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1 MIT OpenCouseWae /ESD.013J Electomagnetics an Applications, Fall 005 Please use the following citation fomat: Makus Zahn, Eich Ippen, an Davi Staelin, 6.013/ESD.013J Electomagnetics an Applications, Fall 005. (Massachusetts Institute of Technology: MIT OpenCouseWae). (accesse MM DD, YYYY). License: Ceative Commons Attibution- Noncommecial-Shae Alike. Note: Please use the actual ate you accesse this mateial in you citation. Fo moe infomation about citing these mateials o ou Tems of Use, visit:

2 6.013/ESD.013J Electomagnetics an Applications Fall 005 Poblem Set 3 - Solutions Pof. Makus Zahn MIT OpenCouseWae Poblem 3.1 A The iea hee is simila to applying the chain ule in a 1D poblem: ( ) [ ( )] [ ] 1 1 f f (x) = = x f(x) f f(x) x f (x), whee f(x) coespons to. So, by iffeentiating f(x) we get pat of the answe to the eivative of 1/f(x). But, we can just o it iectly: = (x x ) + (y y ) + (z z ) [ ] [ ] [ ] [ ] = ê x + ê y + ê z x y z So, we can apply the tick above by just consieing x, y, an z components sepaately. ( ) x = (x x x ) + (y y ) + (z z ) x x = (x x ) + (y y ) + (z z ) Similaly: We have x x = y y y = z z z = = (x x ) + (y y ) + (z z ), so: ( ) 1 [(x x ) ê x + (y y ) ê y + (z z ) ê z ] = [(x x ) + (y y ) + (z z ) ] 3/ The enominatos ae clealy 3, thus ( ) 1 ( ) 1 ( ) = = 3 ê = 1

3 Poblem Set , Fall 005 B This follows fom pat A immeiately by substitution. Remembe is eivatives in tems of the unpime cooinates x, y, an z; oes not opeate on x, y, o z. C ρ( ) V Φ() = = λ 0 a φ V 4πε 0 4πε 0 (a + z ) 1/ whee we consie the infinitesimal chages q = (a φ)λ 0 aoun the ing. y φ a x aφ Figue 1: Diagam fo Poblem 3.1 Pat C. Diffeential length aφ in a cicula hoop of line chage. (Image by MIT OpenCouseWae.) We only cae about the z-axis in the poblem, so, by symmety, thee is no fiel in the x an y iections. π λ 0 (a φ) Φ() =, 4πε 0 (a + z ) 1/ 0 whee (a + z ) 1/ is the istance fom the chage λ 0 a φ to the point z on the z-axis. λ 0 a Φ() = ε 0 (a + z ) 1/ on the z-axis Check the limit as z λ 0 a q Φ(z ) = ε0 z = 4πε 0 z (same fom as point chage whee q = λ 0 πa) Now, 0 0 ( ) Φ Φ Φ λ E = Φ() = 0 a (ê x + êy + êz x y z ) = ê z z ε 0 (a + z ) 1/ aλ 0 z E = ê z ε 0 (a + z ) 3/ Again, we check the limit as z : { λ ê 0 a { q z ; z > 0 ê z ; z > 0 = ε 0 z 4πε = 0 z E(z ) λ 0 a q (same fom as point chage) ê z ε 0 z ; z < 0 ê z 4πε 0 z ; z < 0

4 Poblem Set , Fall 005 D Fom pat C λ 0 Φ = ε 0 ( + z ) 1/ fo a ing of aius. But now we have σ 0, not λ 0. How o we expess λ 0 in tems of σ 0? a Figue : Diagam fo Poblem 3.1 Pat D. Fining the scala electic potential an electic fiel of a chage cicula isk by aing up contibutions fom chage hoops of iffeential aial thickness. (Image by MIT OpenCouseWae.) Take a ing of with in the isk (see figue). We have Total chage = ()(π)()σ 0 }{{} cicum. total chage Line chage ensity = λ 0 = = σ 0 length So, λ 0 = σ 0 an σ 0 Φ = ε 0 ( + z ) 1/ Integating gives a a σ 0 σ 0 σ 0 [ ] =a Φ = + z total = = 0 ε 0 ( + z ) 1/ ε 0 0 ( + z ) 1/ ε 0 =0 σ 0 [ ] = a ε + z z 0 [ ] σ 0 z 1 1 E = Φ total = ε0 z ê z a + z As a, z in a + z can be neglecte, so: σ Φ total (a ) = ε 0 (z a) } 0 σ z > 0, just like sheet chage E(a ) = Φ = ê z ε 0 0 3

5 Poblem Set , Fall 005 Poblem 3. A z + (x,y,z) +q -q - x Figue 3: Diagam fo Poblem 3. Pat A. (Image by MIT OpenCouseWae.) We can simply a the potential contibutions of each point chage: q q Φ =, 4πε 0 + 4πε0 ( ) + = x + y + z ( ) = x + y + z + q 1 1 Φ = 4πε 0 ( ) ( ) x + y + z x + y + z + B +q -q z + - x θ z + - a= cosθ x Figue 4: Diagams fo Poblem 3.1 Pat B. (Image by MIT OpenCouseWae.) p = q, whee p is the ipole moment. We must make some appoximations. As, +,, an 4

6 Poblem Set , Fall 005 become nealy paallel. Thus: + a = cos θ ( ) + 1 cos θ. Similaly, ( ) 1 + cos θ By pat A, [ ] q 1 1 Φ =. 4πε0 + If x 1, then 1/(1 + x) 1 x. In aition, cos θ 1, so ( ) cos θ + 1 cos θ ( ) cos θ 1 + cos θ = cos θ = cos θ + q cos θ p cosθ Φ = 4πε0 4πε 0 C Φ 1 Φ 1 Φ E = Φ = ê θ ê θ sin θ φ ê φ Φ p cosθ Φ p sin θ Φ = =, πε0 3, θ 4πε0 = 0 φ p cosθ 1 p sin θ E = ê + ê θ πε 0 3 4πε 0 p E = [ cosθ ê + sin θ ê θ ] 4πε 0 3 D 1 E cos θ = = = cotθ θ E θ sin θ 1 1 = cotθ θ = = cotθ θ ln = ln(sinθ) + k = = 0 sin θ (when θ = π/, = 0 ) 0 = sin θ 5

7 Poblem Set , Fall 005 Figue 5: The potential at any point P ue to the electic ipole is equal to the sum of potentials of each chage alone. The equi-potential (ashe) an fiel lines (soli) fo a point electic ipole calibate fo 4πε 0 /p = 100. In[1]:= <<Gaphics Gaphics In[]:= [o_,theta_]:= o*sin[theta]^ In[3]:= theta = Pi/ - theta In[4]:= eplot = PolaPlot[[.5, theta], [.5, theta], [1, theta], [, theta] {theta, 0, *Pi}, PlotRange -> All] 6

8 Poblem Set , Fall 005 Out[4]= θ =.5 0 = 1 0 = = 0.5 E Fiel Lines Figue 6: Mathematica Plot 1 Electic fiel lines (Image by MIT OpenCouseWae.) In[5]:= p[phi_,theta_]:= Sqt[Abs[Cos[theta]/(100*Phi)]] In[6]:= pplot = PolaPlot[{p[0.005, theta], p[.01, theta], p[.04, theta], p[.16, theta], p[.64, theta], p[.56, theta], p[10.4, theta], p[40.96, theta]}, {theta, -Pi, Pi}, PlotRange -> All] Out[6]= Φ =.005 Equipotential Lines 1 Φ =.01 Φ = 0.04 Φ = Φ = Figue 7: Mathematica Plot Equipotential lines (Image by MIT OpenCouseWae.) In[7]:= tplot = Show[eplot, pplot] 7

9 Poblem Set , Fall 005 Out[7]= Figue 8: Mathematica Plot 3 Electic fiel an equipotential Lines (Image by MIT OpenCouseWae.) Poblem 3.3 A The bi acquies the same potential as the line, hence has chages inuce on it an conseves chage when it flies away. B The fiels ae those of a chage Q at y = h, x = Ut an an image at y = h an x = Ut. C The potential is the sum of that ue to Q an its image Q. [ ] Q 1 1 Φ = 4πε 0 (x Ut) + (y h) + z (x Ut) + (y + h) + z D Fom this potential { } Φ Q y h y + h E y = =. y 4πε0 [(x Ut) + (y h) + z ] 3/ [(x Ut) + (y h) + z ] 3/ 8

10 Poblem Set , Fall 005 Thus, the suface chage ensity is [ ] Qε 0 h h σ 0 = ε 0 E y y=0 = 4πε z ] 3/ z ] 3/ 0 [(x Ut) + h + [(x Ut) + h + Qh = π[(x Ut) + h + z ] 3/ E The net chage q on the electoe at any given instant is w l Qh xz q =. π[(x Ut) + h + z ] 3/ If w h, z=0 x=0 l Qhw x q =. π[(x Ut) + h ] 3/ x=0 Fo the emaining integation, x = (x Ut), x = x, an l Ut Qhw x q =. π[x + h ] 3/ Ut Thus, [ ] Qw l Ut Ut q = πh (l Ut) + h +. (Ut) + h The ashe cuves (1) an () in the figue 9(a) below ae the fist an secon tems in the above equation. They sum to give (3). q (1) () l (3) Ut v l/u t (a) (b) Figue 9: Cuves fo Poblem 3.3 Pat E. The net chage (a) an voltage (b) as a function of time on the electoe in the y = 0 plane. (Image by MIT OpenCouseWae.) F The cuent follows fom the expession fo q as [ q Qw Uh Uh ] i = = t πh [(l Ut) + h ] + 3/ [(Ut) + h ] 3/ an so the voltage is then V = ir = R q/t. A sketch is shown in figue 9(b) above. 9

11 Poblem Set , Fall 005 Poblem 3.4 Figue 10: Diagam fo Poblem 3.4. The image cuent fom a line cuent Iê z a istance above a pefect conucto. (Image by MIT OpenCouseWae.) A By the metho of images, the image cuent is locate at (0, ) with the cuent I in the opposite iection of the souce cuent. Fo a single line cuent I at the oigin, the magnetic fiel is C I I H = ê φ = π π(x + y ) ( y ê x + x ê y ). Use the supeposition fo a cuent I in the +z iection at y = so that y is eplace by y an fo the cuent I in the z iection at y = so that y is eplace by y +. Then B I I H total = π(x + (y ) ) ( (y ) ê x + x ê y ) π(x + (y + ) ) ( (y + ) ê x + x ê y ) The suface cuent at the y = 0 suface is I K z = H x y=0 + = K = π(x + ) ê z The total cuent flowing on the y = 0 suface is I ê z 1 I ê z 1 ( x ) 1 I total = ê z K z x = x = tan = I ˆe z. π (x + ) π D The foce pe unit length on the cuent I at y = comes fom the image cuent at y = µ 0 I F = (I ê z ) (µ 0 H(x = 0, y = )) = ê y. 4π 10

Note: Please use the actual date you accessed this material in your citation.

Note: Please use the actual date you accessed this material in your citation. MIT OpenCouseWae http://ocw.mit.edu 6.641 Electomagnetic Fields, Foces, and Motion, Sping 5 Please use the following citation fomat: Makus Zahn, 6.641 Electomagnetic Fields, Foces, and Motion, Sping 5.