Chapter 24 Capacitance, Dielectrics, Electric Energy Storage Chapter 24 opener. Capacitors come in a wide range of sizes and shapes, only a few of which are shown here. A capacitor is basically two conductors that do not touch, and which therefore can store charge of opposite sign on its two conductors. Capacitors are used in a wide variety of circuits, as we shall see in this and later Chapters.

23-5 Equipotential Surfaces An equipotential is a line or surface over which the potential is constant. Electric field lines are perpendicular to equipotentials. The surface of a conductor is an equipotential. Figure 23-16. Equipotential lines (the green dashed lines) between two oppositely charged parallel plates. Note that they are perpendicular to the electric field lines (solid red lines).

23-5 Equipotential Surfaces Example 23-10: Point charge equipotential surfaces. For a single point charge with Q = 4.0 × 10-9 C, sketch the equipotential surfaces (or lines in a plane containing the charge) corresponding to V1 = 10 V, V2 = 20 V, and V3 = 30 V. Solution: Equipotential surfaces are spheres surrounding the charge; radii are shown in the figure (in meters).

23-5 Equipotential Surfaces

23-5 Equipotential Surfaces Equipotential surfaces are always perpendicular to field lines; they are always closed surfaces (unlike field lines, which begin and end on charges). Figure 23-18. Equipotential lines (green, dashed) are always perpendicular to the electric field lines (solid red) shown here for two equal but oppositely charged particles.

Equipotential Surfaces Surfaces at same potential like contours on topographic maps Lines link places at same elevation (same Ug) Lines link places at same potential (same V)

Equipotential Surfaces Surfaces that have the same potential (voltage) at every point Electric Field lines are perpendicular to equipotential surfaces Potential difference between surfaces is constant Surfaces are closer where the potential is stronger Electric field always points in the direction of maximum potential DECREASE

Question Which image below best shows the equipotential and Electric field lines A + - B + - C + - D + -

23-6 Electric Dipole Potential The potential due to an electric dipole is just the sum of the potentials due to each charge, and can be calculated exactly. For distances large compared to the charge separation: Figure 23-20. Electric dipole. Calculation of potential V at point P.

23-7 E Determined from V If we know the field, we can determine the potential by integrating. Inverting this process, if we know the potential, we can find the field by differentiating: This is a vector differential equation; here it is in component form:

23-7 E Determined from V Example 23-11: for ring and disk. Use electric potential to determine the electric field at point P on the axis of (a) a circular ring of charge and (b) a uniformly charged disk. Solution: a. Just do the derivatives of the result of Example 23-8; the only nonzero component is in the x direction. b. Same as (a); use the result of Example 23-9.

23-8 Electrostatic Potential Energy; the Electron Volt The potential energy of a charge in an electric potential is U = qV. To find the electric potential energy of two charges, imagine bringing each in from infinitely far away. The first one takes no work, as there is no field. To bring in the second one, we must do work due to the field of the first one; this means the potential energy of the pair is:

23-8 Electrostatic Potential Energy; the Electron Volt One electron volt (eV) is the energy gained by an electron moving through a potential difference of one volt: 1 eV = 1.6 × 10-19 J. The electron volt is often a much more convenient unit than the joule for measuring the energy of individual particles.

24-1 Capacitors A capacitor consists of two conductors that are close but not touching. A capacitor has the ability to store electric charge. Figure 24-1. Capacitors: diagrams of (a) parallel plate, (b) cylindrical (rolled up parallel plate).

24-1 Capacitors Parallel-plate capacitor connected to battery. (b) is a circuit diagram. Figure 24-2. (a) Parallel-plate capacitor connected to a battery. (b) Same circuit shown using symbols.

24-1 Capacitors When a capacitor is connected to a battery, the charge on its plates is proportional to the voltage: The quantity C is called the capacitance. Unit of capacitance: the farad (F): 1 F = 1 C/V.

24-2 Determination of Capacitance For a parallel-plate capacitor as shown, the field between the plates is E = Q/ε0A. Integrating along a path between the plates gives the potential difference: Vba = Qd/ε0A. This gives the capacitance: Figure 24-4. Parallel-plate capacitor, each of whose plates has area A. Fringing of the field is ignored.

24-2 Determination of Capacitance Example 24-1: Capacitor calculations. (a) Calculate the capacitance of a parallel-plate capacitor whose plates are 20 cm × 3.0 cm and are separated by a 1.0-mm air gap. (b) What is the charge on each plate if a 12-V battery is connected across the two plates? (c) What is the electric field between the plates? (d) Estimate the area of the plates needed to achieve a capacitance of 1 F, given the same air gap d. Solution: a. C = 53 pF. b. Q = CV = 6.4 x 10-10 C. c. E = V/d = 1.2 x 104 V/m. d. A = Cd/ε0 = 108 m2.