Electric fields tutorials

1. Tutorial No:- 4
1) Let J = 10y2
z ax – 2x2
y ay + 2x2
z az A/m2
and find : (a) the total current crossing the
surface x = 3, 2<=y<=3, 3.8<=z<=5.2, in the ax direction ; (b) the magnitude of the
current density at the center of this area; (c) the average of Jx over the surface.
Ans:- (a) 399A; (b)296A/m2;(c) 285A/m2
2) In a region near the origin, current density is in the radial (outward) direction with a value
of 10r-1.5
A/m2
. (a) How much current is crossing the spherical surface, r=1mm? (b)
Repeat for r = 2mm. (c) at what rate is ρ increasing at a point where r = 1 mm? (d) At
what rate is the total charge increasing within the sphere r = 1mm?
Ans:- (a) 3.97A; (b) 5.62A (c) -1.581x108 C/m2
s(d) -3.97C/s.
3) Find the magnitude of the current density within an aluminum sample if:(a) the electric
field intensity is 70 mV/m; (b) the free-electron drift velocity is 10-4
m/s ;(c) it is in the
form of a cube, 1 mm on a side, carrying a total current of 2.5A; (c) it is in the form of a
cube, 1 mm on a side, with a potential difference of 0.75µV between opposite faces.
Take Conductivity of Aluminum ( Al) = 3.82 x 107
mho/m. mobility of electrons for
Aluminum (µe) = 0.0012
Ans:- (a) 2.67MA?m2
; (b) 3.18MA/m2
(c) 2.50MA/m2
; (d) 2.86MA/m2
.
4) What is the voltage between the ends of a copper conductor ;(a) if it has a circular cross
section with a diameter of 0.007 inch, is 100 ft long, and carries a current of 8 mA ;(b) if
is hollow circular cylinder, inner radius 2mm and outer radius 3mm, and is 200m long
and has a total current of 20A?
Ans:- (a)0.1693V;(b) 4.39V.
5) The point P (-2, 4, 1) lies on the surface of a conductor and E = 400ax – 2902 ay + 310az
V/m there. Assume the conductor is in free space and find the magnitude of: (a) En at P
(b) Et at P (c) ρs at P.
Ans:- (a) 583V/m; (b) 0; (c) 5.16nC/m2.
6) Find the polarization within a material which: (a) has an electric flux density of 1.5
µC/m2
in an electric intensity of 15KV/m; (b) has D = 2.8µC/m2
and e = 1.7;(c) has 1020
molecules/m3
, each with a dipole moment of 1.5x 10-26
C.m (d) has E= 50 kV/m, and the
relative permittivity is 4.4.
Ans (a) 1.367; (b)1.763; (c) 1.500 (all answer are in µC/m)
7) The region z<0 contain a dielectric material for which ЄR1= 2.5, while the region z<0 is
characterized by ЄR2=4. Let E =-30xax +50ay +70az V/m, and find : (a) En1; (b)Et1;
(c)Et1; (d) E1; (e)1; (f)Dn2; (g)Dt2; (h)D2 (i)P2 (j)2
Ans :- (a) 70.0 V/m ;(b)-30ax +50ay V/m; (c) 58.3 V/m; (d)91.1 V/m; (e)39.8· ;
(f)1.549nC/m2
; (g) 2.07; (h) -1.062ax + 1.771ay+1.549az nC/m2
; (i) -0.79ax +1.328ay
+1.162az nC/m2
; (j) 53.1·
8) Find the capacitance of a parallel plate capacitor having: (a) plates separated by a
distance of 8mm, each with an area of 2 m2
, and a dielectric for which ЄR=520;
(b)d=0.08;mm , S=2m2
, an internal voltage gradient of 10 5
/m, and a charge density on
one plate of 2 µC/m2
; (c) a stored energy of 5µJ with a voltage between plates of 4V.
Ans :- 0.553; (b)0.5; (c) 0.625 (all answer are in F).
9) A point charge of 1nc is located at the origin in free space. Find the equation of the curve
in the x=0 plane along which Ey=1V/m.
Ans :-9y =(y2
+x2
)3/2

2. 10) A given the field D= (2x+1)y2
ax +2x(x+1) y ay C/m2
, compute the total flux crossing the
surface defined by: =4, 0<=< <=,0<=z<=1.
Ans:-402.12C
11) A line charge ρL =10Є0 C/m lies along the x-axis in free space, while a point charge,
Q=40Є0 C is located at (2, 4, -1).Two points are indentified as A (1, 1, 2) and B (4, 0,
5).Find V
Ans:-4.378 V
12) Find the capacitance of: (a) 100 feet (ft) of 58 C/U coaxial cable having an inner
conductor 0.0295inch in diameter, an outer conductor having an inside diameter of 0.116
inch and a polyethylene dielectric; (b) a conducting sphere 1 cm in radius, covered with a
layer of polyethylene 1 cm thick, and a surrounded by a concentric conducting sphere 2
cm in radius.
Ans:- (a) 2800pF; (b) 5.03pF.
13) If = 0 and V = 8x2
yz, find (a) V at P ( 2,-1,3); (b) V at P (c) E at P ; (d) equation of
equipotential surface through P (e) Does V satisfy the Laplace’s equation?
Ans:- (a) -96V (b) 424.992pC/m2
(c) 96ax-96ay+32az V/m (d) x2
yz +12=0 (d) not
14) Use iteration method to estimate the potential at every point in the square trough. (do
upto 2nd
iteration)
15) Use the boundary condition to find E2 in the medium 2 with boundary located at plane
y=0. Medium 1 is perfect dielectric characterized by r1 = 3, medium 2 is perfect
dielectric characterized by r2 = 5, electric field in medium 1 is E1 = 3ax + 2ay + az
16) Calculate the capacitance of spherical capacitor using boundary conditions.
17) Use two dimensional Laplace’s equation to determine potential distribution for the
following boundary condition; V= 0 at x = 0, V = V0 at x=a, V= 0 at y = 0 and V = 0 at
y = b.
18) Explain how the conductivity of metals and semiconductor changes with the increase in
temperature.

3. 19) Given the potential field in cylindrical co-ordinate;
V= 100Cos/[z2
+1] volts and point p at =3m,=60 z=2m Find values at P for (i) Vp (ii) Ep (iii)
|Ep| (iv) dV/dN (v) âN (vi) v in free space.
20) Find the magnitude of current density in a sample of silver for which = 6.17x107
siemen/m and
μe=0.0056m2
/V.S if;
(i) the drift Velocity is 1.5μm/s.
(ii) the electric field intensity is 1mV/m.
(iii) The sample is a cube 2.5mm on a side having a voltage of 0.4mV between opposite faces.
(iv) The sample is cube 5.5mm on a side carrying a total current of 0.5A.
21) Two parallel conducting plates are each 10cm by 10cm and are separated by 2mm. The region
between the plates is filled with a perfect dielectric for which r = (1+500x)2
, where x is the distance from
one plate from one plate to another. Assume a uniform surface charge density to 10nc/m2
on the positive
plate and determine: (i) Q total (ii) Dn (iii) Ex (iv) V0
22) Determine the Capacitance of the parallel plate capacitor having two mediums of thickness 1mm and
2mm with the relative permittivity of 5 and 3 respectively. If the plate has the dimension of 10cm by 5cm,
then also calculate the value of potential for the charge of 10nc.
24) Eight point charge of 1nc each are located in free space at the corners of a cube of side 1m centered at
the origin. Find the electric field intensity at (i) at the center of the cube and (ii) at the center of a face.
25) Given the potential field V=50x2
yz in free space. Find the total energy stored within the cube
0<x,y,z<2.
26) Two spherical conductors are at r=2cm and r=4cm. The region in between is filled with a conducting
material of =80 s/m. If the current density is J

=10/(r2
) âr A/m2 for 2cm<r<6cm find (i) the current
flowing from one conductor to the other (ii) E (iii) the potential difference between the two conductors
and (iv) the total power being dissipated in the material.