Chapter 19 Electric Potential and Electric Field The electrostatic force is a conservative force. Therefore, it is possible to define an electrical potential energy function with this force. Work done by a conservative force is equal to the negative of the change in potential energy Simplest Example There is a uniform field E W = F x = q E x ΔPE = - W = - q E x only for a uniform field As before, the negative of the slope of (any) potential energy plot is the force. Electrical Potential and Potential Energy The potential difference between points A and B is defined as the change in the potential energy (final value minus initial value) of a charge q moved from A to B divided by the amount of the charge ΔV = V B V A = ΔPE / q ΔPE = q ΔV Both electric potential energy and potential difference are scalar quantities. Charge can be positive OR negative. Units of potential difference V = J/C (volt) The electron volt (ev) is defined as the energy that an electron gains when accelerated through a potential difference of 1 V 1 ev = 1.6 x 10-19 J 1
Energy Conversion In general, for a mass moving from A to B due to a conservative force, For the electric force, so that Positive charges accelerate in the direction of decreasing electric potential; Negative charges accelerate in the direction of increasing electric potential. Electric Potential of a Point Charge The point of zero electric potential is taken to be at an infinite distance from the charge The potential created by a point charge q (positive or negative) at any distance r from the charge is V k e q r A potential exists at some point in space whether or not there is a test charge at that point point-charge 2
Electric Potential of a Point Charge Have we been given too many laws? ΔPE = - q E Δx No, both equations are result of the SAME law. The local change of potential with spatial variation is the negative of the local electric field. The sum of the product of the electric field and displacement along a path gives the negative of the potential difference. KeQ KQ e V() r V( r r) r r r ΔV = - E Δx V V( rr) V( r) KQ e 1 1 r 1 ( r/ r) KQr r r e E lim r 0 V r KeQ 2 r Multiple Point Charges Superposition principle applies The total electric potential at some point P due to several point charges is the algebraic sum of the electric potentials due to the individual charges The algebraic sum is used because potentials are scalar quantities What is the electric potential at point P? two point-charges 3
Electrical Potential Energy of Two Charges V 1 is the electric potential due to q 1 at some point P 1 The work required to bring q 2 from infinity to P 1 without acceleration is q 2 V 1 This work is equal to the potential energy of the two particle system q1q 2 PE q2v1 k e r This potential energy is assumed to be shared between the two charges. You can only count this potential energy ONCE in considering the total energy of the system. It is not meaningful to consider the energy of a charge in the electric potential created by itself. Equipotential Surfaces An equipotential surface is a surface on which all points are at the same potential No work is required to move a charge at a constant speed on an equipotential surface The electric field at every point on an equipotential surface is perpendicular to the surface The equipotential surfaces for a point charge are a family of spheres centered on the point charge. 4
Equipotential Surfaces and Electric Fields For two point charges Equipotential lines are shown in blue (positive) and red (negative). Electric field lines are shown in black. Electric Field Lines Field lines point in direction perpendicular to the equipotential surfaces. Field lines point in the directions where the potential decreases the fastest. 5
Examples 38. Figure shows the electric field lines near two charges q 1 and q 2, the first having a magnitude four times that of the second. Sketch the equipotential lines for these two charges, and indicate the direction of increasing potential. 39. Sketch the equipotential lines a long distance from the charges. Indicate the direction of increasing potential. Potentials and Charged Conductors All points on the surface of a charged conductor in electrostatic equilibrium are at the same potential. Therefore, the electric potential is a constant everywhere on the surface of a charged conductor in equilibrium E = 0 inside the conductor The electric field just outside the conductor is perpendicular to the surface The potential everywhere inside the conductor is constant and equal to its value at the surface All of the excess charge of a conductor resides at its surface. 6
Why Do Excess Charge On A Conductor Flock To Sharp Corners? A flat conductor with a net charge Alternative way to see this: How do you put charges in a strait line so that all charges (in the middle) feel no net force? Capacitor And Capacitance A capacitor is a device used in a variety of electric circuits to store charge (and energy). The capacitance, C, of a capacitor is defined as the ratio of the magnitude of the charge on either conductor (plate) to the magnitude of the potential difference between the conductors C Q V The capacitance of a device depends on the geometric arrangement of the conductors For a parallel-plate capacitor whose plates are separated by vacuum: C o A d Capacitance Units: Farad (F) 1 F = 1 C / V A Farad is very large. Often will see µf or pf Charge accumulates on the inside surfaces only. Zero electric field outside. 7
Charging Of A Capacitor When capacitors are first connected in the circuit, electrons are transferred from the left plates through the battery to the right plate, leaving the left plate positively charged and the right plate negatively charged The flow of charges ceases when the voltage across the capacitors equals that of the battery The capacitors reach their maximum charge when the flow of charge ceases The equilibrium charge on the capacitor is q = CV Capacitors in Parallel active figure The flow of charges ceases when the voltage across the capacitors equals that of the battery The total charge is equal to the sum of the charges on the capacitors Q total = Q 1 + Q 2 The potential difference across the capacitors is the same. (Each is equal to the voltage of the battery) The capacitors can be replaced with one capacitor with a capacitance of C eq C eq = C 1 + C 2 The equivalent capacitance of a parallel combination of capacitors is greater than any of the individual capacitors 8
Capacitors in Series When a battery is connected to the circuit, electrons are transferred from the left plate of C 1 to the right plate of C 2 through the battery As this negative charge accumulates on the right plate of C 2, an equivalent amount of negative charge is removed from the left plate of C 2, 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 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 V V1 V2 V3 1 C eq 1 C 1 1 C 2 1 C 3 The equivalent capacitance of a series combination is always less than any individual capacitor in the combination active figure V Q C eq 9
Parallel Plate Capacitors and Dielectrics A dielectric is an insulator; when placed between the plates of a capacitor it gives a lower potential difference with the same charge, due to the polarization of the material. This increases the capacitance. electric field without dielectric electric field with dielectric dielectric constant 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 If we draw Gaussian surface cutting through the middle of the dielectric layer, some net charge due to the dielectric is included. With the same charge on the capacitor plates, the electric field between the plate is reduced by a factor of the dielectric constant, when a dielectric is inserted. V decreases, therefore C increases. 10
Dielectrics The dielectric constant is a property of the material; here are some examples: What is the dielectric constant for metals? Dielectric Breakdown If the electric field in a dielectric becomes too large, it can tear the electrons off the atoms, thereby enabling the material to conduct. This is called dielectric breakdown; the field at which this happens is called the dielectric strength. 11
Energy Stored in a Capacitor Energy stored = ½ Q ΔV (Why is it not Q V?) From the definition of capacitance, this can be rewritten in different forms 1 C( V ) W QV 2 2 2 2 Q 2C Electrical Energy Storage The energy stored in a capacitor can be put to a number of uses: a camera flash; a cardiac defibrillator; and others. In addition, capacitors form an essential part of most electrical devices used today. If we divide the stored energy by the volume of the capacitor, we find the energy per unit volume; this result is valid for any electric field: 12
Example 61. Find the total capacitance of the combination of capacitors shown in the figure. 54. (a) What is the capacitance of a parallel plate capacitor having plates of area 1.50 m 2 that are separated by 0.0200 mm of neoprene rubber? (b) What charge does it hold when 9.00 V is applied to it? Chapter 19 Summary 1. Electric field = force/charge = potential/distance 2. Electric potential due to a point charge. 3. Superposition of potential due to multiple charges. 4. Capacitance 5. Energy stored in capacitor. 6. Capacitance in parallel and in series. 7. Dielectric material. 13