Mathematical Methods for Physics and Engineering - Matematica.NET

11.10 EXERCISES11.24 Prove equation (11.22) and, by taking b = zx 2 i + zy 2 j +(x 2 − y 2 )k, show that thetwo integrals∫∫I = x 2 dV and J = cos 2 θ sin 3 θ cos 2 φdθdφ,both taken over the unit sphere, must have the same value. Evaluate both directlyto show that the common value is 4π/15.11.25 In a uniform conducting medium with unit relative permittivity, charge density ρ,current density J, electric field E and magnetic field B, Maxwell’s electromagneticequations take the form (with µ 0 ɛ 0 = c −2 )(i) ∇ · B = 0, (ii) ∇ · E = ρ/ɛ 0 ,(iii) ∇×E + Ḃ = 0, (iv) ∇×B − (Ė/c2 )=µ 0 J.The density of stored energy in the medium is given by 1 (ɛ 2 0E 2 + µ −10 B2 ). Showthat the rate of change of the total stored energy in a volume V is equal to∫− J · E dV − 1 ∮(E × B) · dS,µ 0Vwhere S is the surface bounding V .[ The first integral gives the ohmic heating loss, whilst the second gives theelectromagnetic energy flux out of the bounding surface. The vector µ −10 (E × B)is known as the Poynting vector. ]11.26 A vector field F is defined in cylindrical polar coordinates ρ, θ, z by( )x cos λz y cos λzF = F 0 i + j +(sinλz)k ≡ F 0ρa aa (cos λz)e ρ + F 0 (sin λz)k,where i, j and k are the unit vectors along the Cartesian axes and e ρ is the unitvector (x/ρ)i +(y/ρ)j.(a) Calculate, as a surface integral, the flux of F through the closed surfacebounded by the cylinders ρ = a and ρ =2a and the planes z = ±aπ/2.(b) Evaluate the same integral using the divergence theorem.11.27 The vector field F is given byF =(3x 2 yz + y 3 z + xe −x )i +(3xy 2 z + x 3 z + ye x )j +(x 3 y + y 3 x + xy 2 z 2 )k.Calculate (a) directly, and (b) by using Stokes’ theorem the value of the lineintegral ∫ F · dr, whereL is the (three-dimensional) closed contour OABCDEOLdefined by the successive vertices (0, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 1), (1, 1, 0), (0, 1, 0),(0, 0, 0).11.28 A vector force field F is defined in Cartesian coordinates by[( y3F = F 03a + y ) ( xy23 a exy/a2 +1 i + + x + y )e xy/a2 j + z ]a 3 aa exy/a2 k .Use Stokes’ theorem to calculate∮F · dr,Lwhere L is the perimeter of the rectangle ABCD given by A =(0, 1, 0), B =(1, 1, 0),C =(1, 3, 0) and D =(0, 3, 0).S413