Circuit Elements and Their Combinations

DC Circuits Ch. 27 These circuit elements and many others can be combined to produce a limitless variety of useful devices Two devices are in series if they are connected at one end, and nothing else is connected there wire open switch closed switch 2-way switch + ideal battery 1.5 V Two devices are in parallel if they are connected at both ends 47 F capacitor 4.7 k resistor

Resistors in Parallel and in Series When resistors are in series, the same current must go through both of them The total voltage difference is V V V = + = R1 R2 = R = V ) 2 IR V IR 1 1 2 2 ( + I R 1 1 2 = + R R R The two resistors act like one with resistance 1 2 When resistors are in parallel, the same potential is across both of them The total current through them is V R R1 R2 V = + = + I I I = = V I R I R 1 2 R 1 1 2 2 1 2 The two resistors act like one with resistance V R I 1 R 1 R 1 R 1 R 1 = + 1 1 R = = + 1 2 2

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Parallel and Series - Formulas Capacitor Resistor Inductor* 1 C 1 C 1 = + = + = + L L L R R R Series 1 2 1 2 C 1 2 1 L 1 L 1 L 1 R 1 R 1 R = + = + C C C = + Parallel 1 2 1 2 1 2 Q C dI Fundamental Formula = = = V IR V E Ldt L * To be defined in a later chapter

The Voltage Divider Many circuits can be thought of as a voltage divider Intentionally or unintentionally What s the voltage drop across each of the resistors? R1 = + R R R R + R + 1 E 2 = = E V IR 1 1 1 R R + = I 1 2 + R R 1 2 = = E V IR 2 2 2 The larger resistor gets most of the voltage R2 R R 1 2 If Mr. Curious has a resistance of 10 k and the light bulb has a resistance of 240 , how bright is Mr. Curious? 10000 10240 + curious V = = E= 117 V Not very bright 120 V

CT 1 Consider two identical resistors wired in series (one behind the other). If there is an electric current through the combination, the current in the second resistor is A.equal to B. half C. smaller than, but not necessarily half the current through the first resistor. CT 2 As more identical resistors R are added to the parallel circuit shown here, the total re-sistance between points P and QB Ans A Ans C A.increases. B. remains the same. C. decreases.

CT 3 Charge flows through a light bulb. Suppose a wire is connected across the bulb as shown. When the wire is connected, A.Practically all the charge continues to flow through the bulb. B. half the charge flows through the wire, the other half continues through the bulb. C. Practically all the charge flows through the wire. D. none of the above Ans C

CT - 4-The circuit below consists of two identical light bulbs burning with equal brightness and a single 12 V battery. When the switch is closed, the brightness of bulb A Ans A A.increases. B.remains unchanged. C.decreases. CT - 5-If the four light bulbs in the figure are identical, which circuit puts out more light? Ans A A.I. B.The two emit the same amount of light. C.II.

Quick Quiz 27.2 JIT With the switch in the circuit of the top figure closed, there is no current in R2 because the current has an alternate zero-resistance path through the switch. There is current in R1, and this current is measured with the ammeter (a device for measuring current) at the bottom of the circuit. If the switch is opened (the bottom figure), there is current in R2. What happens to the reading on the ammeter when the switch is opened? (a) The reading goes up. (b) The reading goes down. (c) The reading does not change. Ans b

Quick Quiz 27.3 JIT With the switch in the circuit of the top figure is open, there is no current in R2. There is current in R1, however, and it is measured with the ammeter at the right side of the circuit. If the switch is closed (the bottom figure), there is current in R2. What happens to the reading on the ammeter when the switch is closed? (a) The reading increases. (b) The reading decreases. (c) The reading does not change. Ans a

Which resistors are in series and which are in parallel

Ex- (a) Find the equivalent resistance between points a and b in the figure below. (b) If a potential difference of 34 V is applied between points a and b, calculate the current in Solve on Board each resistor.

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Ideal vs. Non-Ideal Batteries Up until now, we ve treated a battery as if it produced a fixed voltage, no matter what we demand of it Real batteries also have resistance It limits the current and therefore the power that can be delivered If the internal resistancer is small compared to other resistances in the problem, we can ignore it 30 V A 30 V battery with 10 of resistance is connected to a 50 resistor. What is the actual voltage across the 50 resistor? A) 30 V D) 25 V + ideal battery r realistic battery + The maximum potential difference across the battery is called electromotive force (emf) 10 ?????= ?? + 50 ? B) 36 V E) 24 V C) 6 V = 25 V ? = 1 ? + ?

Quick Quiz 27.4 Part I JIT In the figure, a third resistor is added in series with the first two. What happens to the current in the battery? (a) increases, (b) decreases, (c) remains the same. Ans b

Quick Quiz 27.4 Part II JIT What happens to the terminal voltage of the battery? (a) increases, (b) decreases, (c) remains the same. Ans a

Kirchoffs First Law The total current into any vertex equals the current out of that vertex How to apply it: First, assign a current and a direction to every pathway Two components in series will always have the same current At every vertex, write the equation: I I = 12 V I1 + 3 I2 in out + B A 5 Which equation do you get for point A? A) I1 + I2 = I3 C) I1 + I3 = I2 The equation from point B is 6 V B) I2 + I3 = I1 D) I1 + I2 + I3 = 0 4 I3 You always get one redundant equation = + I I I 3 1 2

Kirchoffs Second Law The total voltage change around a loop is always zero How to apply it: First, assign a direction to every loop I often pick clockwise Start anywhere, and set 0 equal to sum of potential change from each piece: For batteries: V = It is an increase if you go from to + It is a decrease if you go from + to For resistors: V = IR It is a decrease if you go with the current It is an increase if you go against the current 12 V I1 + 3 I2 + 5 6 V + + 4 0 = 12 + 5I 6 3I = I3 2 1 + 0 18 3 5 I I 1 2

Kirchoffs Second Law (2) What is Kirchoff s Second Law for the purple loop? A) 0 = +5I2 6 4I3 B) 0 = +5I2 + 6 4I3 C) 0 = 5I2 6 4I3 D) 0 = 5I2 + 6 4I3 12 V I1 + 3 = + + I I I = = 4I 0 5I 6 3 1 2 3 2 I2 0 18 3 5 I I + 1 2 5 Three equations in three unknowns: solve it We can let Maple do it for us 6 V > solve({i3=i1+i2,0=-5*i2-6-4*i3, 0=18-3*i1+5*i2},[i1,i2,i3]); 4 I3 Negative currents means we guessed the wrong way Not a problem

Ex- Using Kirchoffs' rules, (a) find the current in each resistor shown below. (b) Find the potential difference between points c and f. Which point is at a higher potential? Solve on Board

Kirchoffs Laws with Capacitors I If know which side is positive then that is high potential work like battery. Q The voltage change is given by V = Q/C It is a decrease if (+)Q is the side you are going in It is an increase if Q is the side you are going out The current is related to the time change of Q Add minus sign if Idoesn t enter from the same side as Q it is minus if decreasing If you are in a steady state, the current through a capacitor is always zero C + dQ dt = I In this circuit, in the steady state, where is current flowing? + It s really just a battery and two resistors in series! +

The Simplest RC Circuit R Q0 In the circuit shown at left, the capacitor starts with charge Q0. At time t = 0, the switch is closed. What happens to the charge Q? I C Current begins to flow around the loop, so the charge Q will change Q RI C dt This is a differential equation, and therefore hard to solve dQ Q RC t RC Q e Check the units: dQ Q RC = = = 0 I dt t dQ Q dt RC = = + ln Q k = RC C V A C Q e = t RC Q ?? F C s s 0 V

Charging and Discharging Capacitors Q e = t RC Q The combination RC = is called the time constant It s the characteristic time it takes to discharge We can work out the current from dQ I dt t C e RC 0 = t Q Q e 0 Q = RC = t = e 0 = = t e 0 0 R Q I + R C In this circuit, the capacitor is initially uncharged, but at t = 0 the switch is closed ??= ? ? ? = ? 1 ? ? ?? dQ dt = I ?? ??+ 0 =? ?+ ??

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CT 6 A simple circuit consists of a resistor R, a capacitor C charged to a potential Vo (not shown), and a switch that is initially open but then thrown closed. Immediately after the switch is thrown closed, the current in the circuit is A.V o /R. B.zero. C.need more information Ans A

Quick Quiz 27.5 Part I JIT Consider the circuit in the figure and assume the battery has no internal resistance. Just after the switch is closed, what is the current in the battery? (a) 0 (b) /2R (c) 2 /R (d) /R (e) impossible to determine Ans c

JIT Quick Quiz 27.5 Part II After a very long time, what is the current in the battery? (a) 0 (b) /2R (c) 2 /R (d) /R (e) impossible to determine Ans d

Example. Consider a RC circuit consisting of a Emf = 30.0V, a resistor = 1.00 M a capacitor = 5.00 F and a switch like in Figure 28.34 (charging a capacitor). Find (a) the time constant of the circuit and (b) the maximum charge on the capacito after the switch is closed. (c) If the switch is closed at t = 0, find the current in the resistor 10.0 s later. Solve on Board

Ammeters and Voltmeters An ammeter is a device that measures the current (amps) anywhere in a circuit To use it, you must route the current through it A perfect ammeter should have zero resistance A voltmeter is a device that measures the potential difference (volts) between any two points in a circuit To use it, you can simply connect to any two points A perfect voltmeter has infinite resistance A Which meter is installed incorrectly? A) Left voltmeter C) Left ammeter E) All are correct Voltmeters should be connected to two places in an existing circuit The left voltmeter is placed correctly A voltmeter has infinite resistance The right one effectively blocks the current on the right A V B) Right voltmeter D) Right ammeter + V A V

Household Wiring *Actually, this is alternating current, later chapter All household appliances consume electrical power Think of them as resistors with fixed resistance R Devices are designed to operate at 120 V* Often, they give the wattage at this voltage Can easily get the effective resistance from To make sure power is given to each device, they are all placed in parallel Fuse ( ) 2 = P V R A + box Inside House If you put too many things on at once, a lot of current is drawn The wires, which have some resistance, will start to get hot To avoid setting the house on fire, add a fuse (or a circuit breaker)

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Why three wires? If a device is functioning properly, you need only two wires Live and Neutral wires Toaster If the live wire accidentally touches the casing, the person can be electrocuted The wrong solution connect the neutral to the casing Now imagine the neutral wire breaks The person again can be electrocuted The right solution: Add a third ground wire connected directly to ground Normally no current will flow in this wire If the hot wire touches the casing, it will trigger the fuse/circuit breaker and protect the person

MCAT Practice Problems Ans 14 D Ans 15. C Ans 12. A Ans 13. D