Transcript Lecture 6
Lecture 6 Capacitance Electric Current Circuits Resistance and Ohms law Capacitors in Series When a battery is connected to the circuit, electrons are transferred from the left plate of C1 to the right plate of C2 through the battery As this negative charge accumulates on the right plate of C2, an equivalent amount of negative charge is removed from the left plate of C2, leaving it with an excess positive charge All of the right plates gain charges of –Q and all the left plates have charges of +Q More About Capacitors in Series An equivalent capacitor can be found that performs the same function as the series combination The potential differences add up to the battery voltage Fig. 16-19, p.551 Fig. 16-20, p.552 Capacitors in Series, cont V V1 V2 1 1 1 Ceq C1 C2 The equivalent capacitance of a series combination is always less than any individual capacitor in the combination Demo Fig. P16-34, p.564 Fig. P16-35, p.564 Problem-Solving Strategy Be careful with the choice of units Combine capacitors following the formulas When two or more unequal capacitors are connected in series, they carry the same charge, but the potential differences across them are not the same The capacitances add as reciprocals and the equivalent capacitance is always less than the smallest individual capacitor Problem-Solving Strategy, cont Combining capacitors When two or more capacitors are connected in parallel, the potential differences across them are the same The charge on each capacitor is proportional to its capacitance The capacitors add directly to give the equivalent capacitance Problem-Solving Strategy, final Repeat the process until there is only one single equivalent capacitor A complicated circuit can often be reduced to one equivalent capacitor Replace capacitors in series or parallel with their equivalent Redraw the circuit and continue To find the charge on, or the potential difference across, one of the capacitors, start with your final equivalent capacitor and work back through the circuit reductions Problem-Solving Strategy, Equation Summary Use the following equations when working through the circuit diagrams: Capacitance equation: C = Q / V Capacitors in parallel: Ceq = C1 + C2 + … Capacitors in parallel all have the same voltage differences as does the equivalent capacitance Capacitors in series: 1/Ceq = 1/C1 + 1/C2 + … Capacitors in series all have the same charge, Q, as does their equivalent capacitance Fig. 16-21, p.553 Fig. P16-57, p.566 Energy Stored in a Capacitor Energy stored = ½ Q ΔV From the definition of capacitance, this can be rewritten in different forms 2 1 1 Q Energy QV CV 2 2 2 2C Fig. 16-22, p.554 Applications Defibrillators When fibrillation occurs, the heart produces a rapid, irregular pattern of beats A fast discharge of electrical energy through the heart can return the organ to its normal beat pattern In general, capacitors act as energy reservoirs that can slowly charged and then discharged quickly to provide large amounts of energy in a short pulse Capacitors with Dielectrics A dielectric is an insulating material that, when placed between the plates of a capacitor, increases the capacitance Dielectrics include rubber, plastic, or waxed paper C = κCo = κεo(A/d) The capacitance is multiplied by the factor κ when the dielectric completely fills the region between the plates Capacitors with Dielectrics Dielectric Strength For any given plate separation, there is a maximum electric field that can be produced in the dielectric before it breaks down and begins to conduct This maximum electric field is called the dielectric strength An Atomic Description of Dielectrics Polarization occurs when there is a separation between the “centers of gravity” of its negative charge and its positive charge In a capacitor, the dielectric becomes polarized because it is in an electric field that exists between the plates More Atomic Description The presence of the positive charge on the dielectric effectively reduces some of the negative charge on the metal This allows more negative charge on the plates for a given applied voltage The capacitance increases Fig. 16-30, p.560 Table 16-1, p.557 Fig. 16-1, p.532 Fig. 16-23, p.557 Fig. 16-26, p.558 Fig. 16-28, p.560 Fig. 16-29a, p.560 Fig. 16-29b, p.560 Electric Current Whenever electric charges of like signs move, an electric current is said to exist The current is the rate at which the charge flows through this surface Look at the charges flowing perpendicularly to a surface of area A Q I t The SI unit of current is Ampere (A) 1 A = 1 C/s Electric Current, cont The direction of the current is the direction positive charge would flow This is known as conventional current direction In a common conductor, such as copper, the current is due to the motion of the negatively charged electrons It is common to refer to a moving charge as a mobile charge carrier A charge carrier can be positive or negative Current and Drift Speed Charged particles move through a conductor of crosssectional area A n is the number of charge carriers per unit volume n A Δx is the total number of charge carriers Current and Drift Speed, cont The total charge is the number of carriers times the charge per carrier, q The drift speed, vd, is the speed at which the carriers move ΔQ = (n A Δx) q vd = Δx/ Δt Rewritten: ΔQ = (n A vd Δt) q Finally, current, I = ΔQ/Δt = nqvdA Current and Drift Speed, final If the conductor is isolated, the electrons undergo random motion When an electric field is set up in the conductor, it creates an electric force on the electrons and hence a current Charge Carrier Motion in a Conductor The zig-zag black line represents the motion of charge carrier in a conductor The net drift speed is small The sharp changes in direction are due to collisions The net motion of electrons is opposite the direction of the electric field Demo Electrons in a Circuit The drift speed is much smaller than the average speed between collisions When a circuit is completed, the electric field travels with a speed close to the speed of light Although the drift speed is on the order of 10-4 m/s the effect of the electric field is felt on the order of 108 m/s c = 3 x 108 m/s Meters in a Circuit – Ammeter An ammeter is used to measure current In line with the bulb, all the charge passing through the bulb also must pass through the meter p.578 Fig. A17-1, p.591 Meters in a Circuit – Voltmeter A voltmeter is used to measure voltage (potential difference) Connects to the two ends of the bulb Resistance In a conductor, the voltage applied across the ends of the conductor is proportional to the current through the conductor The constant of proportionality is the resistance of the conductor V R I Fig. 17-CO, p.568 Resistance, cont Units of resistance are ohms (Ω) 1Ω=1V/A Resistance in a circuit arises due to collisions between the electrons carrying the current with the fixed atoms inside the conductor Georg Simon Ohm 1787 – 1854 Formulated the concept of resistance Discovered the proportionality between current and voltages Ohm’s Law Experiments show that for many materials, including most metals, the resistance remains constant over a wide range of applied voltages or currents This statement has become known as Ohm’s Law ΔV = I R Ohm’s Law is an empirical relationship that is valid only for certain materials Materials that obey Ohm’s Law are said to be ohmic