1. EMLAB Chapter 4. Potential and energy 1

2. EMLAB 2 Solving procedure for EM problems Known charge distribution Coulomb’s law Known boundary condition Gauss’ law differential form Vector calculation Known charge distribution Integration of Coulomb’s law Scalar calculation Known charge distribution Poisson equation Scalar calculation

3. EMLAB Gravitational field Earth Moon 3

4. EMLAB 4 Gravitational potential Instead of field lines, potential energy levels can imply the direction and magnitude of gravitational force.

5. EMLAB Work in a gravitational field To move an object in the gravitational field, an external force must be applied that compensates the force due to gravity. 5

6. EMLAB Potential energy in a gravitational field The scalar field quantity of potential energy is introduced to represent energy levels inherent to positions in the space. Differences between the energy levels can be obtained from the work that must applied for the object to move from the initial position to the final position. The position that corresponds to zero energy level is one that is located far away from the earth. 6

7. EMLAB 7 Potential energy in an electric field Electric field Electric potential Electric potential (3D)

8. EMLAB Electric potential energy (V) +q +q t As in the gravitational field, a potential energy for the electric field can be introduced. The potential energy in an electric field is defined as the energy levels of the test charge with +1C. The unit of potential energy is “voltage” named after the physicist Volta. The position far away from the source charge has zero potential energy. Electric potential energy defined as the work to move a test charge with (+1C) 8

9. EMLAB Electric potential due to a point charge +q 1 9 If the position A is infinitely distant from the charge q 1, V A approaches zero.

10. EMLAB +q 1 Distribution of electric potentials due to a point charge 10

11. EMLAB +q-q Potential distribution due to a dipole 11

12. EMLAB A positive charge moves from the position of high potential energy to that of lower potential energy. +q 1 Movement of a charge in a potential field +q 1 12

13. EMLAB 13 Structure of a cathode ray tube

14. EMLAB Scalar field V Rectangular coordinate Cylindrical coordinate Spherical coordinate To obtain the potential difference V, we should integrate the electric field from A to B. For every position, the potential V is defined. So the potential is a scalar field quantity. Only the potential difference has physical significance. So voltage reference point should be specified always. 14

15. EMLAB Example 4.1 In this example, a different integration path is used with the same start point A and end point B. Although the integration paths are different, the voltage difference is the same. A field that has this property is called as a conservative field. 15

16. EMLAB Conservative field All electric field in electrostatic problems are conservative field. In conservative field, the voltage difference depends on only the start and the end point. The result of the integral is independent of the path. The condition for a conservative field is that curl of the field should be zero. The electric field in the example 4.1 is conservative. 16

17. EMLAB 17 Conservative field Non-conservative field Circulating electric field is non-conservative.

18. EMLAB Relation between E and V We have learned how to obtain the voltage difference from an electric field. The reverse process is also possible. That is, we can obtain the electric field from the potential distribution. As in the derivation process for divergence operator, the relation between the electric field and the potential can be derived from the integral equation with the integration path infinitesimally small. Compared with electric field calculations which contain vector operations, voltage calculations are easier and simpler as the voltage is scalar quantity. For ease of operation, we calculate first potential functions and then electric field can be derived from potentials. This operation is called “gradient V”. 18

19. EMLAB Gradient operators in other coordinate systems can be derived from the following relation. Cylindrical coordinateSpherical coordinate Gradient operator in other curvilinear coordinates 19

20. EMLAB Properties of gradient operator +q 1 -q 1 Equi-potential surface Because the derivatives of voltage on the equi-potential surfaces are zero, Gradient V is perpendicular to those surfaces. Electric field line is directed from the higher potential region to lower region. Gradient V is directed to higher potential region. Using chain rule, gradient operation becomes simpler. For the function f of argument R 20

21. EMLAB Potential of multiple charge distribution +q 1 +q 2 +q 3 +q n Continuous charge distribution If potential contributions of separate charges are added, the potential of the multiple charges are obtained. If the charge distribution is continuous, the sum (Σ) symbol is replaced with integral (∫) symbol. Line charge:Surface charge: Point charges 21

22. EMLAB 22 Example : potential due to a charged annular disk Find voltage on the z-axis, and electric field using the voltage.

23. EMLAB Electric dipole +q -q (Equi-potential surface) 23

24. EMLAB 24 Poisson’s & Laplace’s equations for homogeneous medium Laplace operator has different forms for different coordinate systems.

25. EMLAB 25 The differential equation for source-free region becomes a Laplace equation. (rectangular coordinate) (cylindrical coordinate) (spherical coordinate) Laplace’s equations

26. EMLAB 26 Example 1 : Laplace eqs. Unlike the procedures in the previous chapters, the potential V is first obtained solving Laplace equation. Then, using the potential, E, D, , Q, C are obtained. If the plates are wide enough to ignore the variation of electric field along x and y directions Using the boundary conditions on the two plates,

27. EMLAB 27 Example 2

28. EMLAB 28 Charge storage QV +V-+V- If charges are accumulated, potential difference increases.

29. EMLAB 29 Capacitor

30. EMLAB 30 Due to potential difference, positive charges rush to the capacitor. As the amount of charges increases, the voltage increases. If the voltage difference between the terminals of the capacitor is equal to the supply voltage, net flow of charges becomes zero. Charging capacitor

31. EMLAB 31

32. EMLAB 32 Potential distribution near parallel plates

33. EMLAB Electrostatic energy +q 1 +q 2 +q 3 The work to assemble charges q 1, q 2 ~ q n. The work for charges q 1, q 2,~,q n to be separated to infinitely distant points. V n is a potential due to N-1 charges other than n-th charge The magnitudes of works to assemble or disassemble are the same. 33 Potential energy of q i due to q j.

34. EMLAB Electrostatic energy If the product V*D becomes zero on the surface S, the surface integral becomes zero. (1) The integral is performed on the surfaces marked by red lines. (2) The integral is performed over the volume marked by blue lines. 34

35. EMLAB 35 Capacitance The magnitude of an electric field is proportional to charges, and voltages are proportional to electric field. Hence, charges are proportional to voltages. This proportionality constant is called capacitance. Example: Capacitance of a parallel plate capacitor

36. EMLAB 36 Capacitance from electrostatic energy Example : parallel plate capacitor