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 39 Problems curl operation i i i V x Vf det a ax a ay az af af af ax ay az L f af ayaz azay a 2f i I2 azax axaz a2f 0 axay ayax 28 Each bracketed term in 28 is zero because the order of differentiation 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 to the surface integral in Stokes theorem of 25 However the divergence 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 shrink to zero giving a zero result for the line integral The divergence theorem applied to the closed surface with vector V X A is then VxA 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 differentiations V VxA a aA aA a aA az y az ax y 2 xa A a2 A xay avax a2 A aA a aA aA ax I az ax ay 2 A a2 A a2 Ax ayaz azay azax ax z where again the order of differentiation does not matter PROBLEMS Section 1 1 1 Find the area of a circle in the xy plane centered at the origin using a rectangular coordinates x 2 y a2 Hint J x dx A x a2 x a 2 sin 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 the origin using a rectangular coordinates x 2 y 2 z 2 R 2 Hint JI I x x dx a sin x a b cylindrical coordinates r z 2 R2 c spherical coordinates r R Which coordinate system is easiest Section 1 2 3 Given the three vectors A 3ix 2i i B 3i 4i 5i C i i i find the following a b c d e f A EB B C A C A B BC AC AxB BxC AxC A x B C A B x C Are they equal Ax B x C B A C C A B Are they equal What is the angle between A and C and between B and AxC 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 component of B parallel to A is B A A Bll A A Hint Bi aA What is a b If the vectors are A i 2i i B 3i 5i 5i what are the components of B parallel and perpendicular to A Problems 41 6 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 vectors A 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 vector A Ai A i Aii the directional cogines are defined as the cosines of the angles between A and each of the Cartesian coordinate axes Find each of these directional cosines and show that Cos 2 a Cos2 Cos 2 y 1 Y 9 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 lengths of A and B and the enclosed angle 0c The result is known as the law of cosines Hint C C B A B A b For the same triangle prove the law of sines sin 0 sin Ob A Hint BxA C A M M B sin 0 C A 42 Review of Vector Analysis 10 a Prove that the dot and cross can be interchanged in the scalar triple product AxB C BxC A CxA B b Show that this product gives the volume of a parallelepiped whose base is defined by the vectors A and B and whose height is given by C c If A i 2i B i 2i C i i verify the identities of a and find the volume of the parallelepiped formed by the vectors d Prove the vector triple product identity A x B x C B A C C A B I A x B CI IA x BI A Volume A x B C B x C A C x A B 11 a Write the vectors A and B using Cartesian coordinates in terms of their angles 0 and 4 from the x axis b Using the results of a derive the trigonometric expansions sin O sin 0 cos d sin 0 cos 0 cos 0 4 cos 0 cos 4 sin 0 sin 4 ProbLms 43 x Section 1 3 12 Find the gradient of each of the following functions where a and b are constants a f axz bx y b f a r sin 4 brz 2 cos 30 c f ar cos 0 b r 2 sin 0 13 Evaluate the line integral of the gradient of the function f r sin 0 over each of the contours shown x Section 1 4 14 Find the divergence of the following vectors a b c d A xi i zi ri A xy 2 i i i A rcos Oi z r sin 0 i A r 2 sin 0 cos 4 i ie ii 15 Using the divergence theorem prove the following integral identities a JVfdV fdS 44 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 vector integrated 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 dS J fV2g gV2f dV f Vg gVf Hint V fVg fV 2g 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 vector A ri xiA yi zi through the surface of a c Verify the divergence theorem by also evaluating the flux as 4 IV AdV 2J 4 b A Section 1 5 18 Find …
View Full Document
Unlocking...