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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

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