Laplace's Equation & Electrostatics: Poisson's Equation, Uniqueness, Images, Polynomials, Study notes of Physics

Download Laplace's Equation & Electrostatics: Poisson's Equation, Uniqueness, Images, Polynomials and more Study notes Physics in PDF only on Docsity! Chapter 3 concept questions LAPLACE’S EQUATION AND UNIQUENESS Poisson’s equation tells us that If the charge density throughout some volume is zero, what else must be true throughout that volume: ! "2V = # $ % 0 A) V=0 B) E=0 C) Both V and E must be zero D) None of the above is necessarily true A region of space contains no charges. What can I say about V in the interior?3.1 A) Not much, there are lots of possibilities for V(r) in there B) V(r)=0 everywhere in the interior. C) V(r)=constant everywhere in the interior ρ=0 throughout this interior region What must be true for you to know that you’ve found the potential in a region? a) It satisfies b) It satisfies the boundary conditions c) a and b! "2V = # $ % 0 3.6 METHOD OF IMAGES A) Simple Coulomb’s law: B) Something more complicated A point charge +Q sits above a very large grounded conducting slab. What is E(r) for points above the slab? z x y +Q d r E( r r ) = Q 4!" 0 r # #3 with r # = ( r r $ d ẑ) 3.7 r A) 0 B) C) D) Something more complicated A point charge +Q sits above a very large grounded conducting slab. What's the electric force on +Q? +Q Q 2 4!" 0 d 2 downwards Q 2 4!" 0 (2d) 2 downwards 3.8 z x y d A) B) Something else. A point charge +Q sits above a very large grounded conducting slab. What's the energy of this system? +Q 3.8b z x y d ! "Q 2 4#$ 0 (2d) 3.9 Two ∞ grounded conducting slabs meet at right angles. How many image charges are needed to solve for V(r)? +Q A) one B) two C) three D) more than three E) Method of images won't work here r where C1+C2 = 0. Which coordinate should be assigned to the negative constant (and thus the sinusoidal solutions)? ! 1 Y d2Y dy 2 =C 2 ! 1 X d 2 X dx 2 = C 1 A) x B) y C) C1= C2=0 here D) It doesn’t matter 3.11 b Given the two diff. eq's: A) A=0 (pure cosh) B) B=0 (pure sinh) C) Neither: you should rewrite this in terms of A ekx + B e-kx ! D) Other/not sure? 3.11 h The X(x) equation in this problem involves the "positive constant" solutions: A sinh(kx) + B cosh(kx) What do the boundary conditions say about A and B? where C1+C2 = 0. Which coordinate should be assigned to the negative constant (and thus the sinusoidal solutions)? ! 1 Y d2Y dy 2 =C 2 ! 1 X d 2 X dx 2 = C 1 A) x B) y C) C1= C2=0 here D) It doesn’t matter 3.11 c Given the two diff. eq's: What is the value of ? A) Zero B) π C) 2π D) π/2 E) Something else/how could I possibly know this? ! sin(2x)sin(3x)dx 0 2" # 3.12 /0 1,3,5... 4 1 ( , ) sin( / )n x a n V V x y e n y a n ! ! ! " # = = $ 2 troughs (∞ in z, i.e. out of page) have grounded sidewalls. The base of each is held at V0. 3.14 The Rodrigues formula for generating the Legendre Polynomials is If the Legendre polynomials are orthogonal, are the leading coefficients necessary to maintain orthogonality? A) Yes, fm(x) must be properly scaled for it to be orthogonal to fn(x). B) No, the constants will only rescale the integral 21( ) ( 1) 2 ! l l l l dy P x x l dx ! " = #$ % & ' 1 2 ! l l 3.16 A) Pm(θ) B) Pm(cosθ) C) Pm(θ) sinθ D) Pm(cosθ) sinθ E) something entirely different Orthogonality Given 3.17 (The Pl's are Legendre polynomials.) If we want to isolate/determine the coefficients Cl in that series, first multiply both sides by:! V (") = C l P l (cos") l= 0 # $ Orthogonality ! P l (cos") 0 # $ Pm (cos")sin"d" = 2 2l +1 if l = m 0 if l %m & ' ( ) ( ! P l (x) "1 1 # Pm (x)dx = 2 2l +1 if l = m 0 if l $m % & ' ( ' Suppose V on a spherical shell is constant, i.e. V(R, θ) = V0. Which terms do you expect to appear when finding V(outside) ? A) Many Al terms (but no Bl's) B) Many Bl terms (but no Al's) C) Just A0 D) Just B0 E) Something else!! ! V (r,") = A l r l + B l r l+1 # $ % & ' ( Pl l= 0 ) * (cos") 3.18 Suppose V on a spherical shell is constant, i.e. V(R, θ) = V0. Which terms do you expect to appear when finding V(inside) ? A) Many Al terms (but no Bl's) B) Many Bl terms (but no Al's) C) Just A0 D) Just B0 E) Something else! ! V (r,") = A l r l + B l r l+1 # $ % & ' ( Pl l= 0 ) * (cos") 3.18 b Can you write the function as a sum of Legendre Polynomials? ! P0(cos") =1, P1(cos") = cos" P2(cos") = 3 2 cos 2" # 1 2 , P3(cos") = 5 2 cos 3" # 3 2 cos" ! V 0 (1+ cos 2") A)No, it cannot be done B) It would require an infinite sum of terms C) It would only involve P2 D) It would involve all three of P0, P1 AND P2 E) Something else/none of the above 3.19a ! V 0 (1+ cos 2") = ??? C l P l l= 0 # $ (cos") Does the previous answer change at all if you’re asked for V outside the sphere? a) yes b) no 3.21 b Since the electric field is zero inside this conducting sphere, and V , is V=0 inside as well? a) Yes b) No ! = " r E # d r l$ ! " 0 MULTIPOLE EXPANSION Which of the following is correct (and "coordinate free")? A) B) C) D) E) None of these A small dipole (dipole moment p=qd) points in the z direction. We have derived ! V ( v r ) " 1 4#$ 0 qd z r 3 ! V ( v r ) = 1 4"# 0 v p $ ˆ r r 2 ! V ( v r ) = 1 4"# 0 v p $ r r r 2 ! V ( v r ) = 1 4"# 0 v p $ ˆ r r 2 3.22 a ! V ( v r ) = 1 4"# 0 v p $ ˆ r r 3 An ideal dipole (tiny dipole moment p=qd) points in the z direction. We have derived ! r E ( v r ) = p 4"#0r 3 2cos$ ˆ r + sin$ r $ ( ) Sketch this E field... (What would change if the dipole separation d was not so tiny?) 3.22 b You have a physical dipole, +q and -q a finite distance d apart. When can you use the expression: A) This is an exact expression everywhere. B) It's valid for large r C) It's valid for small r D) ? ! V ( v r ) = 1 4"# 0 v p $ ˆ r r 2 3.22 c Which charge distributions below produce a potential which looks like C/r2 when you are far away? E) None of these, or more than one of these! (Note: for any which you did not select, how DO they behave at large r?) 3.26 What is the magnitude of the dipole moment of this charge distribution? A) qd B) 2qd C) 3qd D) 4qd E) It's not determined (To think about: How does V(r) behave as r gets large?) 3.27 What is the magnitude of the dipole moment of this charge distribution? A) qd B) 2qd C) 3qd D) 4qd E) It's not determined (To think about: How does V(r) behave as r gets large?) 3.27 In which situation is the dipole term the leading non-zero contribution to the potential? ! "(r) A) A and C B) B and D C) only E D) A and E E) Some other combo ! +" 0 ! " = k cos(#) 3.28 A) V(r) = V(mono) + V(dip) + higher order terms B) V(r) = V(dip) + higher order terms C) V(r) = V(dip) D) V(r)=only higher order terms than dipole E) No higher terms, V(r)=0 for this one. = +q = - q In terms of the multipole expansion V(r) = V(mono) + V(dip) + V(quad) + … the following charge distribution has the form: 3.29 What is the direction of the dipole moment of the blue sphere? a) b) c) d) e) the dipole moment is zero (or is ill defined) ! " = k sin(#)! " ! y ! z ! r r ! P ! " ! x ! ˆ " ! ˆ " ! ˆ r ! ˆ z 3.30