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Dr. Hugh Blanton ENTC 3331

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Electrostatics in Media

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Dr. Blanton - ENTC 3331 - Energy & Potential 3 A medium (air, water, copper, sapphire, etc.) is characterized by its relative permittivity, ( r ). Medium rr vacuum1 air1.0006 conductors 11 glass4.5 - 10

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Dr. Blanton - ENTC 3331 - Energy & Potential 4 Media can be grouped in two classes: conductorsdielectrics (insulators, semiconductors, etc.) free chargesno free charges charges will move until the conductor is field free charges in the material are polarized by external fields everywhere (this assumes we are dealing with an electrostatic problem with electric flux density field strength polarization field

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

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

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

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

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

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Dr. Blanton - ENTC 3331 - Energy & Potential 10 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Dr. Blanton - ENTC 3331 - Energy & Potential 30 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

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

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Dr. Blanton - ENTC 3331 - Energy & Potential 32 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.

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

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Dr. Blanton - ENTC 3331 - Energy & Potential 34 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.

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Dr. Blanton - ENTC 3331 - Energy & Potential 35 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

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

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

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Dr. Blanton - ENTC 3331 - Energy & Potential 38 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

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Dr. Blanton - ENTC 3331 - Energy & Potential 39 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.

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Dr. Blanton - ENTC 3331 - Energy & Potential 40 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 ( ).

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Dr. Blanton - ENTC 3331 - Energy & Potential 41 Potential and Fields can be calculated for a given charge distribution, 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:

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