1 ENE 325 Electromagnetic Fields and Waves Lecture 4 Electric potential, Gradient, Current and Conductor, and Ohm’s law

2 Review Gauss’s law is another approach to evaluate the electric field and is proper for highly symmetrical configuration. Divergence is defined by Point form of Gauss’s law:

3 Outline Electric potential Electric potential Gradient Gradient Current and Conductor Current and Conductor Ohm’s law Ohm’s law

4 Electric work A work done to move one charge from one to point to another is defined by A work done by the field in moving the charge from point a to point b is A work done by an external force against the field is

5 Ex1 Calculate work required to move a 5 nC charge from the origin to point P (1, 1, 0) against the static field given V/m.

6 Electric potential (1) The electric potential difference V ba is a work done by an external force to move a charge from point a to point b in an electric field divided by the amount of charge moved. V a and V b are the absolute potentials measured with respect to the reference potential at ground plane.

7 Electric potential (2) If then the electric potential is independent of the chosen path. If a closed path is chosen and no work is done. An absolute potential at some finite radius from a point charge fixed at the origin is

8 Electric potential (3) The electric potential resulted from N charges is found by adding the potential for each charge. If a collection of charges become a continuous distribution, the total potential is then

9 Ex2 Let V/m and  L is 100 nC/m a) a) Find the work done in moving a 10 nC from  = 3 m to  = 1 m. b) b) Determine the potential difference V ba.

10 Ex3 Find a work done in moving a charge Q = 5  C from the origin to point P (2,  /4,  /2) in spherical coordinates by giving V/m. Note: line different element.

11 Gradient A plot of the electrostatic potential superimposed over the field lines for a point charge. The electric field can be found by finding the maximum rate and direction of spatial change of the potential field.

12 A gradient equation For a Cartesian coordinate system: For a cylindrical coordinate system: For a spherical coordinate system:

13 Introduction to electromagnetic material The properties of electromagnetic material is specified by , , . Homogeneous material is the material that possesses the same properties at every point in the material. Isotropic material is the material that its properties are independent of direction.

14 Conductor The material that electrons move freely. The conductivity, , depends on charge density and scattering of electrons by their interactions with crystal lattice.  decreases with increasing temperature. Perfect conductor The conductor that has an infinite conductivity.

15 Current and current density Current, current density, resistance, and capacitance can be explained using electromagnetics. Current, I is defined as the amount of charge that passes through a reference plane in a given amount of time. Amperes (A) Current density, J is defined as the amount of current per unit area A/m2 total current can be expressed as

16 Current density and volume charge density Consider At time  t, charges move for a distance  x crossing a reference plane that is normal to the direction of charge movement. Since then where n x is a charge velocity (m/s) or The movement of charge creates “Convection current”.

17 Continuity of current The principle of conservation of charge “charges can be neither created nor destroyed, although equal amounts of positive and negative charge may be simultaneously created, obtained by separation, destroyed, or lost by recombination.” The integral form of the continuity equation, indicates an outward-flowing current where Q i is the charge inside the closed surface. We can show its point form as

18 Conduction current arises from free electrons in a conductor. Electrons in valence band have high enough energy to get into the conduction band. Conduction current

19 Drift velocity where  e is the mobility of the electron in the given material (m 2 /V-s)  aluminum = m 2 /V-s  copper = m 2 /V-s  silver = m 2 /V-s Then where  e is the free-electron charge density, a negative value.

20 where. The conductivity  is measured in siemens per meter (S/m).  aluminum = 3.82x10 7 S/m  copper = 5.8x10 7 S/m  silver = 6.17x10 7 S/m The conductivity depends on the temperature. The point form of Ohm’s law

21 The application of Ohm’s law Assume and are uniform,  we can write Ohm’s law as where is the resistance with the measured unit of Ohm (  ). General form:

22 a)  d = 1 mm/s b) J = 10 7 A/m 2 Ex4 Determine the magnitude of in silver when

23 c) A current of 80 A flows through a silver dice of width = 3 mm and length = 3 mm. d) A same silver dice with a 0.5 mV drop across a top and a bottom face.

24 Ex5 An aluminum rod with the length of 1000 feet has a cross section with the diameter of 0.8 inch. There is 1.2 V potential drop across both ends, determine a) J b) current

25 c) power dissipated in the rod