TUTORIAL 7. Discussion of Quiz 2 Solution of Electrostatics part 1

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1 TUTORIAL 7 Discussion of Quiz 2 Solution of Electrostatics part 1

2 Quiz 2 - Question 1! Postulations of Electrostatics %&''()(*+&,-$'.)/ : % (1)!! E # $$$$$$$$$$ & # (2)!" E # #! Static Electric field is not solenoidal! Static Electric field is irrotational

3 Quiz 2 - Question 1! Electric potential - the line integral of E z " $ E!d l # c # r! The Scalar line integral of the static electric field intensity around any close path vanishes. x I1 I2 I3 y

4 Quiz 2 - Question 2 Gauss s law (int) $(x,y,z) " )# E % 4 (# #(x,y,z) ' % ' &!" " #!! 2 ) % '! 4(# 1 E ds " E % ')!! % & V d q r!(x,y,z)

5 Quiz 2 - Question 2 1. Total charge Q(r) = ### dv # V r "( r ) = rd( 4 3 $r3 ) # 0 r = 4$r 3 dr 0 = $r 4 Q(r) = ### dv # V 1 "( r ) = rd( 4 3 $r3 ) # 0 1 = 4$r 3 dr = $ 0

6 Quiz 2 - Question 2 2. Electric field - direction v E ( v r ) = "V ( v r ) = #V (v r ) #x ˆ x + #V (v r ) #y y ˆ + #V(v r ) #z = dv dr (#r x #x ˆ ) + dv dr (#r y #y ˆ ) + dv dr (#r z #z ˆ ) = dv dr xˆ x + yˆ y + zˆ z r = dv dr ˆ a r ˆ z

7 Quiz 2 - Question 2 2. Electric field - magnitude r ## E " d r s = E 0 (r)ds S ## S = 4$r 2 E 0 (r) 3. Electric potential = Q % 0 V (0) "V (r) = $ r 0 r E # dr

8 Outline for Solution of electrostatics! Poisson s and Laplace s equation! Uniqueness theorem! Method of images! Method of separation of variables (Next tutorial)

9 What are we dealing with? Given: Charge source Background material Q Boundary condition! Question: Find the E and V V Governing equation: " 2 V = # $ %

10 Poisson s equation! Laplacian! Laplacian stands for the divergence of the gradient of " 2 V = ( a r # x #x + a r # y #y + a r # y #y ) $ (r #V a x #x + a r #V y #y + a r #V y #y ) = # 2 V #x + # 2 V 2 #y + # 2 V 2 #y 2! Poisson Equation " 2 V = " # "V! is a second order differential equation holds at every points in space.! Laplace Equation! is the governing equation for problems involving a set of conductors " 2 V = # $ % " 2 V = 0

11 Uniqueness of Electrostatic solutions! A solution of Poisson s equation that satisfies the given boundary conditions is an unique solution.! A solution of an electrostatic problem satisfying its boundary conditions is the only possible solution! Method of image : Field E is uniquely defined by its normal components over the surface which confines this region.

12 Method of images! Replace the original boundary by appropriate image charges in lieu of formal solution of Poisson s or Laplace s equation.! Reminder:! Image charges are virtual charges! Use image charges to set up the boundary conditions.! The images charge should be located outside the region in which the field is to be determined.

13 Boundary condition for one typical case! Laplace s equation in Cartesian coordinate system! Boundary conditions

14 Example 1! Determine the system of image charges that will replace the conduction boundaries that are maintained at zero potential for! a) A point charge Q located between two large, grounded, parallel conducting planes as shown in fig (a) V=0 d d/3 V=0

15 Q1 Q2 Q1 Q1 Q2 Q2 In the first iteration We will have Q1 =-Q at 2/3d up of plane S1 Q2 =-Q at 1/3 below the plane S2 In the second iteration We will have Q1 =+Q at d+1/3d up of plane S1 Q2 =+Q at d+2/3d up of plane S2 Finally, at below of plane S2 ("1) n +1 Q [2n ("1)n ]d and ("1) n +1 Q at [2n + 2 up of plane S1 3 ("1)n ]d

16 Example 1! Determine the system of image charges that will replace the conduction boundaries that are maintained at zero potential for! b) An infinite line charge located midway between two large, intersecting conducting planes forming a 60 degree angle V=0 60 V=0

17 60 When a point charge located between two paralleled conducting planes, the number of image charges goes to infinite. When a point charge located between two intersecting conducting plane forming a " degree if 360/"=n, the number of image charges is n-1.

18 Example 2 Two dielectric media with dielectric constant!1 and!2 are separated by a plane boundary at x=0. A point charge Q exists in medium 1 at distance d from the boundary. (a) Verify that the field in medium 1 can be obtained from Q and an image charge -Q1 both acting in medium 1. (b) Verify that the field in medium 2 can be obtained from Q and an image charge +Q2 coinciding with Q, both acting in medium 2. (c) Determine Q1 and Q2!2!1 -Q1 (Image charge) Q (Point charge) +Q2 (Image charge)

19 Example 2!2!1 -Q1 (Image charge) Q (Point charge) +Q2 (Image charge)

20 Example 2

21 Example 2

22 ! Method of image! Discrete charges Conclusion! Conducting bodies of simple boundary! Method of separation of variables! Potential of conducting bodies is given.! Solve differential equations subject to boundary conditions.

Notes 19 Gradient and Laplacian

ECE 3318 Applied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of ECE Notes 19 Gradient and Laplacian 1 Gradient Φ ( x, y, z) =scalar function Φ Φ Φ grad Φ xˆ + yˆ + zˆ x y z We can