Report

Chapter 18 Direct Current Circuits 18.1 Sources of emf The source that maintains the current in a closed circuit is called a source of emf Emf=electromotive force Any devices that increase the potential energy of charges circulating in circuits are sources of emf Examples include batteries and generators Emf and Internal Resistance A real battery has some internal resistance Therefore, the terminal voltage is not equal to the emf More About Internal Resistance The schematic shows the internal resistance, r The terminal voltage is V = ε – Ir (source minus the internal loss) For the entire circuit, ε = IR + Ir =I (R+r) V Internal Resistance and emf, cont ε is equal to the terminal voltage when the current is zero (open circuit) Also called the open-circuit voltage R is called the load resistance The current depends on both the resistance external to the battery and the internal resistance 18.2 Resistors in Series When two or more resistors are connected end-to-end, they are said to be in series The current is the same in resistors because any charge that flows through one resistor flows through the other The sum of the potential differences across the resistors is equal to the total potential difference across the combination Resistors in Series, cont V1=IR1 Voltages add V =IR1 + IR2 V =I (R1+R2) Req= R1+R2 The equivalent resistance Req has the same effect on the circuit as the original combination of resistors V2=IR2 V Equivalent Resistance – Series Req = R1 + R2 + R3 + … The equivalent resistance of a series combination of resistors is the algebraic sum of the individual resistances and is always greater than any of the individual resistance ( see parallel connection of capacitors!) Equivalent Resistance – Series An Example Four resistors are replaced with their equivalent resistance 18.3 Resistors in Parallel The potential difference across each resistor is the same because each is connected directly across the battery terminals The current, I, that enters a point must be equal to the total current leaving that point I = I1 + I2 The currents are generally not the same Consequence of Conservation of Charge Equivalent Resistance – Parallel, cont V I1=V/R1 and I2=V/R2. The complete current provided by the source is given by, I=I1+I2=V(1/R1+1/R2)=V/Req. 1/Req=1/R1+1/R2 Req is the equivalent resistance for a parallel circuit Household circuits are wired so the electrical devices are connected in parallel Circuit breakers may be used in series with other circuit elements for safety purposes Equivalent Resistance – Parallel Equivalent Resistance 1 1 1 1 Req R1 R2 R3 The inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistance ( see series connection of capacitors) The equivalent is always less than the smallest resistor in the group Problem-Solving Strategy, 1 When two or more resistors are connected in series, they carry the same current, but the potential differences across them are not the same. The resistors add directly to give the equivalent resistance of the series combination Problem-Solving Strategy, 2 When two or more resistors are connected in parallel, the potential differences across them are the same. The currents through them are not the same. The equivalent resistance of a parallel combination is found through reciprocal addition The equivalent resistance is always less than the smallest individual resistor in the combination Problem-Solving Strategy, 3 A complicated circuit consisting of several resistors and batteries can often be reduced to a simple circuit with only one resistor Replace any resistors in series or in parallel using steps 1 or 2. Sketch the new circuit after these changes have been made Continue to replace any series or parallel combinations Continue until one equivalent resistance is found Problem-Solving Strategy, 4 If the current in or the potential difference across a resistor in a complicated circuit is to be identified, start with the final circuit found in step 3 and gradually work back through the circuits Use V = I R and the procedures in steps 1 and 2 Example Compute the equivalent resistance and the current of the network (a) below. Solution Step Step Step Step Step 1: 2: 3: 4: 5: Rp=Req for parallel connection and Rs=Req for series connection 1/(3 W)+1/(6 W)=1/Rp Rp=2 W (b) 4 W+2 W=Rs Rs=6 W (c) I=V/Rs=18 V/6 W=3 A (d) V=(3 A)(4 W+2 W)12 V+6 V=18 V (e) 6 V/6 W=1 A and 6 V/3 W=2 A (f) Shortcut n parallel equal resistors, the equivalent Rp is expressed by Rp=R1/n The same shortcut is valid for n For equal capacitors in series Cs=C1/n 18.4 Kirchhoff’s Rules There are ways in which resistors and batteries can be connected so that the circuits formed cannot be reduced to a single equivalent resistor (two examples are shown on the right) Two rules, called Kirchhoff’s Rules can be used instead Statement of Kirchhoff’s Rules Junction Rule ( I = 0) The sum of the currents entering any junction must equal the sum of the currents leaving that junction A statement of Conservation of Charge Loop Rule ( U = 0) The sum of the potential differences across all the elements around any closed circuit loop must be zero A statement of Conservation of Energy More About the Junction Rule I1 = I2 + I3 From Conservation of Charge Diagram (b) shows a mechanical analog More About the Loop Rule The voltage across a battery is taken to be positive (a voltage rise) if traversed from – to + and and negative if traversed in the opposite direction. (b) The voltage across a resistor is taken to be negative (a drop) if the loop is traversed in in the direction of the assigned current and positive if traversed in the opposite direction (a) Setting Up Kirchhoff’s Rules Assign symbols and directions to the currents in all branches of the circuit If a direction is chosen incorrectly, the resulting answer will be negative, but the magnitude will be correct When applying the loop rule, choose a direction for transversing the loop Record voltage drops and rises as they occur Problem-Solving Strategy – Kirchhoff’s Rules Draw the circuit diagram and assign labels and symbols to all known and unknown quantities. Assign directions to the currents. Apply the junction rule to any junction in the circuit Apply the loop rule to as many loops as are needed to solve for the unknowns Solve the equations simultaneously for the unknown quantities. Example: Find the current I, r and e. Junction rule at a: 2 A+1 A-I=0 I=3 A Loop (1): 12 V-Ir -(3 W)(2 A)=0 r=12 V/(3 A)-6 V/(3 A) r=2 W Loop (2): -e+(1 W)(1 A)-( 3 W)(2 A)=0 e=-5 V (polarity is opposite!) Check with loop (3): 12 V-(2 W)(3 A)-(1 W)(1 A)+e =0 e=-5 V