MIT OpenCourseWare http://ocw.mit.edu Electromagnetic Field Theory: A Problem Solving Approach For any use or distribution of this textbook, please cite as follows: Markus Zahn, Electromagnetic Field Theory: A Problem Solving Approach. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-NonCommercial-Share Alike. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Problems 39curl operation:i. i, i,V x (Vf)= det a aax ay azaf af afax ay az.=(•-•L f.+,( af a/2f) +i(.I2-a2f).0ayaz azay azax axaz axay ayax)(28)Each bracketed term in (28) is zero because the order ofdifferentiation does not matter.(b) The Divergence of the Curl of a Vector is Zero[V* (V x A)=0]One might be tempted to apply the divergence theorem tothe surface integral in Stokes' theorem of (25). However, thedivergence theorem requires a closed surface while Stokes'theorem is true in general for an open surface. Stokes'theorem for a closed surface requires the contour L to shrinkto zero giving a zero result for the line integral. The diver-gence theorem applied to the closed surface with vector V X Ais thenVxA dS= •IV> (VxA) dV= O V -(VxA) = 0(29)which proves the identity because the volume is arbitrary.More directly we can perform the required differentiationsV. (VxA)a,aA, aA,\ a aA aA) a aA, aA\-ax\y az / y az ax I az\ ax ay /xa2A, a2A , a2A,2A a2A, a2Ax (-\xay avax \ayaz azay) \azax ax(zwhere again the order of differentiation does not matter.PROBLEMSSection 1-11. Find the area of a circle in the xy plane centered at theorigin using:(a) rectangular coordinates x2+y = a2 (Hint:J -x dx = A[x •a2-x a2sin-'(x/a)l)40 Review of Vector Anaysis(b) cylindrical coordinates r= a.Which coordinate system is easier to use?2. Find the volume of a sphere of radius R centered at theorigin using:(a) rectangular coordinates x2+y2+z2 = R2 (Hint:JI I x dx==[x •-ý + a sin-(x/a)])(b) cylindrical coordinates r + z 2= R2;(c) spherical coordinates r = R.Which coordinate system is easiest?Section 1-23. Given the three vectorsA = 3ix + 2i, -i.B = 3i, -4i, -5i,C= i.-i,+i,find the following:(a) A+EB, B C, A±C(b) A-B, BC, AC(c) AxB, BxC, AxC(d) (A x B) -C, A -(B x C) [Are they equal?](e) Ax (B x C), B(A C)-C(A -B) [Are they equal?](f) What is the angle between A and C and between B andAxC?4. Given the sum and difference between two vectors,A+B= -i. +5i, -4i,A-B = 3i. -i, -2i,find the individual vectors A and B.5. (a) Given two vectors A and B, show that the componentof B parallel to A isB'ABll = AA*A(Hint: Bi = aA. What is a?)(b) If the vectors areA = i. -2i, + i"B 3i, + 5i, -5i,what are the components of B parallel and perpendicular toA?Problems 416. What are the angles between each of the following vectors:A = 4i. -2i, + 2i,B= -6ix + 3i, -3i,C= i. + 3,+i,7. Given the two vectorsA=3i,+4i, and B=7ix-24i,(a) What is their dot product?(b) What is their cross product?(c) What is the angle 0 between the two vectors?8. Given the vectorA = Ai, +A,i, +Aiithe directional cogines are defined as the cosines of the anglesbetween A and each of the Cartesian coordinate axes. Findeach of these directional cosines and show thatCos2a + Cos2 / + Cos2y = 1Y9. A triangle is formed by the three vectors A, B, and C=B-A.(a) Find the length of the vector C in terms of the lengthsof A and B and the enclosed angle 0c. The result is known asthe law of cosines. (Hint: C C = (B -A) (B -A).)(b) For the same triangle, prove the law of sines:sin 0. sin Ob sin 0,A B C(Hint: BxA=(C+A) A.)M M ý ý ý42 Review of Vector Analysis10. (a) Prove that the dot and cross can be interchanged inthe scalar triple product(AxB) .C=(BxC) A= (CxA) B(b) Show that this product gives the volume of a parallele-piped whose base is defined by the vectors A and B and whoseheight is given by C.(c) IfA=i.+2i,, B=-i.+2i,, C=i,+i.verify the identities of (a) and find the volume of the paral-lelepiped formed by the vectors.(d) Prove the vector triple product identityA x (B x C) = B(A- C)- C(A B)I(A x B) -CIIA x BIA Volume = (A x B) C= (B x C) A= (C x A) - B11. (a) Write the vectors A and B using Cartesian coordinatesin terms of their angles 0 and 4 from the x axis.(b) Using the results of (a) derive the trigonometricexpansionssin(O +) = sin 0 cos d +sin 0 cos 0cos (0 + 4) =cos 0 cos 4 -sin 0 sin 4ProbLms 43xSection 1-312. Find the gradient of each of the following functionswhere a and b are constants:(a) f = axz +bx-y(b) f= (a/r)sin 4 +brz2 cos 30(c) f = ar cos 0 + (b/r2) sin 013. Evaluate the line integral of the gradient of the functionf= r sin 0over each of the contours shown.xSection 1-414. Find the divergence of the following vectors:(a) A= xi, + i,+zi, = ri,(b) A= (xy 2)[i. +i, + i](c) A= rcos Oi,+[(z/r) sin 0)]i,(d) A= r2 sin 0 cos 4 [i, +ie +ii15. Using the divergence theorem prove the followingintegral identities:(a) JVfdV= fdS44 Review of Vector Analysis(Hint: Let A = if, where i is any constant unit vector.)(b) tVxFdV= -FxdS(Hint: LetA=ixF.)(c) Using the results of (a) show that the normal vectorintegrated over a surface is zero:dS= 0(d) Verify (c) for the case of a sphere of radius R.(Hint: i, = sin 0 cos Oi, + sin 0 sin Oi, +cos Oi,.16. Using the divergence theorem prove Green's theorem[f Vg-gVf] dS= J[fV2g-gV2f] dV(Hint: V (fVg)= fV2g+ Vf Vg.)17. (a) Find the area element dS (magnitude and diirection)on each of the four surfaces of the pyramidal figure shown.(b) Find the flux of the vectorA = ri,= xiA +yi, +zi,through the surface of (a).(c) Verify the divergence theorem by also evaluating theflux as4 =IV -AdV2J-4bSection 1-518. Find the curl of the following vectors:(a) A= x2yi +2 Yi, +yiAProblems 45z sin 4(b) A = r cos i, +z sinrcos 0 sin .(c) A= r2 sin 0 cos 4i, + 2 r 619. Using Stokes' theorem prove thatfdl= -Vf xdS(Hint: Let A = if, where i is any constant unit vector.)20. Verify Stokes' theorem for the rectangular boundingcontour in the xy plane with a vector fieldA = (x + a)(y + b)(z + c)i.Check the result for (a) a flat rectangular surface in the xyplane, and (b) for the rectangular cylinder.21. Show that the order of differentiation for the mixedsecond derivativeX kay ay kxdoes not matter for the functionx2 I nyy22. Some of the unit vectors in cylindrical and sphericalcoordinates change direction in space and thus, unlikeCartesian unit vectors, are not constant vectors. This meansthat spatial derivatives of
View Full Document
Unlocking...