PHY 201 Chapter 24, Fall 2017

CHAPTER 24 Electric Potential Electric Potential 24-1 24-1 Electric Potential PHY 201  If we release particle 1 at P, it begins to move and thus has kinetic energy.  Energy cannot appear by magic, so from where does it come? It comes from the electric potential energy U associated with the force between the two particles  To account for the potential energy U (which is a scalar quantity), we define an electric potential V (also a scalar quantity) that is set up at P by particle 2.  The electric potential exists regardless of whether particle 1 is at P. PHY 305  By analogy to the gravitational potential energy, we have for the electric potential energy a 1/r relationship.  Where the zero of potential energy is at very large separation r (r >>1)  Potential energy can be positive or negative. It is positive for two like charges and negative or two unlike charges. PHY 201 24-1 Electric Potential PHY 305  Two positive charges: q1 is fixed at the origin, and q2 is free to move.  If charge q2 is released from rest on the +x-axis, describe its subsequent motion. Tell what happens to the kinetic energy K, potential energy U, and total energy E Attraction or repulsion  After release, q2 accelerates to the right  K increases, U decreases  E = K + U is conserved and stays constant. PHY 201 24-1 Electric Potential PHY 305  Two opposite charges: q1 is fixed at the origin, and q2 is free to move.  If charge q2 is released from rest on the +x-axis, describe its subsequent motion. Tell what happens to the kinetic energy K, potential energy U, and total energy E Attraction or repulsion  After release, q2 accelerates to the left, what about K, U & E?;  K increases, U decreases (it becomes even more negative)  E = K + U is conserved and stays constant. PHY 201 24-1 Electric Potential 24-1 Electric Potential  The electric potential V at a point P in the electric field of a charged object is  The electric potential is the amount of electric potential energy per unit charge PHY 201  The SI unit for potential that follows is the joule per coulomb. This combination occurs so often that a special unit, the volt (abbreviated V), is used to represent it. 1 volt = 1 joule per coulomb. 24-1 Electric Potential Change in Electric Potential. If the particle moves through a potential difference ΔV, the change in the electric potential energy is Work by the Field. The work W done by the electric force as the particle moves from i to f: Conservation of Energy. If a particle moves through a change ΔV in electric potential without an applied force acting on it, applying the conservation of mechanical energy gives the change in kinetic energy as Work by an Applied Force. If some force in addition to the electric force acts on the particle, we account for that work PHY 201 24-1 Electric Potential PHY 201  Definition: The electric potential, V, is the potential energy per unit charge.  Thus, the electric potential a distance r for a point charge q is: 24-1 Electric Potential PHY 201 Equipotential Surfaces and the Electric Field 24-2 24-2 Equipotential Surfaces and the Electric Field Adjacent points that have the same electric potential form an equipotential surface, which can be either an imaginary surface or a real, physical surface. Figure shows a family of equipotential surfaces associated with the electric field due to some distribution of charges. The work done by the electric field on a charged particle as the particle moves from one end to the other of paths I and II is zero because each of these paths begins and ends on the same equipotential surface and thus there is no net change in potential. The work done as the charged particle moves from one end to the other of paths III and IV is not zero but has the same value for both these paths because the initial and final potentials are identical for the two paths; that is, paths III and IV connect the same pair of equipotential surfaces. PHY 201 24-2 Equipotential Surfaces and the Electric Field PHY 201 24-2 Equipotential Surfaces and the Electric Field The electric potential difference between two points i and f is where the integral is taken over any path connecting the points. If the integration is difficult along any particular path, we can choose a different path along which the integration might be easier. If we choose Vi = 0, we have, for the potential at a particular point, In a uniform field of magnitude E, the change in potential from a higher equipotential surface to a lower one, separated by distance Δx, is A test charge q0 moves from point i to point f along the path shown in a non-uniform electric field. During a displacement ds, an electric force q0 E acts on the test charge. This force points in the direction of the field line at the location of the test charge. PHY 201 Potential due to a Charged Particles 24-3 24-3 Potential due to a Charged Particle The potential due to a collection of charged particles is Thus, the potential is the algebraic sum of the individual potentials, with no consideration of directions. Potential due to a group of Charged Particles PHY 201 Potential due to a Electric Dipole 24-4 24-4 Potential due to a Electric Dipole (a) Point P is a distance r from the midpoint O of a dipole. The line OP makes an angle θ with the dipole axis. (b) If P is far from the dipole, the lines of lengths r(+) and r(-) are approximately parallel to the line of length r, and the dashed black line is approximately perpendicular to the line of length r(-). The net potential at P is given by We can approximate the two lines to P as being parallel and their length difference as being the leg of a right triangle with hypotenuse d (Fig. b). Also, that difference is so small that the product of the lengths is approximately r 2. We can approximate V to be where θ is measured from the dipole axis as shown in Fig. a. And since p=qd, we have PHY 201 Potential due to a Continuous Charge Distribution 24-5 24-5 Potential due to a Continuous Charge Distribution For a continuous distribution of charge (over an extended object), the potential is found by (1) dividing the distribution into charge elements dq that can be treated as particles and then (2) summing the potential due to each element by integrating over the full distribution: We now examine two continuous charge distributions, a line and a disk. PHY 201 24-5 Potential due to a Continuous Charge Distribution Line of Charge Fig. a has a thin conducting rod of length L. As shown in fig. b the element of the rod has a differential charge of This element produces an electric potential dV at point P (fig c) given by We now find the total potential V produced by the rod at point P by integrating dV along the length of the rod, from x = 0 to x = L (Figs.d and e) Simplified to, PHY 201 Calculating the Field from the Potential 24-6 24-6 Calculating the Field from the Potential PHY 201 Electric Potential Energy of a System of Charged Particles 24-7 The electric potential energy of a system of charged particles is equal to the work needed to assemble the system with the particles initially at rest and infinitely distant from each other. For two particles at separation r, Two charges held a fixed distance r apart. 24-7 Electric Potential Energy of a System of Charged Particles Three charges are fixed at the vertices of an equilateral triangle. What is the electric potential energy of the system? PHY 201 PHY 305 Electric Potential Energy Multiple charges  For 3 charges and more, each pair will have a potential energy. The total potential energy is the sum over all three pairs.  The more charges there are the bigger the number of charge pairs (for 4 charges, there exist 6 pairs for 5 charges, 10 pairs, and so on…) The bigger the computation above becomes.  It becomes necessary to describe energy in terms of the electric potential. PHY 201 Summary 24 Summary Electric Potential • The electric potential V at point P in the electric field of a charged object: Electric Potential Energy • Electric potential energy U of the particle-object system: • If the particle moves through potential ΔV: Eq. 24-2 Eq. 24-3 Eq. 24-4 Mechanical Energy • Applying the conservation of mechanical energy gives the change in kinetic energy: • In case of an applied force in a particle • In a special case when ΔK=0: Eq. 24-9 Eq. 24-11 Eq. 24-12 Finding V from E • The electric potential difference between two point I and f is: Eq. 24-18 PHY 201 24 Summary Potential due to a Charged Particle • due to a single charged particle at a distance r from that particle : • due to a collection of charged particles Eq. 24-26 Eq. 24-27 Eq. 24-30 Potential due to a Continuous Charge Distribution • For a continuous distribution of charge: Eq. 24-32 Eq. 24-40 Calculating E from V • The component of E in any direction is: Eq. 24-46 Potential due to an Electric Dipole • The electric potential of the dipole is Electric Potential Energy of a System of Charged Particle • For two particles at separation r: PHY 201 CHAPTER 25 Capacitance Introduction PHY 201  The capacitor: a device in which electrical energy can be stored The batteries can supply energy at only a modest rate, too slowly for the photoflash unit to emit a flash of light. However, once the capacitor is charged, it can supply energy at a much greater rate when the photoflash unit is triggered—enough energy to allow the unit to emit a burst of bright light.  The first step in our discussion of capacitors is to determine how much charge can be stored. This “how much” is called capacitance. 25-1 Capacitance A capacitor consists of two isolated conductors (the plates) with charges +q and -q. Its capacitance C is defined from where V is the potential difference between the plates. A parallel-plate capacitor, made up of two plates of area A separated by a distance d. The charges on the facing plate surfaces have the same magnitude q but opposite signs As the field lines show, the electric field due to the charged plates is uniform in the central region between the plates. The field is not uniform at the edges of the plates, as indicated by the “fringing” of the field lines there. PHY 201 PHY 201 • Electric Circuit: • Battery: PHY 201 • Electric Circuit: An electric circuit is a path through which charge can flow. • Battery: A battery is a device that maintains a certain potential difference between its terminals 25-1 Capacitance When a circuit with a battery, an open switch, and an uncharged capacitor is completed by closing the switch, conduction electrons shift, leaving the capacitor plates with opposite charges. In Fig. a, a battery B, a switch S, an uncharged capacitor C, and interconnecting wires form a circuit. The same circuit is shown in the schematic diagram of Fig. b, in which the symbols for a battery, a switch, and a capacitor represent those devices. The battery maintains potential difference V between its terminals. The terminal of higher potential is labeled + and is often called the positive terminal; the terminal of lower potential is labeled - and is often called the negative terminal. Charging Capacitor PHY 201 25-2 Calculating the Capacitance To relate the electric field E between the plates of a capacitor to the charge q on either plate, we shall use Gauss’ law: the potential difference between the plates of a capacitor is related to the field E by Letting V represent the difference Vf - Vi , we can then recast the above equation as: Calculating electric field and potential difference A charged parallel-plate capacitor. A Gaussian surface encloses the charge on the positive plate. The integration is taken along a path extending directly from the negative plate to the positive plate. PHY 201 25-2 Calculating the Capacitance We assume, as Figure suggests, that the plates of our parallel-plate capacitor are so large and so close together that we can neglect the fringing of the electric field at the edges of the plates, taking E to be constant throughout the region between the plates. We draw a Gaussian surface that encloses just the charge q on the positive plate where A is the area of the plate. And therefore, A charged parallel-plate capacitor. A Gaussian surface encloses the charge on the positive plate. The integration is taken along a path extending directly from the negative plate to the positive plate. Parallel-Plate Capacitor Now if we substitute q in the above relations to q=CV, we get, PHY 201 25-2 Calculating the Capacitance Figure shows, in cross section, a cylindrical capacitor of length L formed by two coaxial cylinders of radii a and b. We assume that L >> b so that we can neglect the fringing of the electric field that occurs at the ends of the cylinders. Each plate contains a charge of magnitude q. Here, charge and the field magnitude E is related as follows, Solving for E field: Cylindrical Capacitor A cross section of a long cylindrical capacitor, showing a cylindrical Gaussian surface of radius r (that encloses the positive plate) and the radial path of integration. This figure also serves to illustrate a spherical capacitor in a cross section through its center. From the relation C= q/V, we then have PHY 201 25-2 Calculating the Capacitance For spherical capacitor the capacitance is: Capacitance of an isolated sphere: Others… A cross section of a long cylindrical capacitor, showing a cylindrical Gaussian surface of radius r (that encloses the positive plate) and the radial path of integration. This figure also serves to illustrate a spherical capacitor in a cross section through its center. PHY 201 PHY 305 Capacitors in Parallel Capacitors in Series q1 q2 q3 q = q1 = q2 = q3 V = V1 + V2 + V3 V = V1 = V2 = V3 q = q1 + q2 + q3 PHY 201 25-3 Capacitors in Parallel and in Series Capacitors in Parallel PHY 201 25-3 Capacitors in Parallel and in Series Capacitors in Series PHY 201 25-4 Energy Stored in an Electric Field The electric potential energy U of a charged capacitor, and, is equal to the work required to charge the capacitor. This energy can be associated with the capacitor’s electric field E. Every electric field, in a capacitor or from any other source, has an associated stored energy. In vacuum, the energy density u (potential energy per unit volume) in a field of magnitude E is PHY 201 25-5 Capacitor with a Dielectric PHY 201  A dielectric material (dielectric for short) is an electrical insulator that can be polarized by an applied electric field. 25-5 Capacitor with a Dielectric (b) If the charge on the capacitor plates is maintained, as in this case by isolating the capacitor, the effect of a dielectric is to reduce the potential difference between the plates. The scale shown is that of a potentiometer, a device used to measure potential difference (here, between the plates). A capacitor cannot discharge through a potentiometer. (a)If the potential difference between the plates of a capacitor is maintained, as by the presence of battery B, the effect of a dielectric is to increase the charge on the plates. If the space between the plates of a capacitor is completely filled with a dielectric material, the capacitance C in vacuum (or, effectively, in air) is multiplied by the material’s dielectric constant κ, (Greek kappa) which is a number greater than 1. PHY 201 25-5 Capacitor with a Dielectric PHY 201  The magnitude of the electric field produced by a point charge inside a dielectric is given by this modified form of:  Because k is always greater than unity, both these equations show that for a fixed distribution of charges, the effect of a dielectric is to weaken the electric field that would otherwise be present. 25-5 Capacitor with a Dielectric An Atomic View PHY 201  Polar dielectrics: The molecules of some dielectrics, like water, have permanent electric dipole moments. In such materials (called polar dielectrics), the electric dipoles tend to line up with an external electric field  Nonpolar dielectrics: Regardless of whether they have permanent electric dipole moments, molecules acquire dipole moments by induction when placed in an external electric field. Thus, the effect of both polar and nonpolar dielectrics is to weaken any applied field within them, as between the plates of a capacitor. 25-5 Capacitor with a Dielectric An Atomic View (b) An electric field is applied, producing partial alignment of the dipoles. Thermal agitation prevents complete alignment. Nonpolar Dielectrics PHY 201 25-6 Dielectrics and Gauss’ Law When a dielectric is present, Gauss’ law may be generalized to where q is the free charge. Any induced surface charge is accounted for by including the dielectric constant k inside the integral. A parallel-plate capacitor (a) without and (b) with a dielectric slab inserted. The charge q on the plates is assumed to be the same in both cases. • Inserting a dielectric into a capacitor causes induced charge to appear on the faces of the dielectric and weakens the electric field between the plates. • The induced charge is less than the free charge on the plates. Note: The flux integral now involves κE, not just E. The vector ε0κE is sometimes called the electric displacement D, so that the above equation can be written in the form PHY 201 25 Summary Capacitor and Capacitance • The capacitance of a capacitor is defined as: Determining Capacitance • Parallel-plate capacitor: • Cylindrical Capacitor: • Spherical Capacitor: • Isolated sphere: Eq. 25-1 Eq. 25-9 Eq. 25-14 Capacitor in parallel and series • In parallel: • In series Eq. 25-19 Eq. 25-17 Eq. 25-18 Eq. 25-20 Potential Energy and Energy Density • Electric Potential Energy (U): • Energy density (u) Eq. 25-21&22 Eq. 25-25 PHY 201 25 Summary Capacitance with a Dielectric • If the space between the plates of a capacitor is completely filled with a dielectric material, the capacitance C is increased by a factor κ, called the dielectric constant, which is characteristic of the material. Gauss’ Law with a Dielectric • When a dielectric is present, Gauss’ law may be generalized to Eq. 25-36 PHY 201