Capacitors (Chapter 26)


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1 Capacitance, C Simple capacitive circuits Parallel circuits Series circuits Combinations Electric energy Dielectrics Capacitors (Chapter 26)
2 Capacitors What are they? A capacitor is an electric device used in a variety of electric circuits Its functionality is based on the storage of energy associated with the electric field between two symmetric distributions of unlike charges insulated from one another Any two closely separated conductors will form a capacitor: in order to charge it, one can use a battery to do work in order to transfer a charge Q from one conductor to the other, such that one conductor will have a deficit +Q and the other a surplus Q of electrons. As a result an electric field will appear between the conductors: electric field means ability to do work, or stored energy Capacitors come in various arrangements of conductors: parallel plates, concentric spheres, coaxial cylinders etc While developing the generic ideas about any such capacitor architectures, in PHY 182 we ll focus on the simplest (and most common) type: the parallel plate capacitor
3 Capacitors Capacitance The ability of a capacitor to store charge (that is electric field and energy) is given by its capacitance: Def: The capacitance, C, of a capacitor is defined as the ratio between the amount of electric charge Q it holds and the potential difference V between its plates C Q V C 1 Farad (F) 1C 1V SI Comments: One Farad is a very large capacitance: so most often we ll see µf, nf or pf The capacitance of a capacitor is a characteristic of the device, and does not depend on the difference of potential applied across the plates Thus, according to the definition, if a difference of potential V is applied across the plates of a capacitor of capacitance C, it will store a maximum charge Q = CV +Q V E Q Q CV C V V CEd d
4 Problem: 1. Spherical capacitor: A spherical capacitor consists in an interior sphere of radius r a in the center of a spherical shell of inner radius r b. Calculate the capacitance in terms of r a, r b and constants.
5 For a parallelplate capacitor filled with air, we can easily derive the capacitance by applying the definition to a capacitor as on the adjacent figure If under a potential difference V ab = Ed, the plates will store a charge of density σ, such that Q = σa. Then ab Capacitors The parallelplate capacitor So, the capacitance depends on the geometric arrangement of the conductors and the electric properties of the insulating material between them. But how? C Q Q A V Ed d Ex: Consider a parallelplate capacitor of area A, of plate separation d When connected to the battery of voltage V across battery, charge is pulled off one plate and transferred to the other plate The transfer stops when across capacitor across battery Then the charge stored on the capacitor plates will be V A Q CV V d 0 C V 0 across capacitor 0 A d
6 Capacitors Revisit the field between the parallel plates If the plate separation is much smaller than the size of the plates, the electric field inside is well approximated by the field of two infinite parallel sheets of charge However, the approximation works only close to the center of the plates, not near the edges where the field is not uniform Since the potential difference across the plates is V = Ed, if we apply a larger voltage V, a larger field is produces corresponding to more charge Q deposited Since the field is constant, the equipotential surfaces are equally spaced flat surfaces parallel with the plates The equipotential surface close to the positive plate has the largest potential, and the potential decreases uniformly to the surface on the negative plate: Ex: The potential differences V i V i 1 between equipotential surfaces at 4 equal steps Δx between the plates of a capacitor are EΔx, such that potential difference across the plates is V V V V V V V V V V V Ex Ed ab a b a b 4 Potential V a V 1 V 2 V 3 V b V a > V 1 > V 2 > V 3 > V b slope = ΔV/Δx = E E Δx d x
7 Electric Circuits Capacitors in circuits A circuit is a network of electric devices usually containing a source of electrical energy (such as a battery) connected to electric elements (such as capacitors) A circuit diagram can be used to show the path of the real circuit If a capacitor is connected in a circuit across two points with an electric potential difference, the electrons are transferred through wires from one plate to the other plate, leaving one plate positively charged and the other plate negatively charged The flow of charges ceases when the voltage across the capacitor equals that across the two points in the circuits. Then, as long the potential difference remains unchanged, the capacitor stays inactive: a storage of charge (that is, electric energy) The capacitors are represented in circuits using a symbol for the two plates. The battery needed to produce potential differences is represented by a similar symbol The similarity is due to the fact that both devices are sources of electric charge; however, while the battery is ideally a limitless source, the capacitor is limited by its capacity C + _ Capacitors can be combined in circuits: the simplest combinations are in parallel and series. Let s find the equivalent capacitance that performs the same function as these elementary combinations V
8 Electric Circuits Capacitors in parallel Capacitors in parallel are all connected across the same two points. For illustration, consider two capacitor in parallel Therefore, all capacitors will be connected across the battery, so they will be under the same voltage V The total charge is equal to the sum of the charges on the capacitors Q Q Q net 1 2 This net charge can be considered as being stored on the parallel combination seen as only one capacitor with an equivalent capacitance C p : Q Q Q net 1 2 C C C Cp V CV 1 CV 2 p 1 2 The result can be extrapolated to for n capacitors in parallel: C C C C C p Notice that the parallel equivalent capacitance is larger than any of the individual capacitances n
9 Electric Circuits Capacitors in series Capacitors in series chained negative plate to positive plate, such that each plate holds the same charge, and the charge on the combination is the same as on each capacitor However, the potential difference delivered by the battery across the equivalent capacitance C s is imparted across the capacitors in series. Hence V V V 1 2 Q Q Q Cs C1 C CC 1 2 Cs C C C C C s For n capacitors in series: C C C C s Q Q Q n Notice that the series equivalent capacitance is smaller than any of the individual capacitances
10 Problems: 2. Mixed combinations of capacitors: A capacitive circuit combines capacitors as in the figure (the numbers are capacitances in μf). a) Find the equivalent capacitance of the capacitive circuit in the figure. b) Say that a 12V battery is connected between points ab. What is the amount of charge stored on the combination of capacitors? 3. Capacitive circuit analysis: Three capacitors are connected across a 12V battery as on the figure. a) Find the equivalent capacitance of the circuit b) Find the charge on each capacitor in the circuit and the potential difference across it
11 Energy Stored in a Capacitor The energy stored in a capacitor is equal to the energy necessary to increase the charge on the plates from zero to Q: Q 1 W U Vdq qdq C Q 0 0 U Q 2C From the definition of capacitance, this can be rewritten in different forms U CV QV q +Q Q V Therefore, we see that a capacitor can be seen a charge or energy storage device When connected across a conductive medium this energy is released. 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 We can define the energy density u as the energy per unit volume. For a parallel plate capacitor (but with a result valid for any capacitor) 1 A Ed Vacuum U 2 CV d u 1 2 u 2 0E Volume Ad Ad
12 Dielectrics Capacitors with dielectrics A dielectric is an insulating material that, when placed between the plates of a capacitor, increases the capacitance Ex: Dielectrics can be rubber, plastic, or waxed paper If the capacitance of a capacitor with air between the plates is C 0, when a dielectric completely fills the region between the plates, the capacitance increases by the factor κ > 1 called dielectric constant: C C A A 0 0 d d ε =κε 0 is called the electric permittivity of the dielectric Ex: A dielectric improves the performance of a capacitor: a) Say that a capacitor without dielectric stores a certain amount of charge Q 0. The voltage across the plates is V 0. b) With the dielectric inside, the charge stays the same but C increases and V decreases: the same charge is held with a lower V. Dielectric E0 1 E u 2 E Quiz 1: Is the energy increasing or decreasing when a dielectric is inserted in between the charged plates? a) Increases b) Decreases c) Stays the same 2
13 Dielectrics Polarization If we decrease V we also decrease E which is done by a polarization in the dielectric resulting in an electric field opposite to the initial field. Polarization occurs when there is a separation between the centers of gravity of negative and positive charge of the molecules In a capacitor, the dielectric becomes polarized because it is in an electric field that exists between the plates 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 The polarization results in an induced surface charge density σ i which decreases the net charge density σ σ i E 1 1 i 0 E i 0 0 Therefore, for a high κ dielectric, the induced density can almost cancel out the density on the plates, so a small potential difference will hold a large charge density σ on the plates
14 Problems: 4. Dielectrics in series and parallel: A parallel plate capacitor of capacitance C 0 has the space between the plates filled with two slabs of dielectric, with constants κ 1 and κ 2. What is the capacitance in terms of C 0, d, κ 1 and κ 2 when the space is filled as in figure a) and then as in figure b): a) b) d κ 1 κ 2 ½d d κ 1 κ 2 5. Energy in capacitors: Three parallelplate capacitors are networked as in the figure with given geometrical characteristics. Together they hold a net charge Q. A particle of mass m and charge q hangs above a very long wire of static charge density λ. The particle is released from rest and pulls the dielectric out of a capacitor. Sketch the steps necessary to calculate the speed of the particle at the moment when the dielectric is completely out, using conservation of energy.
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