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## Chapter 24 Capacitance, Dielectrics, Electric Energy Storage

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**24-4 Electric Energy Storage**A charged capacitor stores electric energy; the energy stored is equal to the work done to charge the capacitor:**24-4 Electric Energy Storage**Conceptual Example 24-9: Capacitor plate separation increased. A parallel-plate capacitor carries charge Q and is then disconnected from a battery. The two plates are initially separated by a distance d. Suppose the plates are pulled apart until the separation is 2d. How has the energy stored in this capacitor changed?**24-4 Electric Energy Storage**The energy density, defined as the energy per unit volume, is the same no matter the origin of the electric field: The sudden discharge of electric energy can be harmful or fatal. Capacitors can retain their charge indefinitely even when disconnected from a voltage source – be careful!**24-4 Electric Energy Storage**Heart defibrillators use electric discharge to “jump-start” the heart, and can save lives.**24-4 Electric Energy Storage**National Ignition Facility (NIF) Lawrence Livermore National Laboratory**NIF**Laser system driven by 4000 300 μF capacitors which store a total of 422 MJ. They take 60 s to charge and are discharged in 400 μs. • What is the potential difference across each capacitor? • What is the power delivered during the discharge?**NIF**Laser system driven by 4000 300 μF capacitors which store a total of 422 MJ. They take 60 s to charge and are discharged in 400 μs. • What is the potential difference across each capacitor? • What is the power delivered during the discharge? Solution: • U = CV2/2 →V = (2U/C)1/2 →V = [2(422x106)/4000/300x10-6] ½ = 26.5 kV • P = W/t = U/t = 422x106 /400x10-6 ~ 1012 W = 1000 GW! cf. 1.0-1.5 GW for power plant**24-6 Molecular Description of Dielectrics**The molecules in a dielectric, when in an external electric field, tend to become oriented in a way that opposes the external field. A dielectric is an insulator, and is characterized by a dielectric constant K.**24-6 Molecular Description of Dielectrics**This means that the electric field within the dielectric is less than it would be in air, allowing more charge to be stored for the same potential. This reorientation of the molecules results in an induced charge – there is no net charge on the dielectric, but the charge is asymmetrically distributed. The magnitude of the induced charge depends on the dielectric constant:**24-5 Dielectrics**Capacitance of a parallel-plate capacitor filled with dielectric: Using the dielectric constant, we define the permittivity:**24-5 Dielectrics**Dielectric strength is the maximum field a dielectric can experience without breaking down.**24-5 Dielectrics**Here are two experiments where we insert and remove a dielectric from a capacitor. In the first, the capacitor is connected to a battery, so the voltage remains constant. The capacitance increases, and therefore the charge on the plates increases as well.**24-5 Dielectrics**In this second experiment, we charge a capacitor, disconnect it, and then insert the dielectric. In this case, the charge remains constant. Since the dielectric increases the capacitance, the potential across the capacitor drops.**24-5 Dielectrics**Example 24-11: Dielectric removal. A parallel-plate capacitor, filled with a dielectric with K= 3.4, is connected to a 100-V battery. After the capacitor is fully charged, the battery is disconnected. The plates have area A= 4.0 m2 and are separated by d = 4.0 mm. (a) Find the capacitance, the charge on the capacitor, the electric field strength, and the energy stored in the capacitor. (b) The dielectric is carefully removed, without changing the plate separation nor does any charge leave the capacitor. Find the new values of capacitance, electric field strength, voltage between the plates, and the energy stored in the capacitor.**Summary of Chapter 24**• Capacitor: nontouching conductors carrying equal and opposite charge. • Capacitance: • Capacitance of a parallel-plate capacitor:**Summary of Chapter 24**• Capacitors in parallel: • Capacitors in series:**Summary of Chapter 24**• Energy density in electric field: • A dielectric is an insulator. • Dielectric constant gives ratio of total field to external field. • For a parallel-plate capacitor:**25-1 The Electric Battery**Volta discovered that electricity could be created if dissimilar metals were connected by a conductive solution called an electrolyte. This is a simple electric cell.**25-1 The Electric Battery**Several cells connected together make a battery, although now we refer to a single cell as a battery as well.**25-2 Electric Current**Electric current is the rate of flow of charge through a conductor: The instantaneous current is given by: Unit of electric current: the ampere, A: 1 A = 1 C/s.**25-2 Electric Current**A complete circuit is one where current can flow all the way around. Note that the schematic drawing doesn’t look much like the physical circuit!**25-2 Electric Current**Example 25-1: Current is flow of charge. A steady current of 2.5 A exists in a wire for 4.0 min. (a) How much total charge passed by a given point in the circuit during those 4.0 min? (b) How many electrons would this be?**ConcepTest 25.1 Connect the Battery**4) all are correct 5) none are correct Which is the correct way to light the lightbulb with the battery? 1) 2) 3)**ConcepTest 25.1 Connect the Battery**4) all are correct 5) none are correct Which is the correct way to light the lightbulb with the battery? 1) 2) 3) Current can flow only if there is a continuous connection from the negative terminal through the bulb to the positive terminal. This is the case for only Fig. (3).**25-2 Electric Current**By convention, current is defined as flowing from + to -. Electrons actually flow in the opposite direction, but not all currents consist of electrons.**25-3 Ohm’s Law: Resistance and Resistors**Experimentally, it is found that the current in a wire is proportional to the potential difference between its ends:**25-3 Ohm’s Law: Resistance and Resistors**The ratio of voltage to current is called the resistance:**25-3 Ohm’s Law: Resistance and Resistors**In many conductors, the resistance is independent of the voltage; this relationship is called Ohm’s law. Materials that do not follow Ohm’s law are called nonohmic. Unit of resistance: the ohm, Ω: 1 Ω = 1 V/A.**25-3 Ohm’s Law: Resistance and Resistors**Conceptual Example 25-3: Current and potential. Current I enters a resistor R as shown. (a) Is the potential higher at point A or at point B? (b) Is the current greater at point A or at point B?**25-3 Ohm’s Law: Resistance and Resistors**Example 25-4: Flashlight bulb resistance. A small flashlight bulb draws 300 mA from its 1.5-V battery. (a) What is the resistance of the bulb? (b) If the battery becomes weak and the voltage drops to 1.2 V, how would the current change?**25-3 Ohm’s Law: Resistance and Resistors**Some clarifications: • Batteries maintain a (nearly) constant potential difference; the current varies. • Resistance is a property of a material or device. • Current is not a vector but it does have a direction. • Current and charge do not get used up. Whatever charge goes in one end of a circuit comes out the other end.**ConcepTest 25.2Ohm’s Law**1) Ohm’s law is obeyed since the current still increases when V increases 2) Ohm’s law is not obeyed 3) this has nothing to do with Ohm’s law You double the voltage across a certain conductor and you observe the current increases three times. What can you conclude?**ConcepTest 25.2Ohm’s Law**1) Ohm’s law is obeyed since the current still increases when V increases 2) Ohm’s law is not obeyed 3) this has nothing to do with Ohm’s law You double the voltage across a certain conductor and you observe the current increases three times. What can you conclude? Ohm’s law, V = IR, states that the relationship between voltage and current is linear. Thus, for a conductor that obeys Ohm’s law, the current must double when you double the voltage. Follow-up: Where could this situation occur?**25-4 Resistivity**The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area: The constant ρ, the resistivity, is characteristic of the material.**25-4 Resistivity**This table gives the resistivity and temperature coefficients of typical conductors, semiconductors, and insulators.**25-4 Resistivity**Example 25-5: Speaker wires. Suppose you want to connect your stereo to remote speakers. (a) If each wire must be 20 m long, what diameter copper wire should you use to keep the resistance less than 0.10 Ω per wire? (b) If the current to each speaker is 4.0 A, what is the potential difference, or voltage drop, across each wire?**25-4 Resistivity**For any given material, the resistivity increases with temperature: Semiconductors are complex materials, and may have resistivities that decrease with temperature.**25-4 Resistivity**Example 25-7: Resistance thermometer. The variation in electrical resistance with temperature can be used to make precise temperature measurements. Platinum is commonly used since it is relatively free from corrosive effects and has a high melting point. Suppose at 20.0°C the resistance of a platinum resistance thermometer is 164.2 Ω. When placed in a particular solution, the resistance is 187.4 Ω. What is the temperature of this solution?**ConcepTest 25.3aWires I**1) dA = 4dB 2) dA = 2dB 3) dA = dB 4) dA = 1/2dB 5) dA = 1/4dB Two wires, A and B, are made of the same metal and have equal length, but the resistance of wire A is four times the resistance of wire B. How do their diameters compare?**ConcepTest 25.3aWires I**1) dA = 4dB 2) dA = 2dB 3) dA = dB 4) dA = 1/2dB 5) dA = 1/4dB Two wires, A and B, are made of the same metal and have equal length, but the resistance of wire A is four times the resistance of wire B. How do their diameters compare? The resistance of wire A is greater because its area is less than wire B. Since area is related to radius (or diameter) squared, the diameter of A must be two times less than the diameter of B.**25-5 Electric Power**Power, as in kinematics, is the energy transformed by a device per unit time: or**25-5 Electric Power**The unit of power is the watt, W. For ohmic devices, we can make the substitutions:**25-5 Electric Power**Example 25-8: Headlights. Calculate the resistance of a 40-W automobile headlight designed for 12 V.**25-5 Electric Power**What you pay for on your electric bill is not power, but energy – the power consumption multiplied by the time. We have been measuring energy in joules, but the electric company measures it in kilowatt-hours, kWh: 1 kWh = (1000 W)(3600 s) = 3.60 x 106 J.**25-5 Electric Power**Example 25-9: Electric heater. An electric heater draws a steady 15.0 A on a 120-V line. How much power does it require and how much does it cost per month (30 days) if it operates 3.0 h per day and the electric company charges 9.2 cents per kWh?**25-6 Power in Household Circuits**Conceptual Example 25-12: A dangerous extension cord. Your 1800-W portable electric heater is too far from your desk to warm your feet. Its cord is too short, so you plug it into an extension cord rated at 11 A. Why is this dangerous?**25-7 Alternating Current**Current from a battery flows steadily in one direction (direct current, DC). Current from a power plant varies sinusoidally (alternating current, AC).**25-7 Alternating Current**, The voltage varies sinusoidally with time: , as does the current: