Electric field tutorials

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.