Dr Hugh Blanton ENTC 3331 Electrostatics in Media

Dr. Hugh Blanton ENTC 3331

Electrostatics in Media

• A medium (air, water, copper, sapphire, etc. ) is characterized by its relative permittivity, (er). er Medium vacuum air 1 1. 0006 1 conductors glass 4. 5 - 10 Dr. Blanton - ENTC 3331 - Energy & Potential 3

• Media can be grouped in two classes: conductors dielectrics (insulators, semiconductors, etc. ) free charges no free charges will move until charges in the material the conductor is field free are polarized by external fields everywhere (this assumes we are dealing with an electrostatic problem with electric flux field polarization density strength field Dr. Blanton - ENTC 3331 - Energy & Potential 4

+ + + + + Orientation of dipoles inside medium Dr. Blanton - ENTC 3331 - Energy & Potential 5

• and are defined to be parallel. • A dielectric with field has positive and negative surface charges on opposites. dielectric Dr. Blanton - ENTC 3331 - Energy & Potential + + 6

• The polarization field is antiparallel to the polarization . • The field inside the medium is smaller than the external field. Dr. Blanton - ENTC 3331 - Energy & Potential 7

Microscopic Reasons for Induced Polarization • Deformation polarization in non-polar materials such as glass: + atom polarized atom Dr. Blanton - ENTC 3331 - Energy & Potential 8

• Orientation polarization in polar materials. O before dipoles line up H H after dipoles line up O H + + H Dr. Blanton - ENTC 3331 - Energy & Potential 9

• Note: • Isotropic implies that the , , and fields are in the same direction. • Anisotropic implies that the , , and fields may have different directions. • We limit the media to those that are linear, isotropic, and homogeneous. • For such media, the polarization field is: Electric susceptibility Dr. Blanton - ENTC 3331 - Energy & Potential 10

• Since • It follows that • Materials with large permittivity also have a large susceptibility! Dr. Blanton - ENTC 3331 - Energy & Potential 11

Boundaries Between Dielectrics dielectric 2 What fields are at the boundary? dielectric 1 Different amounts of surface charge at the boundary. • Maxwell’s equations are of general validity • In particular Dr. Blanton - ENTC 3331 - Energy & Potential 12

• Construct a suitable path, C, about the boundary. d c medium 1 medium 2 a b • and split the field into normal (n) and tangential (t) components. Dr. Blanton - ENTC 3331 - Energy & Potential 13

• Now make Dh smaller and smaller • This implies • and • Which implies Below boundary Above boundary Dr. Blanton - ENTC 3331 - Energy & Potential 14

• Now, make Dl smaller and smaller, but not zero Dr. Blanton - ENTC 3331 - Energy & Potential 15

• Boundary conditions for the tangential components of the fields. • Across the boundary between any media, the tangential component of unchanged • in all cases Dr. Blanton - ENTC 3331 - Energy & Potential 16 is

• However • because Dr. Blanton - ENTC 3331 - Energy & Potential 17

• Now use Gauss’s Law • Construct suitable volume, V medium 1 The only charge inside V is the surface charge on the boundary area DS Dr. Blanton - ENTC 3331 - Energy & Potential medium 2 18

medium 1 medium 2 Let Dh go to zero, Now, make the Gaussian surface smaller and smaller, but not zero Dr. Blanton - ENTC 3331 - Energy & Potential 19

• This implies Dr. Blanton - ENTC 3331 - Energy & Potential 20

• Boundary conditions for the normal component of the fields across the boundary between any two media. • which implies Dr. Blanton - ENTC 3331 - Energy & Potential 21

Application of Boundary Conditions • Given that the x-y plane is a chargefree boundary separating two dielectric media with permittivities e 1 and e 2. • If the electric field in medium 1 is • Find • The electric field in medium 2, and • The angles q 1 and q 2. Dr. Blanton - ENTC 3331 - Energy & Potential 22

z • What are the angles between q 1 and q 2 between and , as well as between and. x-y plane • For any two media: • With no charges (charge free) on the boundary plane Dr. Blanton - ENTC 3331 - Energy & Potential 23

• It follows that: • since the z-component of the field is the normal component of the field. Dr. Blanton - ENTC 3331 - Energy & Potential 24

• The tangential components for are: and • Then and Dr. Blanton - ENTC 3331 - Energy & Potential 25

and Dr. Blanton - ENTC 3331 - Energy & Potential 26

• The relation looks very similar to Snell’s law of Refraction Dr. Blanton - ENTC 3331 - Energy & Potential 27

Dielectric with Conductor Boundary • Very important practically: • Capacitor • Coaxial shielded cable shield • External field cannot penetrate inside the shield. Dr. Blanton - ENTC 3331 - Energy & Potential 28

• Boundary conditions: conductor dielectric • Since a conduct is free field Dr. Blanton - ENTC 3331 - Energy & Potential 29

• Field lines at a conductor surface have no tangential component. • They are always perpendicular to the conductor surface! • In addition • The surface charge on the conductor defines the field in the surrounding dielectric Dr. Blanton - ENTC 3331 - Energy & Potential 30

• Conducting slab + + + conductor + + + + • Bottom surface: • Normal and field are in opposite directions. • Top surface: • Normal and field are in same directions. Dr. Blanton - ENTC 3331 - Energy & Potential 31

• Since the conductor is field-free • And since , the magnitude of the surface charge densities is given by the product of permittivity and field strength. Dr. Blanton - ENTC 3331 - Energy & Potential 32

• Dielectric slab capacitor Parallel plate capacitor • Most general capacitor Conductor 1 + + + + + V Dr. Blanton - ENTC 3331 - Energy & Potential Conductor 2 33

• Because the conductors must have inside, • To achieve this, the charges distribute on the two surfaces. • There are equilibrium currents until everything is stationary. • Very fast—speed of light. Dr. Blanton - ENTC 3331 - Energy & Potential 34

• The surface charges on conductor 1 and conductor 2 give rise to the field with • This implies that the total charge on either conductor is: Definition of surface charge density Boundary conditions (no tangential component Dr. Blanton - ENTC 3331 - Energy & Potential 35

• The potential difference V along any one of the field lines is given by: Dr. Blanton - ENTC 3331 - Energy & Potential 36

• Capacitance is the charge per potential difference. Dr. Blanton - ENTC 3331 - Energy & Potential 37

• The capacitance of a parallel-plate capacitor is • proportional to area A. • inversely proportional to separation, d. • proportional to the permittivity of the dielectric filling. • independent of Dr. Blanton - ENTC 3331 - Energy & Potential 38

Summary of Electrostatics • The sources of the electrostatic field are time-independent charge distributions. • That is, the charge distributions are static (derivative is zero). • Electrostatics follows from the empirical facts of • Coulomb’s law • The principle of linear, vectorial superposition of forces and fields. • Energy conservation. Dr. Blanton - ENTC 3331 - Energy & Potential 39

Summary of Electrostatics • Electrostatics can be based on two fundamental Maxwell equations; • • • The electric field is free from circulation ( ) and can always be expressed as the gradient of a potential ( ). Dr. Blanton - ENTC 3331 - Energy & Potential 40

• Potential and Fields can be calculated for a given charge distribution, r • from the field definition • using Gauss’s Law • using image charges • Conducting and dielectric media can be distinguished. • At boundaries between media, the following conditions hold: • • Dr. Blanton - ENTC 3331 - Energy & Potential 41