Chapter 26 & 27. Electric Current and Direct- Current Circuits

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1 Chapter 26 & 27 Electric Current and Direct- Current Circuits

2 Electric Current and Direct- Current Circuits Current and Motion of Charges Resistance and Ohm s Law Energy in Electric Circuits Combination of Resistors Kirchhoff s Rules RC Circuits MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/2013 2

3 Electric Current Under steady state conditions electrons may be fed into the metal at one point and removed at another, producing a current, but the metal as a whole is electrostatically neutral. Strong electrostatic forces keep excess electrons from accumulating at any part of the metal. Similarly a deficiency of electrons is remedied by electrostatic forces of the opposite sign. Excess charge is dissipated extremely rapidly in the conductor. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/2013 3

4 Electric Current The basic laws describing the equilibrium behavior of conductors are altered when static equilibrium is disturbed, i.e. when there is motion or flow of electric charge in the conductor. When the charges in a conductor are in motion, the electric field intensity just at the conductor surface need not be perpendicular to the surface. The conducting body is not an equipotential, as it is for static equilibrium. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/2013 4

5 In A Conductor A sea of electrons (i.e. A Fermi gas) shields the lattice ions from one another. The density of these electrons for Copper (Cu) is 8.5x10 22 /cm 3 These electrons are contributed to the conduction band - one electron per atom. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/2013 5

6 The Definition of Current The definition of current Q I = t where Q is the amount of charge that flows through the cross sectional area in a time t. The units of current are Amperes (A) or coulombs/sec. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/2013 6

7 The Definition of Current In solids, only the electrons flow. For historical reasons, since we didn t know that the electron existed, the conventional current flows opposite to that of the electron current. Conventional current is considered a flow of positive electricity that comes out of the positive terminal of the battery and flows from high potential to low potential. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/2013 7

8 Current and Electron Drift Q = qnav t Q I = t d = qnav d Individual electrons have high speeds (10 6 m/s) but their many collisions with the lattice atoms cause them to execute a random walk through the conductor which is described by a drift velocity v d. When you turn on a light that first electron out of the wall doesn t make it to the light bulb for hours. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/2013 8

9 Wire Dimensions 10 Gauge Cu Wire mm Diameter mm 2 Area I max = 30A The diameter is related to the gauge number n by d = 0.127x92 ( 36-n) 39 MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/2013 9

10 Current and Motion of Charges 2 ( ) carriers coulombs m I = nqv A = n q d 3 vd A m m carrier s Assume 1 carrier per atom: n = n A 2 gm 10 cm x10 3 cm m n = = 8.47x10 gm 63.5 mole 3 28 A 3 ( )( ) d A 8.47x x x10 atoms n = = A 3 m atoms I 30-3 m v = = = 0.420x10 n qa s m ρ m gm atoms N 3 A m mole gm M mole L 100 t = = = 2.38x10 sec = 66.1 hrs -3 v 0.420x10 Time to travel 100m = 5 d MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

11 Current Density Mumbo Jumbo I = JidA = JindA ˆ S S Skip this for now - we will pick it up when we get to magnetism MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

12 Resistance and Ohm s Law Our assumed positive charges will flow in the direction of the electric field in the conductor. V R = I The collisions of the moving charges with the lattice ions gives rise to a resistance to the flow of current. The units of resistance are volts/amp or ohms with the symbol Ω. This is no longer electrostatics so electric fields are allowed inside the conductors. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

13 Ohm s Law V = IR dv R = di Ohm s Law is strictly defined only for materials of constant R but the V-I characteristic is useful even if R is not constant. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

14 Resistance and Resistivity The resistance of a material depends upon the material from which it is made and its physical dimensions. It is customary to separate these two aspects into a material part described by the resistivity ρ and a geometric part. L R =ρ A ρ is the resistivity in Ω-m, L is the length of the conductor in meters and A is the cross sectional area in m 2 MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

15 Resistivity Temperature Dependence The change of the resistivity with temperature is not too important for what we will be doing in this course but it is an important consideration for designing sensitive circuit applications. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

16 Resistivity Temperature Dependence When the switch is first closed the filament in the light is at room temperture and a large current passes through the circuit. After the current has been flowing for a very short time the filament heats up to a high temperature (it s glowing) and the resistance increases. This causes the circuit current to decrease. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

17 Resistivities & Temperature Coefficients MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

18 Resistivities & Temperature Coefficients MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

19 Resistor Color Codes % 30x10 6 Ω at 10% precision (30±3)x10 6 Ω MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

20 Energy in Electric Circuits The rate of loss of potential energy - U = δqi V ( ) b a ( ) a b ( ) U =δq V - V U = - δq V - V - U = δq V - V U δq - = V t t U - = I V t MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/ a b Power delivered to conducting segment

21 Energy in Electric Circuits 2 V P = IV = I R = R 2 In these equations V is understood to be a potential difference. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

22 The Most Basic Electric Circuit is called the emf or Electromotive Force. This is an antiquated term but it is still useful in some discussions. The emf represents the ideal voltage available from the battery. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

23 Internal Resistance In an ideal battery the internal resistance, represented by r, would be equal to zero and the voltage delivered by the battery would be constant as shown by the dashed line. The internal resistance causes the voltage delivered by the battery to be load dependent as shown by the solid line. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

24 Internal Resistance & Power Transfer Question: What should be the value of R, the load resistor, to maximize the power delivered by the battery? The power delivered to R is I 2 R. Therefore we want R to be large. But R is part of the circuit that determines I. As R increases the total resistance in the circuit goes up and the current coming out of the battery goes down. Small R means large I but small I 2 R. Large R means small I and also small I 2 R. Somewhere in between is the max power solution. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

25 Internal Resistance & Power Transfer I = E ( R + r ) 2 P = I R = 2 E R ( R + r ) 2 ( r - R) 2 ( R + r) dp E = = 0 dr R = r 2 This is a concept known as impedance matching. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

26 Resistors in Series Same current - split the voltage MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

27 Resistors in Parallel Same voltage - split the current MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

28 Find the Equivalent Resistance MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

29 Kirchhoff s Rules Loop Rule Whenever any closed loop is traversed, the algebraic sum of the changes in potential around the loop must be equal to zero. Junction Rule At any junction (branch point) in a circuit where the current can divide, the sum of the currents into the junction must equal the sum of the currents out of the junction. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

30 V = - IR - IR - E - Ir - IR + E - Ir = 0 ( ) E - E = I R + R + r + R + r E - E 1 2 I = R 1 + R 2 + r 2 + R 3 + r 1 The text has mistakes in this table above on page 860. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

31 Equivalent to a series circuit with an 8 volts battery and 16 ohms of resistance. The current is I = V/R = 8/16 = 0.5 A MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

32 Multi-loop Circuits Multi-loop means 2 or more circuit loops and 2 or more batteries 1. Combine all series and parallel resistors where possible. 2. Pick a current loop for each circuit loop - make them all go in the same direction. 3. Write a loop equation for each circuit loop. If current through battery is - => + then V is > 0. If current through battery is + => - then V is < 0. For all resistors in a given loop V is < 0. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

33 Multi-loop Circuits 4. Some circuit elements will have more than one loop current passing through them. Opposite direction -i R + i R = -(i - i )R Same direction -i R - i R = -(i + i )R This incorporates the current node equations into the voltage loop equations. The main loop is i 2 MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

34 Multi-loop Circuits 5. There should be as many voltage equations as there are current loops. 6. Solve these equations for each of the currents. 7. Resolve the current in the shared elements using the conservation of current at the appropriate nodes. -i - x + i = x = i - i 2 1 Currents going into the node are negative. Currents coming out of the node are positive. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

35 Multi-loop Circuits 8. Label the currents in the various segments of the circuit. 9. Label the voltage drops across each of the elements of the circuit. 10. Check the numbers. Pick a point in the circuit as ground (i.e. 0 volts). Label the voltages of all the nodes in the circuit relative to ground. Are they consistent? 11. Do the voltage drops agree with the circuit and resistance values. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

36 Multi-loop Circuits - Example I 1 I 1 I 1 MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

37 The Galvanometer A very sensitive ammeter MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

38 The Galvanometer Electrical Symbol Typical Values I max = 200 µa = ma R g = 100 Ω The movable coil galvanometer is the basis for building an ammeter, a voltmeter and an ohmmeter. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

39 Basic Ammeter MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

40 Basic Voltmeter MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

41 Basic Ohmmeter For use as an ohmmeter the scale runs in reverse. The resistor R s is adjusted for max current when a and b are shorted. The nature of the circuit requires that the scale be non linear. It is actually a logarithmic scale. An infinite resistance between a and b results in a zero current. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

42 RC-Circuit Current only flows in a series RC circuit when the capacitor is charging or discharging. The steady state current is zero. MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

43 Charge-Discharge Circuit Position b is the charging position Position a is the discharge position MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

44 RC-Circuit - Discharging Characteristic Initially there is a charge Q 0 on the capacitor and therefore a voltage V 0 = Q 0 /C across the capacitor. When the switch is moved to position a this voltage is applied across the resistor R and an initial current I 0 = Q 0 /(RC) MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

45 RC-Circuit - Discharging Characteristic dq I = - dt The minus sign indicates the current flow direction is opposite the change in Q. Q V = - IR = 0 C Q dq + R = 0 C dt dq 1 = - Q dt RC dq 1 = - dt Q RC Apply Kirchhoff s loop rule Substitute for I Get all the Q s together MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

46 RC-Circuit - Discharging Characteristic Q ' Q 0 0 dq 1 = - dt Q RC ' Q ln = - Q0 Q(t) = Q e τ = RC t t ' ' RC = Q e -t/(rc) 0 0 -t/τ This is the time constant MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

47 RC-Circuit - Discharging Characteristic dq dt Q RC 0 -t/(rc) I = - = e I = I e 0 -t/τ I = V /R = Q /(RC) τ = RC MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

48 RC-Circuit - Charging Q(t) = Q dq I = dt f ( -t/τ 1 - e ) V R 0 -t/(rc) -t/τ I = e = I0e MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

49 Extra Slides MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

50 Some Characteristic Exponential Values t = 0 0 e e 0.0 t =τ e e t = 2τ e e t = 3τ e e t = 4τ e e t = 5τ e e MFMcGraw-PHY 2426 Ch26&27b - Resistors Revised 6/14/

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