Electric Circuits

1. Electric Circuits Now that we have the concept of voltage, we can use this concept to understand electric circuits. Just like we can use pipes to carry water, we can use wires to carry electricity. The flow of water through pipes is caused by pressure differences, and the flow is measured by volume of water per time.

2. Electric Circuits In electricity, the concept of voltage will be like pressure. Water flows from high pressure to low pressure(this is consistent with our previous analogy that Voltage is like height since DP = rgh for fluids) ; electricity flows from high voltage to low voltage. But what flows in electricity? Charges! How do we measure this flow? By Current: current = I = Dq / Dt UNITS: Amp(ere) = Coulomb / second

3. Voltage Sources:batteries and power supplies A battery or power supply supplies voltage. This is analogous to what a pump does in a water system. Question: Does a water pump supply water? If you bought a water pump, and then plugged it in (without any other connections), would water come out of the pump? Question: Does the battery or power supply actually supply the charges that will flow through the circuit?

4. Voltage Sources:batteries and power supplies Just like a water pump only pushes water (gives energy to the water by raising the pressure of the water), so the voltage source only pushes the charges (gives energy to the charges by raising the voltage of the charges). Just like a pump needs water coming into it in order to pump water out, so the voltage source needs charges coming into it (into the negative terminal) in order to “pump” them out (of the positive terminal).

5. Voltage Sources:batteries and power supplies Because of the “pumping” nature of voltage sources, we need to have a complete circuit before we have a current. If we have an air gap (or rubber gap) in the circuit, no current will flow - just like if we have a solid block (like a cap) in a water circuit, no water will flow. If the gap is small, and the voltage is high enough, the current will cross over the gap - somewhat like water, if the pressure is high enough, will break through a plug.

6. Circuit Elements In this first part of the course we will consider two of the common circuit elements: resistor capacitor The resistor is an element that “resists” the flow of electricity. The capacitor is an element that stores charge for use later (like a water tower).

7. Resistance Current is somewhat like fluid flow. Recall that it took a pressure difference to make the fluid flow due to the viscosity of the fluid and the size (area and length) of the pipe. So to in electricity, it takes a voltage difference to make electric current flow due to the resistance in the circuit.

8. Resistance By experiment we find that if we increase the voltage, we increase the current: V is proportional to I. The constant of proportionality we call the resistance, R: V = I*R Ohm’s Law UNITS:R = V/I soOhm = Volt / Amp. The symbol for resistance is W(capital omega).

9. Resistance Just as with fluid flow, the amount of resistance does not depend on the voltage (pressure) or the current (volume flow). The formula V=IR relates voltage to current. If you double the voltage, you will double the current, not change the resistance. As was the case in fluid flow, the amount of resistance depends on the materials and shapes of the wires.

10. Resistance The resistance depends on material and geometry (shape). For a wire, we have: R = r L / A where r is called the resistivity (in Ohm-m) and measures how hard it is for current to flow through the material,L is the length of the wire, and Ais the cross-sectional area of the wire. The second lab experiment deals with Ohm’s Law and the above equation.

11. Electrical Power The electrical potential energy of a charge is: PE = q*V . Power is the change in energy with respect to time: Power = DPE / Dt . Putting these two concepts together we have: Power = D(qV) / Dt = V(Dq) / Dt = I*V.

12. Electrical Power Besides this basic equation for power: P = I*V remember we also have Ohm’s Law: V = I*R . Thus we can write the following equations for power: P = I2*R = V2/R = I*V . To see which one gives the most insight, we need to understand what is being held constant.

13. Example When using batteries, the battery keeps the voltage constant. Each D cell battery supplies 1.5 volts, so four D cell batteries in series (one after the other) will supply a constant 6 volts. When used with four D cell batteries, a light bulb is designed to use 5 Watts of power. What is the resistance of the light bulb?

14. Example We know V = 6 volts, and P = 5 Watts; we’re looking for R. We have two equations: P = I*V and V = I*R which together have 4 quantities: P, I, V & R.. We know two of these (P & V), so we should be able to solve for the other two (I & R).

15. Example Using the power equation we can solve for I: P = I*V, so 5 Watts = I * (6 volts), or I = 5 Watts / 6 volts = 0.833 amps. Now we can use Ohm’s Law to solve for R: V = I*R, so R = V/I = 6 volts / 0.833 amps = 7.2 W .

16. Example extended If we wanted a higher power light bulb, should we have a bigger resistance or a smaller resistance for the light bulb? We have two relations for power that involve resistance: P=I*V; V=I*R; eliminating V gives: P = I2*R and P=I*V; I=V/R; eliminating I gives:P = V2 / R . In the first case, Power goes up as R goes up; in the second case, Power goes down as R goes up. Which one do we use to answer the above question?

17. Example extended Answer: In this case, the voltage is being held constant due to the nature of the batteries. This means that the current will change as we change the resistance. Thus, the P = V2 / R would be the most straight-forward equation to use. This means that as R goes down, P goes up. (If we had used the P = I2*R formula, as R goes up, I would decrease – so it would not be clear what happened to power.) The answer: for more power, lower the resistance. This will allow more current to flow at the same voltage, and hence allow more power!

18. Hooking Resistors Together Instead of making and storing all sizes of resistors, we can make and store just certain values of resistors. When we need a non-standard size resistor, we can make it by hooking two or more standard size resistors together to make an effective resistor of the value we need. The symbol for a resistor is written:

19. Two basic ways There are two basic ways of connecting two resistors: series and parallel. In series, we connect resistors together like railroad cars: + - + - high V low R1 R2

20. Series If we include a battery as the voltage source, the series circuit would look like this: R1 + Vbat R2 Note that there is only one way around the circuit, and you have to go through BOTH resistors in making the circuit - no choice!

21. Parallel In a parallel hook-up, there is a branch point that allows you to complete the circuit by going through either one resistor or the other: you have a choice! High V R1 Low V R2

22. Parallel Circuit If we include a battery, the parallel circuit would look like this: + + + Vbat R1 R2 - -

23. Formula for Series: To see how resistors combine to give an effective resistance when in series, we can look either at V = I*R, or at R = rL/A . R1 I V1 + R2 Vbat V2 -

24. Formula for Series Using V = I*R, we see that in series the current must move through both resistors. (Think of water flowing down two water falls in series.) Thus Itotal = I1 = I2. Also, the voltage drop across the two resistors add to give the total voltage drop: (The total height that the water fell is the addition of the two heights of the falls.) Vtotal = (V1 + V2). Thus, Reff = Vtotal / Itotal = (V1 + V2)/Itotal = V1/I1 + V2/I2 = R1 + R2.

25. Formula for Series Using R = rL/A , we see that we have to go over both lengths, so the lengths should add. The lengths are in the numerator, and so the values should add. This is just like in R = V/I(from V = IR) where the V’s are in the numerator and so add!

26. Formula for Parallel Resistors The result for the effective resistance for a parallel connection is different, but we can start from the same two places: (Think of water in a river that splits with some water flowing over one fall and the rest falling over the other but all the water ending up joining back together again.) V=I*R, or R = rL/A . Itotal + Vbat I1 R1 R2 I2 -

27. Formula for Parallel Resistors V=I*R, or R = rL/A For parallel, both resistors are across the same voltage, so Vtotal = V1 = V2. The current can go through either resistor, so: Itotal = (I1 + I2 ). Since the I’s are in the denominator, we have: R = Vtotal/Itotal = Vtotal/(I1+I2); or 1/Reff = (I1+I2)/Vtotal = I1/V1 + I2/V2= 1/R1 + 1/R2.

28. Formula for Parallel Resistors If we start from R = rL/A , we can see that parallel resistors are equivalent to one resistor with more Area. But A is in the denominator (just like the current I was in the previous slide), so we need to add the inverses: 1/Reff = 1/R1 + 1/R2 .

29. Review: Resistors: V = IR Power = IV; R = rL/A Series: Reff = R1 + R2 Parallel: 1/Reff = 1/R1 + 1/R2 series gives largest Reff , parallel gives smallest Reff .

30. Computer Homework The Computer Homework, Vol 3, #6, gives both an introduction and problems dealing with resistors. (For PHYS 202 you only need to do the first 5 questions.)

31. Capacitance A water tower holds water.A capacitor holds charge. The pressure at the base of the water tower depends on the height (and hence the amount) of the water. The voltage across a capacitor depends on the amount of charge held by the capacitor.

32. Capacitance We define capacitance as the amount of charge stored per volt: C = Qstored / DV. UNITS: Farad = Coulomb / Volt Just as the capacity of a water tower depends on the size and shape, so the capacitance of a capacitor depends on its size and shape. Just as a big water tower can contain more water per foot (or per unit pressure), so a big capacitor can store more charge per volt.

33. Capacitance While we normally define the capacity of a water tank by the TOTAL AMOUNT of water it can hold, we define the capacitance of an electric capacitor as the AMOUNT OF CHARGE PER VOLT instead. There is a TOTAL AMOUNT of charge a capacitor can hold, and this corresponds to a MAXIMUM VOLTAGE that can be placed across the capacitor. Each capacitor DOES HAVE A MAXIMUM VOLTAGE.

34. Capacitance • What happens when a water tower is over-filled? It can break due to the pressure of the water pushing on the walls. • What happens when an electric capacitor is “over-filled” or equivalently a higher voltage is placed across the capacitor than the listed maximum voltage? It will “break” by having the charge “escape”. This escaping charge is like lightning - a spark that usually destroys the capacitor.

35. Capacitors As we stated before, the capacitance of a capacitor depends on its size and shape. Basically a capacitor consists of two separated (at least electrically separated) conductors (usually pieces of metal) so that we can pull charge from one and deposit it on the other. In the next slide we look at a common type of capacitor, the parallel plate capacitor where the two conductors are plates that are aligned parallel to each other; each of area, A; separated by a distance,d; and containing anon-conductingmaterial betweenthe plates. Top plate A d Material between plates Bottom plate

36. Parallel Plate Capacitor For a parallel plate capacitor, we can pull charge from one plate (leaving a -Q on that plate) and deposit it on the other plate (leaving a +Q on that plate). Because of the charge separation, we have a voltage difference between the plates, DV. The harder we pull (the more voltage across the two plates), the more charge we pull: C = Q /DV. Note that C is NOT CHANGED by either Q or DV; C relates Q and DV! The same applied to resistance: the resistance did not depend on the current and voltage – the resistance related the two. Top plate +Q A d DV Material between plates Bottom plate -Q

37. V or DV ? When we deal with height, h, we usually refer to the change in height, Dh, between the base and the top. Sometimes we do refer to the height as measured from some reference point. It is usually clear from the context whether h refers to an actual h or a Dh. With voltage, the same thing applies. We often just use V to really mean DV. You should be able to determine from the context whether we really mean V or DV when we say V.

38. Parallel Plate Capacitor For this parallel plate capacitor, the capacitance is related to charge and voltage (C = Q/V), but the actual capacitance depends on the size and shape: Cparallel plate = K A / (4 p k d) where K (called dielectric constant) depends on the material between the plates, A is the area of each plate, d is the distance between the plates, and k is Coulomb’s constant (9 x 109 Nt-m2 / Coul2). Top plate +Q A d DV Material between plates Bottom plate -Q

39. Example: Parallel Plate Capacitor Consider a parallel plate capacitor made from two plates each 5 cm x 5 cm separated by 2 mm with vacuum in between. What is the capacitance of this capacitor? Further, if a power supply puts 20 volts across this capacitor, what is the amount of charged stored by this capacitor?

40. Example: Parallel Plate Capacitor The capacitance depends on K, A, k and d: Cparallel plate = K A / (4 p k d) where K = 1 for vacuum, A = 5 cm x 5 cm = 25 cm2 = 25 x 10-4 m2, d = 2 mm = 2 x 10-3 m, and k = 9 x 109 Nt-m2/Coul2 , so C = [(1) * (25 x 10-4 m2) ] / [4 * 3.14 * 9 x 109 Nt-m2/Coul2 * 2 x 10-3 m] = 1.10 x 10-11 F = 11 pF .

41. Other types of capacitors Note: We can have other shapes for capacitors. These other shapes will have formulas for them that differ from the above formula for parallel plates. These formulas will also show that the capacitance depends on the materials and shape of the capacitor.

42. Example (cont.) We can see from the previous example thata Farad is a huge capacitance! If we have a DV = 20 volts, then to calculate the charge, Q, we can use: C = Q/V to get: Q= C*V = 11 x 10-12 F * 20 volts = 2.2 x 10-10 Coul= 0.22 nCoul = 220 pCoul. Remember that we often drop the D in front of the V since we often are concerned by the change in voltage rather than the absolute value of the voltage - just as we do when we talk about height!

43. Capacitance Note that if we doubled the voltage, we would not do anything to the capacitance. Instead, we would double the charge stored on the capacitor. However, if we try to overfill the capacitor by placing too much voltage across it, the positive and negative plates will attract each other so strongly that they will spark across the gap and destroy the capacitor. Thus capacitors have a maximum voltage!

44. Energy Storage If a capacitor stores charge and carries voltage, it also stores the energy it took to separate the charge. The formula for this is: Estored = (1/2)QV = (1/2)CV2, where in the second equation we have used the relation: C = Q/V .

45. Energy Storage Note that previously we had: PE = q*V , and now for a capacitor we have: E = (1/2)*Q*V . Why the 1/2 factor for a capacitor?

46. Energy Storage The reason is that in charging a capacitor, the first bit of charge is transferred while there is very little voltage on the capacitor (recall that the charge separation creates the voltage!). Only the last bit of charge is moved across the full voltage. Thus, on average, the full charge moves across only half the voltage!

47. Hooking Capacitors Together Instead of making and storing all sizes of capacitors, we can make and store just certain values of capacitors. When we need a non-standard size capacitor, we can make it by hooking two or more standard size capacitors together to make an effective capacitor of the value we need. (Similar to what we saw with resistors.)

48. Two basic ways Just as with resistors, there are two basic ways of connecting two capacitors: series and parallel. In series, we connect capacitors together like railroad cars; using parallel plate capacitors it would look like this: + - + - high V low V C1 C2

49. Series If we include a battery as the voltage source, the series circuit would look like this: C1 + Vbat C2 Note that there is only one way around the circuit, and you have to jump BOTH capacitors in making the circuit - no choice! + - + -

50. Parallel In a parallel hook-up, there is a branch point that allows you to complete the circuit by jumping over either one capacitor or the other: you have a choice! High V C1 Low V C2 + - + -