

• Hint: Be able to do the homework (both theproblems to turn in AND the recommended ones)youll do fine on the exam!Friday, February 26, 1999 in classCh. 15 - 16You may bring one 3X5 index card (hand-writtenon both sides), a pencil or pen, and a scientificcalculator with you.

• U = INTERNAL ENERGY of the capacitor.This is where the energy comes from to power many of our cordless, rechargeable devicesWhen its gone, we have to plug the devices into the wall socket to recharge the capacitors!

• Just so you know...Insulating materials between capacitor plates are known as dielectrics. In the circuits we have dealt with, that material is air. It could be other insulators (glass, rubber, etc.).Dielectric materials are characterized by a dielectric constant k such that when placed between the plates of a capacitor, the capacitance becomesC = k CoCo is the vacuum (air) capacitance.

• Dielectrics, therefore, increase the charge a capacitor can hold at a given voltage, since...Q = C V = k Co V

• Its time to develop an understanding ofelectrical systems that are NOT in electrostaticequilibrium. In these systems, charges move,under the influence of an externally imposedelectric fields. Such systems provide us withthe useful electricity we get out of flashlightbatteries and rechargeable devices.Current and Resistance

• Weve already used this concept, even though we havent formally introduced it.What do you think of when you hear the word current?

• Electrical Current is simply the flow of electrical charges.The current is the number of charges flowing through a surface A per unit time.

• By convention, we say that the direction of the current is the direction in which the positive charge carriers move.Note: for most materials we examine, its really the negative charge carriers that move. Nevertheless, we say that electrons move in a direction opposite to the electrical current.Leftover from Ben Franklin!

• Why do charges move?Well, what happens when you put an electric field across a conductor?electrical force on the charges in the conductor.charges can move in conductors.a current flows!

• EAVd DtThe charge carriers, each with charge q, move with an average speed vd in response to the electric field. If there are n charge carriers per volume in the conductor, then the number of charge carriers passing a surface A in a time interval Dt is given by

• Electric fields exert an electrical force on charges given byAnd I remember from last semester that Newtons 2nd Law saysSo shouldnt the charges be accelerating instead of moving with an average velocity vd?

• EAVd DtLets follow the path of one of the charge carriers to see whats really going on...The thermal motion of the charges in the conductor keep thecharges bouncing around all over the places, hitting thefixed atoms in the conductor.The electric field exerts a force which gently guides thepositive charges toward the right so that over time, theyappear to drift along the electric field.You might think of the collisions as a frictional forceopposing the flow generated by the electric field.

• You are probably asking yourself, So, just how long does it take the average electron to traverse a 1m length of 14 gauge copper wire if the current in the wire is 1 amp?

• How long does it take the averageelectron to traverse a 1m length of14 gauge copper wire if the currentin the wire is 1 amp?Lets try to guess first:a) yearsb) weeksc) daysd) hourse) minutesf) secondsg) microsecondsh) nanoseconds

• 14 gauge wire is a common size of wire having a radius of 0.0814 cmLets assume that atom of copper is able tosupply one free charge to the current.The mass density of copper is 8.92 g/cm31 mole of copper weighs 63.5 g.So, the number density of charge carriers in the copper is...

• n = 8.46 X 1022 atoms/cm3q = 1.6 X 10-19 CA = p r2 = 2.1 X 10-2 cm2

• Describes the degree to which a current through a conductor is impeded.In particular, if a voltage V is applied across a conductor, a current I will flow. The resistance R is defined to be:R = V / I

• R = V / I[R] = [V] / [I][R] = Volt / Amp= Ohm (W)

• Georg Ohm (early 19th century) systematically examined the electrical properties of a large number of materials. He found that the resistance of a large number of objects is NOT dependent upon the applied voltage. That is...V = I R

• Every material has its own characteristic resistivity to the conduction of electric charge. On what does the Resistance (R) of an object depend? (OHMIC CONDUCTORS)lengthcross-sectional area

• Certainly, if two wires have the same cross-sectional area, the longer of the two will have the greater resistance.R ~ LFor two wires of the same length, the one with the larger cross-sectional area will have the smaller resistance. Think of water flowing in a pipe.R ~ 1 / A

• So if we plotted the resistance (R) versus theratio of the length (L) to the cross-sectionalarea (A)We define the slope to be theresistivity (r) of the material.

• R = r L / AThe proportionality constant, r, is the resistivity.r = R A / L[r] = [R] [A] / [L]= W m2 / m= W m

• Resisitivity and Resistance are also a function of Temperature.r = ro [ 1 + a ( T - To) ]R = Ro [ 1 + a ( T - To) ]To is usually taken to be 20o C.a is the temperature coefficient of resistivity and is a characteristic of the particular material.

• The temperature dependence of resistance holds for everyday to warm temperatures. At very low temperatures, for some materials, the resistance can fall to zero. These materials are known as superconductors.The temperature at which their resistance falls off rapidly is known as the critical temperature.

• ChargeInsulators/ConductorsCoulombs LawElectric FieldsPotentials & Potential EnergyCapacitorsSeries & Parallel Circuits

• Conservation of Charge.Charge is QuantizedAn electron carries a charge of -1e.A proton carries a charge of +1e.1e = 1.6 X 10-19 C

• GROUNDRUBBERThe charges remain near the end of the rubber rod--right where we rubbed them on!

• GROUNDCOPPERRub charges on hereThey move down theconductor toward our handEventually ending up in the ground.

• GROUNDCOPPERBring negatively charged rubber ball close to the a copper rod. The copper rod is initially neutral.Negative charges on the copper runaway from the rubber ball and into theground.

• GROUNDCOPPERThe copper rod is now positively charged. The electrons originally on it were forced away into the ground by the negative charges on the rubber ball.+++++++

• GROUNDCOPPERPut a rubber glove on your hand to insolate the copper rod from ground.+++++++Finally, remove the rubber ball...

• GROUNDCOPPER+++++++The excess positive charge is trapped on the copper rod with no path to ground. It redistributes itself uniformly over thecopper rod. We have taken an initially neutral copper rod and induced a positive charge on it!

• k = 9 X 109 N m2/C2Superposition PrincipleElectrostatic Forces

• NOTE: d is the distance alongthe Electric field only!!!Electrical work is Quite Easily Done!Work in a UNIFORM electric fieldAs long as theelectric fieldis uniform, thisis the answer!DPE = -W = -q E d

• DV = Vb-Va = DPE/qThe Electrical Potential:In a uniform field

• Vb - Va = - E dIt decreases in the direction of the electric field,REGARDLESS OF THE SIGN OF THE CHARGE!

• The electrical potential ALWAYSdecreases in the direction of theelectric field! It does not dependupon the sign of the charge.The electrical potential energydepends upon the sign of the charge.It decreases in the direction of theelectrical force.

• What happens to the potential energy of a negative charge (-q) as it moves in the direction of the electric field?DPE = - q E d = - (-q) E d = + q E d

• What happens to the potential of a negative charge (-q) as it moves in the direction of the electric field?DV = - E d

• For pointchargesElectrostatic Potential Energy

• Equipotential surfaces are perpendicular to the electric field lines everywhere! When in electrostatic equilibrium (i.e., no charges are moving around), all points on and inside of a conductor are at the same electrical potential!Work is only done when a charge movesparallel to the electric field lines. So no work is done by the electric field as a charge moves along an equipotential surface.

• 1) no electric field exists inside conductor.2) Excess charges on an isolated conductor are found entirely on its surface.3) The electric field just outside of aconductor must be perpendicularto the surface of the conductor.Insulators and Conductors

• C = eoA/deo = permittivity of free space = 8.85 X 10-12 C2/Nm2C = Q / VCapacitors

• Ceq = C1 + C2Capacitors in parallel ADD.