Lecture 13 Electrostatic Energy and Energy Density

Transcription

1 Lecture 13 Electrostatic Energy and Energy Density Sections: 4.8 Homework: See homework file

2 Energy of System of Point Charges 1 any system of charged bodies held static in relatively close proximity contains potential energy potential energy is the work done to build the system, i.e., to bring the charges together against the Coulomb forces if the charges of the system were set free, this energy would set them into motion and would be converted into kinetic energy z P the potential of a point charge Q 1 R1 Q1 V1 ( P) = = E dl= E dl, V Q 1 4πε R1 R1 x is the work done to bring a unit positive charge from infinity to a point which is R 1 (meters) away from Q 1 R 1 y LECTURE 13 slide

3 Energy of System of Point Charges if a charge Q is brought at a distance R,1 from the first charge Q 1 the energy spent is Q1 W,1 = Q = QV,1, J=C V 4πε R,1 it does not matter which charge is brought close to which W = W,1 1, as long as Q 1 and Q are held apart at a distance R 1, the system holds W,1 =W 1, energy LECTURE 13 slide 3

4 Energy of System of Point Charges 3 the potential of the (Q 1,Q ) system is now z P V 1& ( P) Q1 Q = + 4πε R 4πε R 1 the work to bring over a third charge Q 3 is x Q Q W + W = Q + = Q ( V + V ) = W + W 1 3,1 3, 3 3 3,1 3, 1,3,3 4πε R3,1 4πε R 3, the total energy spent to built the system of three charges is 1 QQ 1 QQ 3 1 QQ 3 W,1 + W3,1 + W3, = + + = 4πε R,1 R3,1 R 3, QV + QV + QV = W + W + W Q 1,1 3 3,1 3 3, 1, 1,3,3 R 1 R LECTURE 13 slide 4 Q y

5 Energy of System of Point Charges 4 for a system of N charges We = ( QV,1 + QV 3 3,1 + + QNVN,1) + ( QV 3 3, + + QNVN, ) + work to bring charges,, N close to charge 1 charges 3,, N close to charge + ( Q V + QV ) + QV N 1 N 1, N N NN, N N, N 1 charges N 1 and N close to charge N charge N close to charge N 1 an alternative expression (due to the reciprocal nature of the energy) We = ( QV 1 1, + QV 1 1,3 + + QV 1 1, N) + ( QV,3 + + QV, N) + bring charge 1 close to charges,, N charge close to charges 3,, N ( QN VN, N 1 QN VN, N) + QN 1VN 1, N + + charge N close to charges N 1 and N charge N 1 close to charge N add both expressions to obtain N N N 1 W = Q V + Q V + + Q V e 1 1, n, n N Nn, n= n= 1 n= 1 n LECTURE 13 slide 5

6 Energy of System of Charges N 1 W = QV where V = V e n n n nk, n= 1 k= 1 k n N The total electrostatic energy W e of a system of discrete charges is the sum of the energies of all possible pairs of charges. if charge is distributed in a volume W e 1 = ρ v v Vdv LECTURE 13 slide 6

7 Energy of System of Charges: Example (a) Find the energy of a system of two charges Q 1 = Q = 10 1 C in vacuum located at P 1 ( 1,0,0) mm and P (1,0,0) mm, respectively. (b) What is the work necessary to bring over a third charge Q 3 = 10 1 C from infinity to P 3 (0,0,0)? (c) What is the energy stored in the system of 3 charges? LECTURE 13 slide 7

8 Energy Density We = V Ddv = [ ( VD) D V ] dv v ρv v ( VD) = V D+ V D 1 1 We = ( VD) ds ( D V ) dv S [ v] v to account for all energy allow the volume v to expand to infinity D 1/ R, V 1/ R, ds R lim ( VD) ds= 0 E R S [ v] W e 1 = ( ) dv, J DE v energy density w e LECTURE 13 slide 8

9 Energy Density energy integral in an isotropic medium where D = εe W e 1 v ε = E dv W energy density and the field vectors w e = 1 DE e 1 ε = D, J/m v energy density in an isotropic medium we 1 = ε E w e = 3 1 D ε dv LECTURE 13 slide 9

10 Energy Density: Example The plates of a parallel-plate capacitor have an area A = 10 cm. The distance between the plates is d = 1 mm. The relative permittivity of the insulator is ε r = 10. Find the stored electrical energy W e in the capacitor and its density w e if the voltage is V = 100 V. LECTURE 13 slide 10

11 Did You Know? A typical AA battery has a capacity of about 500 mah (or.5 Ah). With a cell voltage of 1.5 V, this corresponds to about 3.75 Wh. What is then the energy in joules stored in the AA battery? LECTURE 13 slide 11

12 Technology Brief: Energy Storage Devices energy density of storage devices (aka specific energy) is defined as stored energy per unit mass energy [J] [Wh] 3600 W = = mass [kg] [kg] SOME ELECTRIC ENERGY STORAGE DEVICES [Ulaby&Ravaioli, Fundamentals of Applied Electromagnetics, 7 th ed.] Compare with other sources of energy: gasoline ~46.4 MJ/kg ~1,900 Wh/kg coal ~4.0 MJ/kg ~6,700 Wh/kg uranium (nuclear fission) ~80,60,000 MJ/kg ~, Wh/kg LECTURE 13 slide 1

13 Illustration of Energy Density: Coaxial Line LECTURE 13 slide 13

14 Illustration of Energy Density: Parallel-plate Line LECTURE 13 slide 14

15 Illustration of Energy Density: Twin-lead Line LECTURE 13 slide 15

16 You have learned: that the total electrostatic energy of a system of point charges is the sum of the potential energy of all charge pairs in the system how to calculate the total electrostatic energy from the field vectors E and D what energy density is and how to compute it from E and D the energy density is proportional to the square of the field magnitude LECTURE 13 slide 16