Capacitance. A different kind of capacitor: Work must be done to charge a capacitor. Capacitors in circuits. Capacitor connected to a battery


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1 Capacitance The ratio C = Q/V is a conductor s self capacitance Units of capacitance: Coulomb/Volt = Farad A capacitor is made of two conductors with equal but opposite charge Capacitance depends on shape and distance apart between the conductors Special case: parallel plate capacitor (derivation) A different kind of capacitor: Find the expression for the capacitance of a cylindrical capacitor consisting of two conductors, each of length L. One, a cylinder of radius R 1 and the other a shell of larger radius R 2. Strategy for a capacitor problem: 1. Draw the capacitor shape 2. Pretend there is charge + and Q on the plates 3. Determine the E field between the plates 4. Use E to find the potendal difference across the plates 5. Compute C=Q/V 6. Important: Q should cancel out! Work must be done to charge a capacitor Energy stored in a capacitor is related to the potential difference across the capacitor Capacitors in circuits What s a circuit? 1. Wires are represented by lines 2. The battery has its own symbol 3. The capacitor has its own symbol 4. Think of charge as water flowing through the wires from the positive to negative terminals (current) The battery shares its charge with the capacitor plates Capacitor connected to a battery E is zero in a conductor (wires in circuits are conductors) So there is equal potential all along the wires of a circuit, but capacitor plates don t touch, so the potential value can be different across a capacitor. A battery connected to a capacitor will move charge from one plate to the other The battery retains a voltage across its terminals and will impose the same voltage across the capacitor The amount of charge moved is Q=CV, where C= the equivalent capacitance of the whole circuit Parallel and series Parallel there s a fork in the road Series No splitting in the current s path Sometimes it s not obvious what is is parallel and what is in series. Example
2 If the area of the plates of a parallelplate capacitor is doubled, the capacitance is A. not changed. B. doubled. C. halved. D. increased by a factor of 4. E. decreased by a factor of 1/4. Which of the following statements about a parallel plate capacitor is false? A. The two plates have equal charges of the same sign. B. The capacitor stores charges on the plates. C. The capacitance is proportional to the area of the plates. D. The capacitance is inversely proportional to the separation between the plates. E. A charged capacitor stores energy. If C 1 < C 2 < C 3 < C 4 for the combination of capacitors shown, the equivalent capacitance is A. less than C 1. B. more than C 4. C. between C 1 and C 4. Beginning Chapter 25 Direct current vs. Alternating current Alternating current is what comes out of the wall outlet, but direct current is easier to analyze so here we focus on DC circuits (direct current circuits) Definition of current SI Units of current Introducing Resistance Ohms law If there is a potential drop (change in potential) in a wire, that wire has some resistance Resistance of a conducting wire Define resistivity (not to be confused with density, same symbol) A Light bulb is an example of a resistor Example 253 Calculate the resistance per unit length of a copper wire with a diameter of mm (the resistivity is found in table 251, for copper, it is 1.7X108 Ohmmeters). 2
3 If electrons pass a particular point in a wire every second, what is the current in the wire? A. 4.7 ma B. 7.5 A C. 2.9 A D. 7.5 ma E A Energy in Circuits Change in potential energy depends on the charge moved from one potential to another Power = rate of potential energy loss Joule Heating Law Power delivered to a resistor can be found from Ohm s Law Emf (electromotive force) Not a force!! Energy is required to maintain a current in a wire This energy supply is called source of emf Emf has units of volts (like potential) An example of an ideal source of emf is a battery which maintains a constant potential difference across the terminals (converts chemical energy into electrical energy) In an ideal battery, potential and emf are interchangeable Real battery A real battery is not ideal, meaning it has internal resistance r Example 256 (work in groups, students will present) An 11 ohm resistor is connected across a real battery of emf 6.00V and internal resistance 1.00 ohms. Find a) the current b) the terminal voltage of the battery c) power supplied by the battery (total power) d) power delivered to the external resistor e) and the power delivered to the battery s internal resistance. Hint: d and e should add up to c Example 257 For a battery that has an emf equal to E and internal resistance equal to r, what value of external resistance R should be placed across the terminals to obtain maximum power to the resister? 3
4 Resistors in circuits Resistors in series Resistors in parallel Example 258 Example 259 An ideal battery supplies 12 V across the parallel combination of resistors (4.0 Ω and 6.0 Ω) a) Find the equivalent resistance, b) the total current (coming out of the battery), c) the current through each resistor, d) the power delivered to each resistor and e) the power supplied by the battery. Example An ideal battery supplies 12 V across the series combination of resistors (4.0 Ω and 6.0 Ω) a) Find the equivalent resistance, b) the total current in the circuit, c) the potential drop across each resistor, d) the power delivered to each resistor and e) the power supplied by the battery. What if the circuit is just too complicated? How do you analyze a circuit that can t be replaced by equivalent resistances? KIRCHHOFF s RULES You need to be able to solve systems of equations (brush up on your algebra!) Organize your equations, be clear and specific about what you are solving for and what the knowns are PLEASE don t plug in numbers until you have solved completely for the variables in question Kirchhoff s rules (2): 1. junction rule At any junction in a circuit (when wires meet) when current can divide, the sum of the currents in must equal the sum of the currents out of the junction 2. loop rule When any closed loop is traversed in a circuit, the algebraic sum of the changes in potential around the loop must be zero How to use these rules to generate useful equations: Is the junction rule useful in single loop circuits? Steps: 1. Draw the circuit 2. Label current arbitrarily in each branch of circuit 3. Choose the (positive) direction around which you will traverse the loop 4. Generate as many equations as you have unknowns 5. Solve equations with algebra 4
5 Example Don t plug in numbers until the end!!! a) Find the current in each branch of the circuit. (to be drawn) (I ll just set up this problem and leave the algebra for you to practice on your own) Another example for you to try 5
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