Department of Physics United States Naval Academy Lecture 12: Capacitance; Electric Potential Energy & Energy density Learning Objectives • Explain how the work required to charge a capacitor results in the potential energy of the capacitor and apply the relationship between the potential energy U, the capacitance C, and the potential difference V . For any electric field, apply the relationship between the potential energy density u in the field and the field’s magnitude E. Capacitors in Parallel and in Series: When there is a combination of capacitors in a circuit, we can sometimes replace that combination with an equivalent capacitor Capacitors in Parallel: The total charge q stored on the capacitors is the sum of the charges stored on all the ca- pacitors. The equivalent capacitances Ceq is n X Ceq = C1 + C2 + ··· + Cn = Ci i=1 Capacitors in Series: The sum of the potential differ- ences across all the capacitors is equal to the applied po- tential difference V . The equivalent capacitances Ceq is n 1 1 1 1 X 1 = + + ··· = C C C C C eq 1 2 n i=1 i Electric Potential Energy: The electric potential energy U of a charged capacitor, q2 U = = 1 CV 2 2C 2 is equal to the work required to charge the capacitor. This energy can be associated with the capacitor’s electric field E~ . That is, the potential energy of a charged capacitor may be viewed as being stored in the electric field between its plates. 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 1 2 u = 2 "oE Capacitor with a Dielectric: If the space between the plates of a capacitor is completely filled with a dielectric material, which is an insulating material such as mineral oil or plastic, the capacitance C in vacuum (or, effectively, in air) is multiplied by the material’s dielectric constant k, which is a number greater than 1. Another effect of the introduction of a dielectric is to limit the potential difference that can be applied between the plates to a certain value Vmax, called the breakdown potential. If this value is substantially exceeded, the dielectric material will break down and form a conducting path between the plates. Every dielectric material has a characteristic dielectric strength, which is the maximum value of the electric field that it can tolerate without breakdown. In a region that is completely filled by a dielectric, all electrostatic equations containing the permittivity constant "o must be modified by replacing "o with " = κεo where " is the permittivity of the dielectric material and κ is the dielectric constant. When a dielectric material is placed in an external electric field, it develops an internal electric field that is oriented opposite the external field, thus reducing the magnitude of the electric field inside the material. © 2019 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Question 1.0 When a dielectric slab is inserted between the plates of one of the two identical capacitors as shown, do the following properties of that capacitor increase, decrease, or remain the same: (a) capacitance, (b) charge, (c) potential difference, and (d) potential energy? (e) How about the same properties of the other capacitor? © 2019 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Problem 1.0 A potential difference of V = 100 V is applied across a capacitor arrangement (see below) with capacitances C1 = 10:0 µF, C2 = 5:00 µF, and C3 = 15:0 µF. What are (a) charge q3, (b) potential difference V3, and (c) stored energy U3 for capacitor 3, (d) q1, (e) V1, and (f ) U1 for capacitor 1, and (g) q2, (h) V2, and (i) U2 for capacitor 2? © 2019 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Problem 2.0 A parallel-plate capacitor of plate area A = 10:5 cm2 and plate separation 2d = 7:12 mm is shown. The left half of the gap is filled with material of dielectric constant κ1 = 21:0; the top of the right half is filled with material of dielectric con- stant κ2 = 42:0; the bottom of the right half is filled with material of dielectric constant κ3 = 58:0. What is the capacitance? © 2019 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Problem 3.0 A certain parallel-plate capacitor is filled with a dielectric for whichκ = 5:5. The area of each plate is 0.034 m2, and the plates are separated by 2.0 mm. The capacitor will fail (short out and burn up) if the electric field between the plates exceeds 200 kN/C. What is the maximum energy that can be stored in the capacitor? © 2019 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD.