Tutorial no. 3(1)
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This document contains 13 tutorial questions about electrostatics concepts like divergence, electric field, electric flux density, and electric potential. The questions involve calculating values for divergence, volume charge density, work done by electric fields, electric potential, and electric field in various configurations of point charges and dipoles. The answers provided calculate or evaluate the requested quantities for each question.
Tutorial no. 3(1)
- 1. Tutorial No: 2 1) Find a numerical value for the divergence of D at the point indicated if (a) D = 20xy2 (z+1) âx + 20x2 y(z+1) ây + 10x2 y2 âz C/m2 at PA(0.3,0.4,0.5); (b) D = 4ρz sin aρ + 2ρzcos a + 2ρ2 sin âz C/m2 at PB(1,/2,2); (c) D = sin cos ar + cos cos a - sin a C/m2 at PC(2, =/3, =/6). Ans: - (a) 7.5 C/m2 ; (b) 12 C/m3 ; (c) 0. 2) Find the volume charge density that is associated with each of the following fields: (a) D=xy2 âx + yx2 ây + z âz C/m2 ; (b) D = ρz2 sin2 aρ + ρz2 sin cos a + ρ2 z sin2 âz C/m2 ; (c) ar C/m2 Ans:- (a)x2 +y2 +1 C/m3 ; (b)z2 +ρ2 + sin 2 C/m3 3) Given the flux density D=(2 cos /r3 ) ar+ (sin /r3 ) a C/m2 ,evaluate both sides of the divergence theorem for the region defined by 1<r<2, 0<</2, 0< < /2. Ans:- (a) 0; (b) 0. 4) An electric field is given as E=6y2 z âx +12xyz ây +6xy2 âz V/m. An incremental path is represented by L=-3âx+5ây-2âz µm. Find the work done in moving a 2-µC charge along this path if the location of the path is at: (a) P(0,2,5); (b) PB (1,1,1); (c) PC (-0.7, -2, -0.3) Ans:- (a) 720 pJ; (b) -60 pJ; (c) -60 pJ 5) Find the work done in moving a 5-µC charge from the origin to P (2, -1, 4) through the field E = 2xyz âx +x2 z ây +x2 y âz V/m via the path: (a) straight line segments: (0,0,0) to (2,0,0) to (2,-1,0) to (2,-1,4); (b) straight line: x=-2y, z=2x; (c) curve: x= -2y3 , z=4y2 Ans:- (a) 80 µJ; (b) 80 µJ; (c) 80µJ 6) Let E = x ây V/m at a certain instant of time, and calculate the work required to move a 3-C point charge from (1, 3, 5) to (2, 0, 3) along the straight – line segments joining (a) (1, 3, 5 ) to (2, 3, 5) to (2, 0, 5) to (2, 0, 3 ): (b) (1, 3, 5) to (1, 3, 3) to (1, 0, 3 ) to (2, 0, 3) Ans:- (a) 18 J; (b) 9 J 7) Let E= (-6y/x2 ) âx +(6/x) ây +5 âz V/m and calculate : (a) VPQ given P (-7, 2, 1) and Q (4, 1, 2); (b)Vp if V=0 at Q; (c) vp if V=0 at (2, 0, -1) Ans:- (a) 8.21 V; (b) 8.21 V; (c) -8.29 V 8) Assume a zero reference at infinity ,and find the potential at P (0, 0, 10) that is caused by this charge configuration in free space: ( a) 20 nC at the origin ; (b) 10 nC/m along the line x=0, z=0, -1< y< 1; (c) 10 nC/m along the line x=0, y=0, -1 < z < 1 Ans:- (a) 17.98 V; (b) 17.95 V; (c) 18.04 V 9) If V = (60 sin)/r2 V in free space and point P is located at r = 3m, = 60· = 25·, find: (a) Vp; (b)Ep; (c) dV/dN at P ; (d) aN at P; (e) ρV at P Ans:- (a) 5.77 V; (b) 3.58 ar – 1.111a V/m; (c) 4.01 V/m; (d) -0.961 ar + 0.277 a ; (e) - 7.57pC/m3 . 10) A dipole of moment p = -4âx +5ây +3âz nC/m is located at D (1, 2, -1) in free space. Find V at: (a) P ( 0, 0, 0); (b) PB (1, 2, 0); (c) PC (1, 2, -2); (d) PD (2, 6, 1) Ans:- (a) 5.77 V; (b) 3.85 ar -1.111 a V/m; (d) -0.961, ar + 0.277 a ; (c) -7.57 pC/m3 11) Point charges of +3 µC are located at ( 0, 0, 1 mm) and (0, 0, -1 mm) , respectively, in free space. (a) Find p. (b) Find E in spherical components at P ( r=2, = 40, =50 ) (c) Find E in spherical components at (1, 2, 1.5 ) Ans:- (a) 6âz nC.m; (b) 10.33 ar + 4.33 a V/m; (c) 3.08 ar +2.29 a V/m 12) Find the energy stored in free space for the region , 0< p< a, 0< <, 0 < z < 2, given the potential field V = : (a) V0ρ/a; (b) v0 (p/a) cos2 Ans :- (a) 1.571 Є0 V0 2 ; (b) 1.374 Є0 V0 2 13) A point charge of 6nC is located at the origin in free space. Find Vp if point P is located at P(0.2,- 0.4,0.4) and : (a) V = 0 at infinity; (b) V = 0 at (1,0,0); (c) V= 20 V at (-0.5,1,-1). Ans:- (a) 89.9 V; (b) 36.0 V (c) 73.9 V.