2/12/07184 lecture 191 phy 184 spring 2007 lecture 19 title: kirchoff’s rules for circuits

2/12/07184 Lecture 191 PHY 184 Spring 2007 Lecture 19 Title: Kirchoffs Rules for Circuits

2/12/07184 Lecture 192 Results of Midterm 1 (Sec 2) Pretty high average; will even be higher after the correction set! 3/9 or 33.3% --> 53% 5/9 or 55.6% --> 69% Correction set: Example

2/12/07184 Lecture 193AnnouncementsAnnouncements Corrections Set 1 will open soon Corrections Set 1 is just Midterm 1 with different numbers You will get 30% credit for those problems you did incorrectly on Midterm 1 However, you must do all the problems in Corrections Set 1 to get credit After Corrections Set 1 is due, we will open Midterm 1 so that you can see your particular questions and answers This week we will continue with circuits and start magnetism.

2/12/07184 Lecture 194 Resistivity of Wires in a Circuit The circuit on the right shows a resistor of resistance R=6 connected to an ideal battery providing 12V by means of two copper wires. Each wire has the length 20 cm and radius 1 mm. We generally neglect their resistance, the voltage drop across them and the energy dissipated. Justify by calculating the voltage across R accounting for the influence of the wire! Idea: For each wire Total resistance of the circuit is R+2R w

2/12/07184 Lecture 195 Resistivity of Wires in a Circuit (2) The circuit on the right shows a resistor of resistance R=6 connected to an ideal battery providing 12V by means of two copper wires. Each wire has the length 20.cm and radius 1 mm. We generally neglect their resistance, the voltage drop across them and the energy dissipated. Justify by calculating the voltage across R accounting for the influence of the wire! The voltage across R is V=Ri to be compared to 12 V if we neglect the wires. Neglecting the resistance of the wire introduces an error of only 0.03%

2/12/07184 Lecture 196 Clicker Question The circuit on the right shows a resistor of resistance R=6 connected to an ideal battery providing 12V by means of two copper wires. Each wire has the length 20.cm and radius 1 mm. We generally neglect their resistance, the voltage drop across them and the energy dissipated. What is the total voltage drop across the two wires (i=1.9993 A, R=0.0011 each)? A) 5.5 10 -4 V B) 0.01875 V C) 4.4 mV D) 2.2 mV One wire: V w = iR = 2.2 mV

2/12/07184 Lecture 197CircuitsCircuits We have been working with simple circuits containing either capacitors or resistors. Capacitors wired in parallel Capacitors wired in series

2/12/07184 Lecture 198 Circuits (2) Resistors wired in parallel Resistors wired in series We dealt with circuits containing either capacitors or resistors that could be resolved in systems of parallel or series components.

2/12/07184 Lecture 199 Complex Circuits Circuits can be constructed that cannot be resolved into series or parallel systems of capacitors or resistors

2/12/07184 Lecture 1910 Kirchhoffs Rules for Multi-loop Circuits To handle these types of circuits, we must apply Kirchhoffs Rules. Kirchhoffs Rules can be stated as Kirchhoffs Junction Rule The sum of the currents entering a junction must equal the sum of the currents leaving a junction. Kirchhoffs Loop Rule The sum of voltage drops around a complete circuit loop must sum to zero.

2/12/07184 Lecture 1911 Kirchhoffs Junction Rule Kirchhoffs Junction Rule is a direct consequence of the conservation of charge. In a conductor, charge cannot be created or destroyed. At a junction: all charges streaming into the junction must also leave the junction i1i1 i2i2 i3i3 i 1 =i 2 +i 3

2/12/07184 Lecture 1912 Kirchhoffs Loop Rule Kirchhoffs Loop Rule is a direct consequence of the conservation of electric potential energy. Suppose that this rule was not valid we could construct a way around a loop in such a way that each turn would increase the potential of a charge traveling around the loop we would always increase the energy of this charge, in obvious contradiction to energy conservation Kirchhoffs Loop Rule is equivalent to the law of energy conservation.

2/12/07184 Lecture 1913 EMF Devices - Directions An emf device (e.g., a battery) keeps the positive terminal (labeled +) at a higher electrical potential than the negative terminal (labeled -). When a battery is connected in a circuit, its internal chemistry causes a net current inside the battery : positive charge carriers move from the negative to the positive terminal, in direction of the emf arrow. Or: Positive charge carriers move from a region of low electric potential (negative terminal) to a region of high electric potential (positive terminal) This flow is part of the current that is set up around the circuit in that same direction.

2/12/0714

2/12/0715 The flow of electrons is always from anodeto--cathode outside of the cell (i.e., in the circuit) and from cathodeto--anode inside the cell. Inside a chemical cell, ions are carrying the electrons from cathodeto--anode inside the cell. Anode (negative terminal): Zinc powder Cathode (positive terminal): Manganese dioxide (MnO 2 ) powder Electrolyte: Potassium hydroxide (KOH) The half-reactions are: At the cathode 2 MnO 2 + H 2 O + 2 e- >Mn 2 O 3 + 2 OH- At the anode Zn + 2 OH- > ZnO + H 2 O + 2 e- The overall reaction is: Zn + 2MnO 2 > ZnO + Mn 2 O 3 + [E=1.5 V]

2/12/0716 Alkaline battery Al Kaline batter

2/12/07184 Lecture 1917 Single Loop Circuits We begin our study of more complicated circuits by analyzing circuits with several sources of emf and resistors connected in series in a single loop. We will apply Kirchhoffs Rules to these circuits. To apply these rules: establish conventions for determining the voltage drop across each element of the circuit depending on the assumed direction of current and the direction of the analysis of the circuit. Because we do not know the direction of the current in the circuit before we start, we must choose an arbitrary direction for the current.

2/12/07184 Lecture 1918 Single Loop Circuits (2) We can determine if our assumption for the direction of the current is correct after the analysis is complete. If our assumed current is negative, then the current is flowing in the direction opposite to the direction we chose. We can also choose the direction in which we analyze the circuit arbitrarily. Any direction we choose will give us the same information.

2/12/07184 Lecture 1919 Single Loop Circuits (3) If we move around the circuit in the same direction as the current, the voltage drop across a resistor will be negative. If we move around the circuit in the direction opposite to the current, the voltage drop across the resistors will be positive. If we move around the circuit and encounter a source of emf pointing in the same direction, we assume that this source of emf contributes a positive voltage. If we encounter a source of emf pointing in the opposite direction, we consider that component to contribute a negative voltage.

2/12/07184 Lecture 1920 Circuit Analysis Conventions i is the magnitude of the assumed current

2/12/07184 Lecture 1921 Single Loop Circuits We have studied circuits with various networks of resistors but only one source of emf. Circuits can contain multiple sources of emf as well as multiple resistors. We begin our study of more complicated circuits by analyzing a circuit with two sources of emf (V emf,1 and V emf,2 ) and two resistors ( R 1 and R 2 ) connected in series in a single loop We will assume that the two sources of emf have opposite polarity.

2/12/07184 Lecture 1922 Single Loop Circuits (2) We assume that the current is flowing around the circuit in a clockwise direction. Starting at point a with V=0, we analyze around the circuit in a clockwise direction. Because the components of the circuit are in series, all components have the same current, i

2/12/07184 Lecture 1923 Start at point a. The first circuit component is a source of emf V emf,1, which produces a positive voltage gain of +V emf,1 Next we find resistor R 1, which produces a voltage drop V 1 given by -iR 1 Continuing around the circuit we find resistor R 2, which produces a voltage drop V 2 given by -iR 2 Next we meet a second source of emf, V emf,2 This source of emf is wired into the circuit with a polarity opposite that of V emf,1 We treat this component as a voltage drop of -V emf,2 rather than a voltage gain We now have completed the circuit and we are back at point a. Single Loop Circuits (3)

2/12/07184 Lecture 1924 Kirchoffs loop rule for the voltage drops states Generalization: the voltage drops across components in a single loop circuit must sum to zero. This statement must be qualified with conventions for assigning the sign of the voltage drops around the circuit. Single Loop Circuits (4)

2/12/07184 Lecture 1925 Now lets analyze the same circuit in the counter- clockwise direction starting at point a The first circuit element is V emf,2 which is a positive voltage gain The next element is R 2 Single Loop Circuits (5)

2/12/07184 Lecture 1926 Single Loop Circuits (6) Because we have assumed that the current is in the clockwise direction and we are analyzing the loop in the counter-clockwise direction, this voltage drop is +iR 2 Proceeding to the next element in the loop, R 1, we use a similar argument to designate the voltage drop as +iR 1 The final element in the circuit is V emf,1, which is aligned in a direction opposite to our analysis direction, so the voltage drop across this element is -V emf,1

2/12/07184 Lecture 1927 Single Loop Circuits (7) Kirchhoffs Loop Rule then gives us Comparing this result with the result we obtained by analyzing the circuit in the clockwise direction we see that they are equivalent. The direction that we choose to analyze the circuit does not matter.

2/12/07184 Lecture 1928 Clicker Question The figure shows the current i in a single-loop circuit with a battery B and a resistance R. How should the emf arrow be drawn? A) pointing to the right B) pointing to the left C) doesnt matter

2/12/07184 Lecture 1929 Clicker Question The figure shows the current i in a single-loop circuit with a battery B and a resistance R. How should the emf arrow be drawn? A) pointing to the right in direction of the current

2/12/07184 lecture 191 phy 184 spring 2007 lecture 19 title: kirchoff’s rules for circuits

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