PHYS 205 Chapter 20, Fall 2017

Chapter 20 Electric Circuits Outline 1. Electromotive Force and Current 2. Ohm’s Law 3. Resistance and Resistivity 4. Electric Power 5. Alternating current 6. Series Wiring 7. Parallel Wiring 8. Circuits Wired Partially in Parallel and Partially in Series 9. Internal Resistance 10. Kirchhoff’s Rules 11. The Measurement of Current and Voltage 12. Capacitors in Series and in Parallel 13. RC Circuits 14. Safety and the Physiological Effects of Current 1- Electromotive Force and Current Within a battery, a chemical reaction occurs that transfers electrons from one terminal to another terminal. The maximum potential difference across the terminals is called the electromotive force (emf). I is the amount of charges passing through A per unit time: C in A Ampere ( ) A Q I t    2 Current density: ( ) I A J A m  Current and Drift Speed - drift speed = speed of e in a conductor dv  . . . d Q I n q v A t          . . . . . . charge in gray portion d Q n A x q n Av t q      n = number of mobile charges unit volume Charge Carrier Motion in a Conductor : drift velocity of e- Sites of collision of e- with atoms in the conductor slowing the motion 2- Ohm’s Law Ohm’s law states that I is proportional to I V R   resistance of the conductor V R I   V R   b - a V V V   Units:  = Ohms, symbol: Ohmic material V I R   V I R   Non-Ohmic material 3-Resistance and Resistivity A L R   and . . . d V E l q I n q v A t      = current density = . . d I J n q v A  Ohm's Law states that: where is the Electric conductivity of the material. J E    thus, but V I A I l A l V V l R R I A          1 Define the resistivity    l in R A    is a property of the material and is independent of the shape Temperature Variation of Resistivity The resistivity of a metal depends on temperature.     0 0 1 T T           0 0 1 R R T T      Where:  is the resistivity at temperature T 0 is the resistivity at temperature T0  is the temperature coefficient of resistivity Superconductors A class of materials and compounds whose resistances fall to virtually zero below a certain temperature, TC, , the critical temperature. The critical temperatures of: Al is 1.2 K, He liquid is 4.2 K. YBa2Cu3O7 is 90 K, N2 liquid is 80 K. Once a current is set up in a superconductor, it persists without any applied voltage, since R = 0 4- Electrical Power Consider the motion of a charge Q, Ground: reference for voltage From A to B, Q gains in potential energy: From C to D, Q loses in potential energy because of the heating of the resistor . Epot q V    The rate at which the energy is lost when going through the resistor is: Q V I V Power t       2 2 ( V ) P I R R    3 6 1 10 3600 3.6 10 1 Kilowatt-hour kWh W s J      From Ohm’s Law, alternate forms of power are: 5- Alternating Current In an AC circuit, the charge flow reverses direction periodically.  ft V V o sin 2  In circuits that contain only resistance, the current reverses direction each time the polarity of the generator reverses.    ft I ft R V R V I o o   sin 2 sin 2    peak current  ft I V IV P o o sin 2 2    ft I I o  sin 2  ft V V o sin 2  rms rms 2 2 2 V I V I I V P o o o o              Root-mean-square (rms) current and voltage R I V rms rms  VrmsIrms P  R I P 2 rms  R V P 2  rms 6- Series Wiring Resistors are connected end to end. Potentials add ΔV = IR1 + IR2 = I (R1+R2) Consequence of Conservation of Energy The equivalent resistance, Requ , has the effect on the circuit as the original combination of resistors Requ = R1 + R2 Example 8 Resistors in a Series Circuit A 6.00 Ω resistor and a 3.00 Ω resistor are connected in series with a 12.0 V battery. Assuming the battery contributes no resistance to the circuit, find (a) the current, (b) the power dissipated in each resistor, and (c) the total power delivered to the resistors by the battery. (a) (b) (c)       .9 00 .3 00 .6 00 RS .1 33 A 00 .9 12 0. V     RS V I     10 6. W .6 00 33 A .1 2 2     I R P     .5 31 W .3 00 33 A .1 2 2     I R P 15 9. W .5 31 W 10 6. W   P  7 – Parallel Wiring Resistors are connected across each other. Equivalent resistance replaces the two original resistances V = I1R1 = I2 R2 = Req I I = I1 +I2 ΔV/Req = ΔV/R1 + ΔV/R2 1 2 1 1 1 Requ R R   8 – Circuits Wired Partially in Parallel and Partially in Series The schematic shows the internal resistance, r The terminal voltage, ΔV = Vb-Va ΔV = ε – Ir For the entire circuit: ε = IR + Ir R = load resistor 9- Internal resistance 10- Kirchhoff’s Rules There are ways in which resistors can be connected so that the circuits formed cannot be reduced to a single equivalent resistor Two rules, called Kirchhoff’s Rules can be used to solve the problem. Statement of Kirchhoff’s Rules Junction Rule The sum of the currents entering any junction must equal the sum of the currents leaving that junction A statement of Conservation of Charge Loop Rule The sum of the potential differences across all the elements around any closed circuit loop must be zero A statement of Conservation of Energy I1 = I2 + I3 From Conservation of Charge Diagram b shows a mechanical analog The junction rule: Potential difference across R and ε Junction Equations from Kirchhoff’s Rules In general, the number of times the junction rule can be used is one fewer than the number of junction points in the circuit The loop rule can be used as often as needed so long as a new circuit element (resistor or battery) or a new current appears in each new equation You need as many independent equations as you have unknowns Junction rule at C: I1 +I2 = I3 Loop rule: Loop abcda: 10V - (6)I1 - (2)I3 = 0 Loop befcb: -14V + (6)I1 - 10V – (4)I2 = 0 So that I1=2 A; I2=-3 A & I3=-1 A When a current is found to have a negative value, its direction on the circuit diagram should be reversed. Example 14 Using Kirchhoff’s Loop Rule Determine the current in the circuit.      potential rises drops potential 24 V 0.8 V 0.6 12                 I I I  .0 90 A Determine the current through the car battery, the headlights, and the alternator 11- The Measurement of Current and Voltage An ammeter must be inserted into a circuit so that the current passes directly through it. To measure the voltage between two points in a circuit, a voltmeter is connected between the points. 12- Capacitors in Series and in Parallel Parallel capacitors     3 2 1 C C C CP  C V C C V C V q q q 2 1 2 1 2 1                   2 1 2 1 2 1 1 1 C q C C q C q V V V Series capacitors     3 2 1 1 1 1 1 C C C CS 13- RC Circuits When the switch is closed, a current is induced and charge accumulate on the capacitor. The capacitor continues to charge until it reaches its maximum charge (Q = Cε) and the current stops. Once the capacitor is fully charged, the current in the circuit is zero. • Kirchhoff’s rule across the loop: ε-Vc-iR = ε -(q/C)-R(dq/dt) = 0 Charging a Capacitor in an RC Circuit The charge on the capacitor varies with time q = Q(1 – e-t/RC) The time constant,  = RC The time constant represents the time required for the charge to increase from zero to 63.2% of its maximum Discharge When the switch is closed, the capacitor discharges until Q = 0 and the current stops. Kirchhoff’s rule across loop: Vc-iR = (q/C)-R(dq/dt) = 0 Discharging Capacitor in an RC Circuit When a charged capacitor is placed in the circuit, it can be discharged q = Qe-t/RC The charge decreases exponentially At t =  = RC, the charge decreases to 0.368 Qmax In other words, in one time constant, the capacitor loses 63.2% of its initial charge 14- Safety and Physiological Effects of Current To reduce the danger inherent in using circuits, proper electrical grounding is necessary.