Transcript lecture19
Chapter 16 Capacitors Batteries Parallel Circuits Series Circuits Friday, February 26, 1999 in class Ch. 15 - 16 Hint: Be able to do the homework (both the problems to turn in AND the recommended ones) you’ll do fine on the exam! You may bring one 3”X5” index card (hand-written on both sides), a pencil or pen, and a scientific calculator with you. 2 1 Q 1 2 W U QV CV 2 2C 2 U = INTERNAL ENERGY of the capacitor. This is where the energy comes from to power many of our cordless, rechargeable devices…When it’s 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 becomes C = k Co Co 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 k Co Current and Resistance It’s time to develop an understanding of electrical systems that are NOT in electrostatic equilibrium. In these systems, charges move, under the influence of an externally imposed electric fields. Such systems provide us with the useful electricity we get out of flashlight batteries and rechargeable devices. We’ve already used this concept, even though we haven’t formally introduced it. What do you think of when you hear the word “current?” Electrical Current is simply the flow of electrical charges. charge carriers moving charges. can be + or - A The current is the number of charges flowing through a surface A per unit time. Q I t Q I t 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, it’s 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? E electrical force on the charges in the conductor. charges can move in conductors. a current flows! Vd t E A The charge carriers, each with charge q, move with v an average speed d 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 t is given by Q nqAvd t Q I nqAvd t Electric fields exert an electrical force on charges given by F qE And I remember from last semester that Newton’s 2nd Law says F ma So shouldn’t the charges be accelerating instead of moving with an average velocity vd? Let’s follow the path of one of the charge carriers to see what’s really going on... Vd t E A The thermal motion of the charges in the conductor keep the charges bouncing around all over the places, hitting the fixed atoms in the conductor. The electric field exerts a force which “gently” guides the positive charges toward the right so that over time, they appear to drift along the electric field. You might think of the collisions as a frictional force opposing 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 average electron to traverse a 1m length of 14 gauge copper wire if the current in the wire is 1 amp? Let’s try to guess first: a) years b) weeks c) days d) hours e) minutes f) seconds g) microseconds h) nanoseconds 14 gauge wire is a common size of wire having a radius of 0.0814 cm Let’s assume that atom of copper is able to supply one free charge to the current. The mass density of copper is 8.92 g/cm3 1 mole of copper weighs 63.5 g. So, the number density of charge carriers in the copper is... n (6.02 10 23 atoms mol g mol (635 . )(8.92 ) g cm3 ) n = 8.46 X 1022 atoms/cm3 Q I nqAvd t I vd nq A q = 1.6 X 10-19 C A = p r2 = 2.1 X 10-2 cm2 1A vd 22 chgs 19 (8.46 10 . 10 3 )(16 C chg cm 2 )(2.1 10 cm ) 3 cm s vd 355 . 10 d 100 cm 4 t 2.8 10 s 3 cm vd 355 . 10 s t = 7.8 hours! 2 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=IR The objects for which Ohm’s Law holds are known as OHMIC. Objects for which resistance IS a function of the applied voltage (i.e., Ohm’s Law is invalid) are known as NON-OHMIC. V Slope = R I V I 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) length cross-sectional area R~L Certainly, if two wires have the same crosssectional area, the longer of the two will have the greater resistance. For 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 the ratio of the length (L) to the cross-sectional area (A) R L/A We define the slope to be the resistivity (r) of the material. R=rL/A The proportionality constant, r, is the resistivity. r=RA/L [r] = [R] [A] / [L] = W m2 / m =Wm 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. R Tc T Charge Insulators/Conductors Coulomb’s Law Electric Fields Potentials & Potential Energy Capacitors Series & Parallel Circuits + Two Kinds of Charge - Charge is Quantized An electron carries a charge of -1e. A proton carries a charge of +1e. 1e = 1.6 X 10-19 C Conservation of Charge. RUBBER The charges remain near the end of the rubber rod--right where we rubbed them on! GROUND COPPER Rub charges on here They move down the conductor toward our hand Eventually ending up in the ground. GROUND Rubber COPPER Bring negatively charged rubber ball close to the a copper rod. The copper rod is initially neutral. Negative charges on the copper run away from the rubber ball and into the ground. GROUND Rubber + ++ COPPER + The 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. GROUND Rubber + ++ Finally, remove the COPPER + + rubber ball... + + Put a rubber glove on your hand to insolate the copper rod from ground. GROUND The excess positive charge is trapped on the copper rod with + no path to ground. It COPPER + + + redistributes itself + uniformly over the + + copper rod. We have taken an initially neutral copper rod and induced a positive charge on it! GROUND Electrostatic Forces kq 1q 2 F r 2 r k = 9 X 109 N m2/C2 Superposition Principle kq E = 2 r r superposition principle applies So…. Etotal E1 E2 E3 ... F = qE + Field lines Far apart. - Field lines close together. Work in a UNIFORM electric field W=qEd Scalar quantity! No directions! Electrical work is “Quite Easily Done!” NOTE: d is the distance along the Electric field only!!! r q d E As long as the electric field is uniform, this is the answer! PE = -W = -q E d The Electrical Potential: V = Vb-Va = PE/q In a uniform field V = - E d, Wilbur! It’s EASY! +q Va E d Vb Vb - Va = - E d It decreases in the direction of the electric field, REGARDLESS OF THE SIGN OF THE CHARGE! The electrical potential ALWAYS decreases in the direction of the electric field! It does not depend upon the sign of the charge. The electrical potential energy depends upon the sign of the charge. It decreases in the direction of the electrical force. -q A Fe d B E What happens to the potential energy of a negative charge (-q) as it moves in the direction of the electric field? PE = - q E d = - (-q) E d = + q E d -q A Fe B d E What happens to the potential of a negative charge (-q) as it moves in the direction of the electric field? V = - E d For point charges The superposition principle applies to potentials! kq V r Vtot = V1 + V2 + V3 + ... Electrostatic Potential Energy kq 1q 2 PE r Equipotential surfaces are perpendicular to the electric field lines everywhere! Work is only done when a charge moves parallel to the electric field lines. So no work is done by the electric field as a charge moves along an equipotential surface. 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! Insulators and Conductors 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 a conductor must be perpendicular to the surface of the conductor. Capacitors C = eoA/d C=Q/V eo = permittivity of free space = 8.85 X 10-12 C2/Nm2 2 1 Q 1 2 W U QV CV 2 2C 2 + V _ Ceq = C1 + C2 C1 Capacitors in parallel ADD. + 1 Ceq C2 = +Q -Q C1, V1 V _ 1 C1 C2 + 1 +Q C2, V2 -Q Capacitors in series ADD INVERSELY.