Physics 2102 Gabriela González Physics 2102 Capacitors
Summary (last class) • Any two charged conductors form a capacitor. • Capacitance : C= Q/V • Simple Capacitors:Parallel plates: C = e0 A/dSpherical : C =e04pab/(b-a)Cylindrical: C =e0 2pL/ln(b/a)
Energy Stored in a Capacitor • Start out with uncharged capacitor • Transfer small amount of charge dq from one plate to the other until charge on each plate has magnitude Q • How much work was needed? dq
General expression for any region with vacuum (or air) volume = Ad Energy Stored in Electric Field • Energy stored in capacitor:U = Q2/(2C) = CV2/2 • View the energy as stored in ELECTRIC FIELD • For example, parallel plate capacitor: Energy DENSITY = energy/volume = u =
DIELECTRIC +Q - Q Dielectric Constant • If the space between capacitor plates is filled by a dielectric, the capacitance INCREASES by a factor • This is a useful, working definition for dielectric constant. • Typical values of k: 10 - 200 C = A/d
dielectric slab Example • Capacitor has charge Q, voltage V • Battery remains connected while dielectric slab is inserted. • Do the following increase, decrease or stay the same: • Potential difference? • Capacitance? • Charge? • Electric field?
dielectric slab Example (soln) • Initial values: capacitance = C; charge = Q; potential difference = V; electric field = E; • Battery remains connected • V is FIXED; Vnew = V (same) • Cnew = kC (increases) • Qnew = (kC)V = kQ(increases). • Since Vnew = V, Enew = E (same) Energy stored? u=e0E2/2 => u=ke0E2/2 = eE2/2
C1 Q1 A B C2 Q2 D C Ceq Qtotal Capacitors in Parallel • A wire is a conductor, so it is an equipotential. • Capacitors in parallel have SAME potential difference but NOT ALWAYS same charge. • VAB = VCD = V • Qtotal= Q1 + Q2 • CeqV = C1V + C2V • Ceq = C1 + C2 • Equivalent parallel capacitance = sum of capacitances • PARALLEL: • V is same for all capacitors • Total charge in Ceq= sum of charges
Q1 Q2 B C A C1 C2 Q Ceq Capacitors in series • Q1 = Q2 = Q (WHY??) • VAC = VAB + VBC • SERIES: • Q is same for all capacitors • Total potential difference in Ceq = sum of V
C1 Q1 C2 Q2 Qeq Ceq C1 C2 Capacitors in parallel and in series • In parallel : • Ceq = C1 + C2 • Veq=V1=V2 • Qeq=Q1+Q2 • In series : • 1/Ceq = 1/C1 + 1/C2 • Veq=V1 +V2 • Qeq=Q1=Q2 Q1 Q2
Summary • Capacitors in series and in parallel: • in series: charge is the same, potential adds, equivalent capacitance is given by 1/C=1/C1+1/C2 • in parallel: charge adds, potential is the same,equivalent capaciatnce is given by C=C1+C2. • Energy in a capacitor: U=Q2/2C=CV2/2; energy density u=e0E2/2 • Capacitor with a dielectric: capacitance increases C’=kC
10 mF 20 mF 30 mF 120V Example 1 What is the charge on each capacitor? • Q = CV; V = 120 V • Q1 = (10 mF)(120V) = 1200 mC • Q2 = (20 mF)(120V) = 2400 mC • Q3 = (30 mF)(120V) = 3600 mC • Note that: • Total charge (7200 mC) is shared between the 3 capacitors in the ratio C1:C2:C3 -- i.e. 1:2:3
20 mF 30 mF 10 mF 120V Example 2 • Q = CV; Q is same for all capacitors • Combined C is given by: What is the potential difference across each capacitor? • Ceq = 5.46 mF • Q = CV = (5.46 mF)(120V) = 655 mC • V1= Q/C1 = (655 mC)/(10 mF) = 65.5 V • V2= Q/C2 = (655 mC)/(20 mF) = 32.75 V • V3= Q/C3 = (655 mC)/(30 mF) = 21.8 V Note: 120V is shared in the ratio of INVERSE capacitances i.e.1:(1/2):(1/3) (largest C gets smallest V)
10 mF 10 mF 10 mF 10V 5 mF 5 mF 10V Example 3 In the circuit shown, what is the charge on the 10F capacitor? • The two 5F capacitors are in parallel • Replace by 10F • Then, we have two 10F capacitors in series • So, there is 5V across the 10F capacitor of interest • Hence, Q = (10F )(5V) = 50C
10mF (C1) 20mF (C2) Example • 10mF capacitor is initially charged to 120V. 20mF capacitor is initially uncharged. • Switch is closed, equilibrium is reached. • How much energy is dissipated in the process? Initial charge on 10mF = (10mF)(120V)= 1200mC After switch is closed, let charges = Q1 and Q2. Charge is conserved: Q1 + Q2 = 1200mC • Q1 = 400mC • Q2 = 800mC • Vfinal= Q1/C1 = 40 V Also, Vfinal is same: Initial energy stored = (1/2)C1Vinitial2 = (0.5)(10mF)(120)2 = 72mJ Final energy stored = (1/2)(C1 + C2)Vfinal2 = (0.5)(30mF)(40)2 = 24mJ Energy lost (dissipated) = 48mJ
Physics 2102 Gabriela González Physics 2102 Current and resistance Georg Simon Ohm (1789-1854)
E Andre-Marie Ampere 1775-1836 Electrical current In a conductor, electrons are free to move. If there is a field E inside the conductor, F=qE means the electrons move in a direction opposite to the field: this is an electrical current. We think of current as motion of imaginary positive charges along the field directions.
E Electrical current Wasn’t the field supposed to be zero inside conductors? Yes, if the charges were in equilibrium. The reasoning was “electrons move until they cancel out the field”. If the situation is not static, that is, if electrons are moving, then the field can be nonzero in a conductor, and the potential is not constant across it! However, “somebody” has to be pumping the electrons: this is the job of the battery we put across a circuit. If there is no source creating the electric field, the charges reach equilibrium at E=0.
Electrical current: Conservation Current is a scalar, NOT a vector, although we use arrows to indicate direction of propagation. Current is conserved, because charge is conserved! “junction rule”: everything that comes in, must go out.
Ohm’s laws Georg Simon Ohm (1789-1854) "a professor who preaches such heresies is unworthy to teach science.” Prussian minister of education 1830 Devices specifically designed to have a constant value of R are called resistors, and symbolized by Resistance Electrons are not “completely free to move” in a conductor. They move erratically, colliding with the nuclei all the time: this is what we call “resistance”. The resistance is related to the potential we need to apply to a device to drive a given current through it. The larger the resistance, the larger the potential we need to drive the same current.