Electrostatics: Electrostatic Devices

Transcription

1 Electrostatics: Electrostatic Devices EE331 Electromagnetic Field Theory Outline Laplace s Equation Derivation Meaning Solving Laplace s equation Resistors Capacitors Electrostatics -- Devices Slide 1

2 Derivation of Laplace s Equation Derivation of Poisson s Equation (1 of ) In electrostatics, the field around charges is described by Gauss law D v In LI media, the constitutive relation is D E so Gauss law can be written in terms of E. Ev In electrostatics, we have E V. We can use this put our equation solely in terms of the electric potential. V v Electrostatics -- Devices Slide 4

3 Derivation of Poisson s Equation ( of ) We arrive at Poisson s equation for inhomogeneous media V v V v Poisson s equation for inhomogeneous media If the medium is homogeneous, is a constant and can be brought to the righthand side of the equation. v v V V Poisson s equation for homogeneous media Electrostatics -- Devices Slide 5 Derivation of Laplace s Equation In the absence of charge, v = 0 and Poisson s equation reduces to Laplace s equation. V v V 0 Laplace s equation for inhomogeneous media v V V 0 is called the Laplacian Laplace s equation for homogeneous media Electrostatics -- Devices Slide 6 3

4 No Charge in Electrostatics? No charge in the region between the plates. V 0 Electrostatics -- Devices Slide 7 Notes Poisson s and Laplace s equations describe how electric potential varies throughout a volume. These are scalar differential equations and usually easier to solve than vector differential equations. Use Poisson s equation when there is charge and Laplace s equation when there is not. Laplace s equation is particularly important in electrostatics because it can be used to calculate electric potential around conductors maintained at different voltages. Uniqueness theorem states that there exists only one solution. Electrostatics -- Devices Slide 8 4

5 Meaning of Laplace s Equation Meaning of Laplace s Equation Laplace s equation is u 0 is a 3D second-order derivative. A second-order derivative quantifies curvature. But, we set the second-order derivative to zero. Functions satisfying Laplace s equation vary linearly. Electrostatics -- Devices 10 5

6 Problem Setup Suppose we know the value of V(x,y) at some points in space. What does the function look like at every other point? Figure it out by solving Laplace s equation. V x, y 0 Electrostatics -- Devices 11 Solution of Laplace s Equation Laplace s equation is sort of a number filler inner. Laplace s equation fills in the numbers so they vary linearly between known regions. Electrostatics -- Devices 1 6

7 Another Example Electrostatics -- Devices 13 Solving Laplace s Equation 7

8 Recipe for Solving Laplace s Equation Laplace s equation is solved as a boundary value problem (i.e. partial differential equation plus boundary conditions). 1. Choose a coordinate system that will simplify the math.. Solve Laplace s equation V = 0 in each homogeneous region. a. When V is a function of only one variable, use direct integration. b. Otherwise, use separation of variables. 3. Apply the boundary conditions at the edges of the homogeneous regions. 4. Calculate E from V using E V. 5. Calculate D from E using D E. Electrostatics -- Devices Slide 15 Suppose we have some medium with permittivity and thickness d. Electrostatics -- Devices Slide 16 8

9 Suppose we have some medium with permittivity and thickness d. Then we apply a voltage V 0. Electrostatics -- Devices Slide 17 Suppose we have some medium with permittivity and thickness d. Then we apply a voltage V 0 which puts charge on the plates. Electrostatics -- Devices Slide 18 9

10 Suppose we have some medium with permittivity and thickness d. Then we apply a voltage V 0 which puts charge on the plates. Calculate the electric potential and electric field between the plates. Electrostatics -- Devices Slide 19 Step 1 Choose a coordinate system. Cartesian Electrostatics -- Devices Slide 0 10

11 Step Solve Laplace s equation V 0 V 0 0 V d V 0 If we assume the device is uniform in the x and y directions, Laplace s equation reduces to V V V x y z 0 dv 0 dz Electrostatics -- Devices Slide 1 Step Solve Laplace s equation We integrate to get dv 0 dz V z azb Electrostatics -- Devices Slide 11

12 Step 3 Apply boundary conditions. First boundary condition V 0 0 V 0 a0 bb0 Electrostatics -- Devices Slide 3 Step 3 Apply boundary conditions. Second boundary condition 0 0 V d V V d a d V a 0 0 V d Electrostatics -- Devices Slide 4 1

13 Step 3 Apply boundary conditions. Altogether, the solution is V0 V z z 0 z d d Electrostatics -- Devices Slide 5 Step 4 Calculate E from V. The electric field intensity is d d V0 V 0 V E V E 0 ˆ z V Ez z E az dz dz d d d Observe that E does not depend on. Electrostatics -- Devices Slide 6 13

14 Step 5 Calculate D from E. Applying the constitutive relation, we get the electric flux density V 0 V0 D E D aˆ ˆ z D a d d z Electrostatics -- Devices Slide 7 Resistors 14

15 What is a Resistor? A resistor is a passive two-terminal electrical component that limits the conductivity so as to limit current flow. Electrostatics -- Devices Slide 9 Analysis Setup I + - V S J R V I? Electrostatics -- Devices Slide 30 15

16 Derivation of Resistance for Uniform Conductivity Ohm s Law J E Electric Field Intensity V E R S S R V I S Electric Current Density I V J E S I S V Electrostatics -- Devices Slide 31 Derivation of Resistance for Nonuniform Conductivity Now we must use electromagnetic analysis to derive V and I. Voltage across conductor V Ed E d V R I E ds S Current through conductor I J ds E ds S S Electrostatics -- Devices Slide 3 16

17 Recipe for Analyzing Resistors 1. Choose a convenient coordinate system.. Assume V 0 as the potential difference across the terminals of the conductor. 3. Calculate electric potential V by solving Laplace s equation V = Calculate E using E V. 5. Calculate I from I Eds. 6. Calculate R using R = V 0 /I. Note: The final equation for R should not contain V 0 or I. Use this as a self-check. S Electrostatics -- Devices Slide 33 The Parallel Plate Resistor S surface area R d S Electrostatics -- Devices Slide 34 17

18 Capacitors What is a Capacitor? A capacitor is a passive two-terminal electrical component that can store and release electric energy. It supplies current so as to keep the voltage across its terminals constant. Electrostatics -- Devices Slide 36 18

19 Capacitance, C Capacitance is defined as the magnitude of the charge on one of the plates to the potential difference between the two plates. We do not care about the signs to calculate capacitance. Q C Q V 0 Q Electrostatics -- Devices Slide 37 Recipe for Analyzing Capacitors 1. Choose a convenient coordinate system.. Let the plates carry charges +Q and -Q. 3. Calculate D using Gauss law. 4. Calculate E using E D. 5. Calculate V 0 using V Ed. 6. Calculate C using C Q V 0. 0 L Note: The final equation for C should not contain Q or V 0. Use this as a self-check. Electrostatics -- Devices Slide 38 19

20 Some Simple Capacitors C S d C L a ln b Electrostatics -- Devices Slide 39 0