# R R V I R. Conventional Current. Ohms Law V = IR

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1 DC Circuits opics EMF and erminal oltage esistors in Series and in Parallel Kirchhoff s ules EMFs in Series and in Parallel Capacitors in Series and in Parallel Ammeters and oltmeters

2 Conventional Current Positive charges attracted to negative terminal Only electrons move from negative to positive Convention is to use positive charges. r Ohms Law = esistance is an expression of olts lost per amp

3 EMF and erminal oltage = = + r + = = - r Battery r = + r ε r Battery is a nearly constant voltage source, but does have a small internal resistance, which reduces the actual voltage from the ideal emf: Problem EMF and erminal oltage A.5- dry cell can be tested by connecting it to a lowresistance ammeter. t should be able to supply at least A. What is the internal resistance of the cell in this case, assuming it is much greater than that of the ammeter? A a r b 3

4 esistors in Series and in Parallel Simple Series Circuit = 60 Cut the resistor into three equal lengths and then separate. = A = 0 esistors in Series and in Parallel Simple Series Circuit = 60 L A = A = A 3 = A = 0 = 0 3 = 0 = 40 C = 40 3 = 40 = A = 0 he sum of the drops must = 3 3 4

5 = A esistors in Series and in Parallel = 60 = 0 Cut the resistor in half and then separate. Parallel Circuit = A esistors in Series and in Parallel = 60 L A A 0 A = 0 Parallel Circuit 5

6 esistors in Series and in Parallel oltage Drops Around a Series Circuit b A c 3 a + d esistors in Series and in Parallel edraw Circuit = A 3 d a b c d b A c 3 a + d 6

7 oltage Drops esistors in Series and in Parallel = A d a b c d esistors in Series and in Parallel Equivalent esistance a 4 8 b 7

8 esistors in Series and in Parallel Equivalent esistance a b esistors in Series and in Parallel a b 8

9 esistors in Series and in Parallel A parallel connection splits the current; the voltage across each resistor is the same: Problem esistors in Series and in Parallel hree 45 lightbulbs and three 75 lightbulbs are connected in series. (a) What is the total resistance of the circuit? (b) What is their resistance if all six are wired in parallel? 9

10 Problem esistors in Series and in Parallel A battery with an emf of.0 shows a terminal voltage of.8 when operating in a circuit with two light bulbs rated at 3.0 W (at.0 ) which are connected in parallel. What is the battery s internal resistance? a e q r b Problem esistors in Series and in Parallel A battery with an emf of.0 shows a terminal voltage of.8 when operating in a circuit with two lightbulbs rated at 3.0 W (at.0 ) which are connected in parallel. What is the battery s internal resistance? eq 0.49 A a r b 0

11 Problem esistors in Series and in Parallel A close inspection of an electric circuit reveals that a 480 resistor was inadvertently soldered in the place where a 30 resistor is needed. How can this be fixed without removing anything from the existing circuit? Problem esistors in Series and in Parallel A 75-W, 0- bulb is connected in parallel with a 40-W, 0- bulb. What is the net resistance?

12 Summary of,, in series and Parallel Series Parallel Equivalent Equivalent otal Equivalent Equivalent otal Kirchhoff s ules Some circuits cannot be broken down into series and parallel connections. For these circuits we use Kirchhoff s rules. Junction rule: he sum of currents entering a junction equals the sum of the currents leaving it. At junction - a 3 You MUS pick a direction for current. f the solution is negative you picked the wrong direction!! Magnitude remains the same.

13 Kirchhoff s ules Loop rule: he sum of the changes in potential around a closed loop is zero. O the sum of voltage drops=sum of voltage rises. Δ 0 + D =0 D is negative (algebraically) = D ε 0 Kirchhoff s ules Problem Solving: Kirchhoff s ules Label each current. dentify unknowns. Apply junction and loop rules; you will need as many independent equations as there are unknowns. Solve the equations, being careful with signs. Use standard algebraic substitutions ery long most flexible A lot of potential for mistakes Use Linear (matrix) algebra where possible ery quick to set up and solve in calculator Not as flexible 3

14 Kirchhoff s ules Problem For the circuit shown in diagram, find the current. Ω 6 + Start Here Ω Matrix Algebra + + = ax by cz d ex fy gz h ix jy kz l = Use the loop rule to get these equations a e i b f j c g k x y z d h l he known voltage values from the loop rule esistor values are coefficients of current he currents you are looking for 4

15 How to solve using matrices he large matrix we will call matrix (for resistor) he next matrix will call matrix (For currents) he lone matrix will call matrix (for voltage) We would like to solve using Ohm s law (=). We are looking for but you can t divide matrices. You must multiply matrix by the reciprocal of matrix. How to solve using matrices Using your calculator o set up matrix: Click on [matrix] Select the  option Select the matrix you want to edit [A], [B], [], [] o solve matrix: Click on [] [x - ]*[] 5

16 Kirchoff s ules Determine how much current is flowing through each part of the diagram below. Step : Label the high potential side of resistors and batteries with a + and the low potential side with a hese will be used to determine the voltage drops and rises. oltage drops occur when the current goes from high potential to low potential. oltage rises occur when the current goes from low potential to high potential Kirchoff s ules Step : Locate all junctions and try to determine direction for ALL currents through each wire. f two different loops pass the same section of wire, then the current along that section of wire must be the same for both loops. Step 3: Write Kirchoff s loop rule for each of the loops Step 4: Write Kirchoff s junction rule for each of the junctions Step 5: Set up your matrices and solve using the equations from above. here should be at least as many equations as you have unknowns. 6

17 EMFs in Series and in Parallel EMFs in series in the same direction: total voltage is the sum of the separate voltages ac ab bc

18 EMFs in Series and in Parallel EMFs in series, opposite direction: total voltage is the difference, but the lower-voltage battery is charged. ac ab bc EMFs in Series and in Parallel EMFs in parallel only make sense if the voltages are the same; this arrangement can produce more current than a single emf. 8

19 Capacitors in Series and in Parallel Capacitors in parallel have the same voltage across each one: C eq C C C Q Q Q Q Capacitors in Series and in Parallel Capacitors in series have the same charge: C eq C C C Q Q Q Q

20 Problem Capacitors in Series and in Parallel n diagram, if C = C = C 3 =.6 F, how much charge is stored on each capacitor when = 45.0? C C C 3 Ammeters and oltmeters An ammeter measures current; a voltmeter measures voltage. Both are based on galvanometers, unless they are digital. he current in a circuit passes through the ammeter; the ammeter should have low resistance so as not to affect the current. 0

21 Ammeters and oltmeters A voltmeter should not affect the voltage across the circuit element it is measuring; therefore its resistance should be very large. Summary A battery is a source of emf in parallel with an internal resistance ab ε r Series: 3 3 Parallel:

22 Summary Kirchhoff s rules: Sum of currents entering a junction equals sum of currents leaving it. 0 otal potential difference around closed loop is zero. Δ 0 Capacitors in series:... Ceq C C C3 Q Q Q Q Summary Capacitors in parallel: C eq C C C Q Q Q Q Ammeter: measures current oltmeter: measures voltage

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