Chapter 26 DC Circuits

Units of Chapter EMF and Terminal Voltage - 1, Resistors in Series and in Parallel - 3, 4, 5, 6, Kirchhoff’s Rules - 8, 9, Circuits Containing Resistor and Capacitor (RC Circuits) - 11, 12, DC Ammeters and Voltmeters - 14, 15 Electrical Hazards

26.1 EMF and Terminal Voltage Electric circuit needs battery or generator to produce current – these are called sources of emf. Battery is a nearly constant voltage source, but does have a small internal resistance, which reduces the actual voltage from the ideal emf:

26.1 EMF and Terminal Voltage This resistance behaves as though it were in series with the emf.

26.1 EMF and Terminal Voltage Application #1: A  resistor is connected to the terminals of a battery whose emf is 12.0 V and whose internal resistance is 0.5 . Calculate (a) the current in the circuit, (b) the terminal voltage of the battery, V ab, and (c) the power dissipated in the resistor R and in the battery’s internal resistance r.

26.2 Resistors in Series and in Parallel A series connection has a single path from the battery, through each circuit element in turn, then back to the battery.

26.2 Resistors in Series and in Parallel The current through each resistor is the same; the voltage depends on the resistance. The sum of the voltage drops across the resistors equals the battery voltage. (26-2)

26.2 Resistors in Series and in Parallel From this we get the equivalent resistance (that single resistance that gives the same current in the circuit). (26-2)

26.2 Resistors in Series and in Parallel A parallel connection splits the current; the voltage across each resistor is the same:

26.2 Resistors in Series and in Parallel The total current is the sum of the currents across each resistor:

26.2 Resistors in Series and in Parallel This gives the reciprocal of the equivalent resistance: (26-3)

26.2 Resistors in Series and in Parallel An analogy using water may be helpful in visualizing parallel circuits:

26.2 Resistors in Series and in Parallel Application #2: (a) The lightbulbs in the example below are identical and have identical resistance R. What configuration produces more light? (b) Which way do you think the headlights of a car are wired?

Application #3: Equivalent Resistance (a) What is the equivalent resistance in the circuit to the right? 26.2 Resistors in Series and in Parallel

Application #4: Two 100-  resistors are connected (a) in parallel, and (b) in series to a 24.0-V battery. What is the current through each resistor and what is the equivalent resistance of each circuit?

26.2 Resistors in Series and in Parallel Application #5: (a) How much current flows from the battery in the circuit? (b) What is the current flowing through the 500-  resistor in the circuit?

26.2 Resistors in Series and in Parallel Application #6: If a circuit has three identical lightbulbs of resistance R, how does the brightness of bulbs A and B compare with that of bulb C when the switch is (a) open and (b) closed?

26.2 Resistors in Series and in Parallel Application #7: (a) Estimate the equivalent resistance of the “ladder” of equal 250-  resistors in the circuit. (b) What is the current flowing through each of the three resistors on the left if a 48.0 V battery is connected between points A and B?

26.3 Kirchhoff’s Rules Some circuits cannot be broken down into series and parallel connections.

26.3 Kirchhoff’s Rules For these circuits we use Kirchhoff’s rules. Junction rule: The sum of currents entering a junction equals the sum of the currents leaving it.

26.3 Kirchhoff’s Rules Loop rule: The sum of the changes in potential around a closed loop is zero.

26.3 Kirchhoff’s Rules Problem Solving: Kirchhoff’s Rules 1. Label each current. 2. Identify unknowns. 3. Apply junction and loop rules; you will need as many independent equations as there are unknowns. 4. Solve the equations, being careful with signs.

26.3 Kirchhoff’s Rules Application #8: Calculate the currents I 1, I 2, and I 3 in each of the branches of the circuit.

26.2 Resistors in Series and in Parallel Application #9: (a) Determine the current in each part of the “ladder” circuit of equal 250-  resistors in the circuit if a 48.0 V battery is connected between points A and B. (b) What would it be if the middle circuit has 200-  resistors and the right circuit has 300-  resistors?

26.2 Resistors in Series and in Parallel Application #10: The Marchand Box (a) Determine the current in each part of the “ladder” circuit of equal 100-  resistors in the circuit if a 48.0 V battery is connected between points A and B?

Pinhead

26.3 Kirchhoff’s Rules A Wheatstone bridge is a type of “bridge circuit” used to make measurements of resistance. The unknown resistance to be measured, R x, is placed in the circuit with accurately known resistances R 1, R 2, and R 3. One of these, R 3, is a variable resistor which is adjusted so that when the switch is closed momentarily, the ammeter shows zero current flow.

26.3 Kirchhoff’s Rules Application #11: (a) Determine R x in terms of R 1, R 2, and R 3. (b) If a Wheatstone bridge is “balanced” when R 1 = 630 , R 2 = 972 , and R 3 = 42.6 , What is the value of the unknown resistance?

26.3 Kirchhoff’s Rules EMFs in series in the same direction: total voltage is the sum of the separate voltages

26.3 Kirchhoff’s Rules EMFs in series, opposite direction: total voltage is the difference, but the lower- voltage battery is charged.

26.3 Kirchhoff’s Rules EMFs in parallel only make sense if the voltages are the same; this arrangement can produce more current than a single emf.

26.3 Kirchhoff’s Rules Application #12: A good car battery is being used to jump start a car with a weak battery. The good battery has an emf of 12.5 V and internal resistance of . Suppose the weak battery has an emf of 10.1 V and internal resistance of 0.10 . -continued on next slide-

26.3 Kirchhoff’s Rules Application (con’t): Each copper jumper cable is 3.0m long and 0.50 cm in diameter, and can be attached as shown. Assume the starter motor can be represented as a resistor R s = 0.15 . Determine the current through the starter motor (a) if only the weak battery is connected to it, (b) if the good battery is also connected as shown.

26.3 Kirchhoff’s Rules Application #13: What would happen if the jumper cables were mistakenly connected in reverse, the positive terminal of each battery connected to the negative terminal of the other battery? Why could this be dangerous?

26.4 Circuits Containing Resistor and Capacitor (RC Circuits) When the switch is closed, the capacitor will begin to charge.

26.4 Circuits Containing Resistor and Capacitor (RC Circuits) The voltage across the capacitor increases with time: This is a type of exponential. (26-5b)

26.4 Circuits Containing Resistor and Capacitor (RC Circuits) This curve has a characteristic time constant: The charge follows a similar curve: (26-5a)

26.4 Circuits Containing Resistor and Capacitor (RC Circuits) as expected for a circuit containing only a resistor. The current I through the circuit at any time t can be obtained by differentiating Eq. 26-5a: (26-6) Thus at t = 0, the current is

26.4 Circuits Containing Resistor and Capacitor (RC Circuits) If an isolated charged capacitor is connected across a resistor, it discharges: (26-7)

26.4 Circuits Containing Resistor and Capacitor (RC Circuits) The current I through the circuit at any time t can be obtained by differentiating Eq. 26-5a: (26-8) At t = 0, the current is equal to I 0. The time it takes to decrease to 37% of its original value is

26.4 Circuits Containing Resistor and Capacitor (RC Circuits) Application #14: The capacitance in the circuit shown is C = 0.30  F, the total resistance is 20 k , and the battery emf is 12 V. Determine (a) the time constant, (b) the maximum charge the capacitor could acquire, (c) the time it takes for the charge to reach 99 % of the is value, (d) the current I when the charge Q is half its maximum value, (e) the maximum current, and (f) the charge Q when the current I is 0.20 its maximum value.

26.4 Circuits Containing Resistor and Capacitor (RC Circuits) Application #15: In the RC circuit shown, the battery has fully charged the capacitor so Q 0 = C E. Then at t = 0 the switch is thrown from position a to b. The battery emf is 20.0 V, and the capacitance C = 1.02  F, the current I is observed to decrease to 0.50 of its initial value in 40  s. (a) What is the value of R? (b) What is the value of Q, the charge on the capacitor at t = 0? (c) What is Q at t = 60  s?

26.5 DC Ammeters and Voltmeters An ammeter measures current; a voltmeter measures voltage. Both are based on galvanometers, unless they are digital. The current in a circuit passes through the ammeter; the ammeter should have low resistance so as not to affect the current.

26.5 DC Ammeters and Voltmeters A voltmeter should not affect the voltage across the circuit element it is measuring; therefore its resistance should be very large.

26.5 DC Ammeters and Voltmeters An ohmmeter measures resistance; it requires a battery to provide a current

26.5 DC Ammeters and Voltmeters If the meter has too much or (in this case) too little resistance, it can affect the measurement.

26.5 DC Ammeters and Voltmeters Application #16: (a) Design an ammeter to read 1.0 A full scale using a galvanometer with a full-scale sensitivity of 50  A and a resistance of r = 30 . Check if the scale is linear. (b) Using the same galvanometer with internal resistance r = 30  and full-scale current sensitivity of 50  A, design a voltmeter that reads from 0 to 15 V. Is the scale linear?

26.5 DC Ammeters and Voltmeters Application #17: Suppose you are testing an electronic circuit which has two resistors, R 1 and R 2, each 15k , connected in series as shown. The battery maintains 8.0 V across them and has negligible internal resistance. A voltmeter whose sensitivity is 10,000  /V is put on the 5.0-V scale. What voltage does the meter read when connected across R 1, and what error is caused by the finite resistance of the meter?

Electric Hazards Even very small currents – 10 to 100 mA can be dangerous, disrupting the nervous system. Larger currents may also cause burns. Household voltage can be lethal if you are wet and in good contact with the ground. Be careful!

Electric Hazards A person receiving a shock has become part of a complete circuit.

Electric Hazards Faulty wiring and improper grounding can be hazardous. Make sure electrical work is done by a professional.

Electric Hazards The safest plugs are those with three prongs; they have a separate ground line. Here is an example of household wiring – colors can vary, though! Be sure you know which is the hot wire before you do anything.

26.3 Kirchhoff’s Rules Warm-Up: Using the circuit to the right, draw a picture showing how you would connect (a) an ammeter to determine the 3 currents, and (b) a voltmeter to determine the potential drop across each of the 5 resistances.

Summary of Chapter 26 A source of emf transforms energy from some other form to electrical energy A battery is a source of emf in parallel with an internal resistance Resistors in series:

Summary of Chapter 26 Resistors in parallel: Kirchhoff’s rules: 1. sum of currents entering a junction equals sum of currents leaving it 2. total potential difference around closed loop is zero

Summary of Chapter 26 RC circuit has a characteristic time constant: To avoid shocks, don’t allow your body to become part of a complete circuit Ammeter: measures current Voltmeter: measures voltage