ELECTRIC FLUX AND GAUSS LAW - Electrical Engineering (EE) PDF Download

 Page 1


 
 
1 
ELECTRIC FLUX AND GAUSS LAW 
2EL-01 
1. The three small spheres as shown in figure, 
carry charges q
1
 = 4.0 nC, q
2
 = -8.0 nC and 
q
3
 = 2.0 nC. Find the net electric flux 
through each of the following closed 
surfaces shown in cross section in the figure.  
  
2. A charge Q is 
distributed uniformly 
on a ring of radius r. A 
sphere of equal radius r 
is constructed with its centre at the periphery 
of the ring. Find the flux of the electric field 
through the surface of the sphere. 
3. Figure shows an imaginary cube of edge L/2. 
A uniformly charged rod of length L moves 
towards left at a small but constant speed v. 
At t = 0, the left end just touches the centre 
of the face of the cube opposite it. Which of 
the graphs shown in figure represents the 
flux of the electric field through the cube as 
the rod goes through it? 
L/2 
L 
v 
(a)
Flux 
a 
b 
c 
d 
time (b)
 
4. A hemispherical body is placed in a uniform 
electric field E. What is the flux linked with 
the curved surface, if field is (a) parallel to 
base figure (a), (b) perpendicular to base 
figure (b). 
 
R 
n 
(A) 
E
      
R 
n 
(B) 
E
 
 
(C ) 
E
n 
R 
 
5. What is the field in the cavity, if a conductor 
having a cavity is charged ? Does the result 
depend on the shape and size of cavity or 
conductor? 
6. Figure shows a closed surface which 
intersects a conducting sphere. If a positive 
charge is placed at the points P.  Find the 
sign of flux passing through curved surface 
‘s’. 
 
7. In which position (A, B, C or D) of second 
charge the flux of the electric field through 
the hemisphere remains uncharged.  
A 
B 
C 
D 
q
 
Ring 
A 
B 
0 
1 
Sphere
0 
Page 2


 
 
1 
ELECTRIC FLUX AND GAUSS LAW 
2EL-01 
1. The three small spheres as shown in figure, 
carry charges q
1
 = 4.0 nC, q
2
 = -8.0 nC and 
q
3
 = 2.0 nC. Find the net electric flux 
through each of the following closed 
surfaces shown in cross section in the figure.  
  
2. A charge Q is 
distributed uniformly 
on a ring of radius r. A 
sphere of equal radius r 
is constructed with its centre at the periphery 
of the ring. Find the flux of the electric field 
through the surface of the sphere. 
3. Figure shows an imaginary cube of edge L/2. 
A uniformly charged rod of length L moves 
towards left at a small but constant speed v. 
At t = 0, the left end just touches the centre 
of the face of the cube opposite it. Which of 
the graphs shown in figure represents the 
flux of the electric field through the cube as 
the rod goes through it? 
L/2 
L 
v 
(a)
Flux 
a 
b 
c 
d 
time (b)
 
4. A hemispherical body is placed in a uniform 
electric field E. What is the flux linked with 
the curved surface, if field is (a) parallel to 
base figure (a), (b) perpendicular to base 
figure (b). 
 
R 
n 
(A) 
E
      
R 
n 
(B) 
E
 
 
(C ) 
E
n 
R 
 
5. What is the field in the cavity, if a conductor 
having a cavity is charged ? Does the result 
depend on the shape and size of cavity or 
conductor? 
6. Figure shows a closed surface which 
intersects a conducting sphere. If a positive 
charge is placed at the points P.  Find the 
sign of flux passing through curved surface 
‘s’. 
 
7. In which position (A, B, C or D) of second 
charge the flux of the electric field through 
the hemisphere remains uncharged.  
A 
B 
C 
D 
q
 
Ring 
A 
B 
0 
1 
Sphere
0 
 
2 
ELECTRIC FLUX AND GAUSS LAW 
2EL-01 
8. A charge q is placed at the center of an 
imaginary hemispherical surface. The flux of 
the electric field due to the charge through 
the surface of the hemisphere.  
q
 
9. Find the flux of the electric field through 
each of the five surfaces of the inclined plane 
as shown in figure. What is the total flux 
through the entire closed surface?  
 
10. Consider two concentric conducting spheres. 
The outer sphere is hollow and initially has a 
charge -7Q on it. The inner sphere is solid 
and has a charge +2Q on it. 
 (a) How much charge is on the outer surface 
and inner surface of the outer sphere. 
 (b) If a wire is connected between the inner 
and outer spheres, after electrostatic 
equilibrium is established, how much total 
charge is on the outer sphere? How much 
charge is on the outer surface and inner 
surface of outer sphere? Does the electric 
field at the surface of the inside sphere 
change when the wire is connected?  
 (c) We return to original condition in (a). We 
now connect the outer sphere to ground will 
be on the outer sphere? How much charge 
will be on the inner surface and outer surface 
of the outer sphere?  
11. A cube of side l has one corner at the origin 
of coordinates and extends along the positive 
x, y and z-axes. Suppose the electric field in 
this region is given by 
ˆ
() E a by j ?? . 
Determine the charge inside the cube.  
 
12. The electric field in a cubical volume is  
 
00
ˆˆ
1
zz
E E i E j
aa
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
 
 Each edge of the cube measures d and one of 
the corners lies at the origin of coordinates. 
Determine the net charge within the cube  
 
13. A point charge is placed at a distance a/2 
from the centre of a square of side ‘a’ as 
shown in the diagram; calculate the electric 
flux passing through the square.  
 
Page 3


 
 
1 
ELECTRIC FLUX AND GAUSS LAW 
2EL-01 
1. The three small spheres as shown in figure, 
carry charges q
1
 = 4.0 nC, q
2
 = -8.0 nC and 
q
3
 = 2.0 nC. Find the net electric flux 
through each of the following closed 
surfaces shown in cross section in the figure.  
  
2. A charge Q is 
distributed uniformly 
on a ring of radius r. A 
sphere of equal radius r 
is constructed with its centre at the periphery 
of the ring. Find the flux of the electric field 
through the surface of the sphere. 
3. Figure shows an imaginary cube of edge L/2. 
A uniformly charged rod of length L moves 
towards left at a small but constant speed v. 
At t = 0, the left end just touches the centre 
of the face of the cube opposite it. Which of 
the graphs shown in figure represents the 
flux of the electric field through the cube as 
the rod goes through it? 
L/2 
L 
v 
(a)
Flux 
a 
b 
c 
d 
time (b)
 
4. A hemispherical body is placed in a uniform 
electric field E. What is the flux linked with 
the curved surface, if field is (a) parallel to 
base figure (a), (b) perpendicular to base 
figure (b). 
 
R 
n 
(A) 
E
      
R 
n 
(B) 
E
 
 
(C ) 
E
n 
R 
 
5. What is the field in the cavity, if a conductor 
having a cavity is charged ? Does the result 
depend on the shape and size of cavity or 
conductor? 
6. Figure shows a closed surface which 
intersects a conducting sphere. If a positive 
charge is placed at the points P.  Find the 
sign of flux passing through curved surface 
‘s’. 
 
7. In which position (A, B, C or D) of second 
charge the flux of the electric field through 
the hemisphere remains uncharged.  
A 
B 
C 
D 
q
 
Ring 
A 
B 
0 
1 
Sphere
0 
 
2 
ELECTRIC FLUX AND GAUSS LAW 
2EL-01 
8. A charge q is placed at the center of an 
imaginary hemispherical surface. The flux of 
the electric field due to the charge through 
the surface of the hemisphere.  
q
 
9. Find the flux of the electric field through 
each of the five surfaces of the inclined plane 
as shown in figure. What is the total flux 
through the entire closed surface?  
 
10. Consider two concentric conducting spheres. 
The outer sphere is hollow and initially has a 
charge -7Q on it. The inner sphere is solid 
and has a charge +2Q on it. 
 (a) How much charge is on the outer surface 
and inner surface of the outer sphere. 
 (b) If a wire is connected between the inner 
and outer spheres, after electrostatic 
equilibrium is established, how much total 
charge is on the outer sphere? How much 
charge is on the outer surface and inner 
surface of outer sphere? Does the electric 
field at the surface of the inside sphere 
change when the wire is connected?  
 (c) We return to original condition in (a). We 
now connect the outer sphere to ground will 
be on the outer sphere? How much charge 
will be on the inner surface and outer surface 
of the outer sphere?  
11. A cube of side l has one corner at the origin 
of coordinates and extends along the positive 
x, y and z-axes. Suppose the electric field in 
this region is given by 
ˆ
() E a by j ?? . 
Determine the charge inside the cube.  
 
12. The electric field in a cubical volume is  
 
00
ˆˆ
1
zz
E E i E j
aa
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
 
 Each edge of the cube measures d and one of 
the corners lies at the origin of coordinates. 
Determine the net charge within the cube  
 
13. A point charge is placed at a distance a/2 
from the centre of a square of side ‘a’ as 
shown in the diagram; calculate the electric 
flux passing through the square.  
 
 
 
3 
ELECTRIC FLUX AND GAUSS LAW 
2EL-01 
14. In figure shown a charge q is placed at a 
distance ? =0 near one of the edges of a cube 
of edge l on a line of symmetry along 
diagonal.  
 
 (a) What is flux through each of the sides 
containing the point a. 
 (b) What is the flux through the other three 
faces? 
15. Find the electric flux through cylindrical 
surface in a uniform electric field E  
  
16. What is the flux if the cylinder of previous 
problem were vertical ? 
17. Two identical metal plates each having 
surface area ‘A’, having charge ‘q
1
’ and ‘q
2
’ 
are placed facing each other at a separation 
‘d’. Find the charge appearing on surface (1), 
(2), (3) and (4). Assume the size of the plate 
is much longer than the separation between 
the plates.  
 
18. A point charge q is placed on the apex of a 
cone of semi-vertex angle ?. Find the electric 
flux through the base of the cone.  
 
19. The cube as shown in figure has sides of 
length L = 10.0 cm. The electric field is 
uniform, has a magnitude E = 4.00 × 10
3
 
N/C, and is parallel to the xy-plane at an 
angle of 37° measured from the +x axis 
toward the +y-axis.  
 
 What is the electric flux through each of the 
six cube faces S
1
, S
2
, S
3
, S
4
, S
5
 and S
6
 ? 
 (b) What is the total electric flux through all 
faces of the cube?  
20. A cube has sides of length L = 0.300 m. It is 
placed with one corner at the origin as shown 
in the above figure. The electric field is not 
uniform, but is given by 
 ?
ˆ .
5.00 / . ) (3.00 N/C ) . E N C m xi m zk ? ? ? 
 Find the electric flux through each of the six 
cube faces S
1
, S
2
, S
3
, S
4
, S
5
 and S
6
 and find 
the total electric charge inside the cube. 
Page 4


 
 
1 
ELECTRIC FLUX AND GAUSS LAW 
2EL-01 
1. The three small spheres as shown in figure, 
carry charges q
1
 = 4.0 nC, q
2
 = -8.0 nC and 
q
3
 = 2.0 nC. Find the net electric flux 
through each of the following closed 
surfaces shown in cross section in the figure.  
  
2. A charge Q is 
distributed uniformly 
on a ring of radius r. A 
sphere of equal radius r 
is constructed with its centre at the periphery 
of the ring. Find the flux of the electric field 
through the surface of the sphere. 
3. Figure shows an imaginary cube of edge L/2. 
A uniformly charged rod of length L moves 
towards left at a small but constant speed v. 
At t = 0, the left end just touches the centre 
of the face of the cube opposite it. Which of 
the graphs shown in figure represents the 
flux of the electric field through the cube as 
the rod goes through it? 
L/2 
L 
v 
(a)
Flux 
a 
b 
c 
d 
time (b)
 
4. A hemispherical body is placed in a uniform 
electric field E. What is the flux linked with 
the curved surface, if field is (a) parallel to 
base figure (a), (b) perpendicular to base 
figure (b). 
 
R 
n 
(A) 
E
      
R 
n 
(B) 
E
 
 
(C ) 
E
n 
R 
 
5. What is the field in the cavity, if a conductor 
having a cavity is charged ? Does the result 
depend on the shape and size of cavity or 
conductor? 
6. Figure shows a closed surface which 
intersects a conducting sphere. If a positive 
charge is placed at the points P.  Find the 
sign of flux passing through curved surface 
‘s’. 
 
7. In which position (A, B, C or D) of second 
charge the flux of the electric field through 
the hemisphere remains uncharged.  
A 
B 
C 
D 
q
 
Ring 
A 
B 
0 
1 
Sphere
0 
 
2 
ELECTRIC FLUX AND GAUSS LAW 
2EL-01 
8. A charge q is placed at the center of an 
imaginary hemispherical surface. The flux of 
the electric field due to the charge through 
the surface of the hemisphere.  
q
 
9. Find the flux of the electric field through 
each of the five surfaces of the inclined plane 
as shown in figure. What is the total flux 
through the entire closed surface?  
 
10. Consider two concentric conducting spheres. 
The outer sphere is hollow and initially has a 
charge -7Q on it. The inner sphere is solid 
and has a charge +2Q on it. 
 (a) How much charge is on the outer surface 
and inner surface of the outer sphere. 
 (b) If a wire is connected between the inner 
and outer spheres, after electrostatic 
equilibrium is established, how much total 
charge is on the outer sphere? How much 
charge is on the outer surface and inner 
surface of outer sphere? Does the electric 
field at the surface of the inside sphere 
change when the wire is connected?  
 (c) We return to original condition in (a). We 
now connect the outer sphere to ground will 
be on the outer sphere? How much charge 
will be on the inner surface and outer surface 
of the outer sphere?  
11. A cube of side l has one corner at the origin 
of coordinates and extends along the positive 
x, y and z-axes. Suppose the electric field in 
this region is given by 
ˆ
() E a by j ?? . 
Determine the charge inside the cube.  
 
12. The electric field in a cubical volume is  
 
00
ˆˆ
1
zz
E E i E j
aa
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
 
 Each edge of the cube measures d and one of 
the corners lies at the origin of coordinates. 
Determine the net charge within the cube  
 
13. A point charge is placed at a distance a/2 
from the centre of a square of side ‘a’ as 
shown in the diagram; calculate the electric 
flux passing through the square.  
 
 
 
3 
ELECTRIC FLUX AND GAUSS LAW 
2EL-01 
14. In figure shown a charge q is placed at a 
distance ? =0 near one of the edges of a cube 
of edge l on a line of symmetry along 
diagonal.  
 
 (a) What is flux through each of the sides 
containing the point a. 
 (b) What is the flux through the other three 
faces? 
15. Find the electric flux through cylindrical 
surface in a uniform electric field E  
  
16. What is the flux if the cylinder of previous 
problem were vertical ? 
17. Two identical metal plates each having 
surface area ‘A’, having charge ‘q
1
’ and ‘q
2
’ 
are placed facing each other at a separation 
‘d’. Find the charge appearing on surface (1), 
(2), (3) and (4). Assume the size of the plate 
is much longer than the separation between 
the plates.  
 
18. A point charge q is placed on the apex of a 
cone of semi-vertex angle ?. Find the electric 
flux through the base of the cone.  
 
19. The cube as shown in figure has sides of 
length L = 10.0 cm. The electric field is 
uniform, has a magnitude E = 4.00 × 10
3
 
N/C, and is parallel to the xy-plane at an 
angle of 37° measured from the +x axis 
toward the +y-axis.  
 
 What is the electric flux through each of the 
six cube faces S
1
, S
2
, S
3
, S
4
, S
5
 and S
6
 ? 
 (b) What is the total electric flux through all 
faces of the cube?  
20. A cube has sides of length L = 0.300 m. It is 
placed with one corner at the origin as shown 
in the above figure. The electric field is not 
uniform, but is given by 
 ?
ˆ .
5.00 / . ) (3.00 N/C ) . E N C m xi m zk ? ? ? 
 Find the electric flux through each of the six 
cube faces S
1
, S
2
, S
3
, S
4
, S
5
 and S
6
 and find 
the total electric charge inside the cube. 
 
4 
ELECTRIC FLUX AND GAUSS LAW 
2EL-01 
21. A cube of side a is placed such that the 
nearest face which is parallel to the y-z plane 
is at a distance ‘a’ from the origin. The 
electric field component are E
x
 = ax
1/2
,         
E
y
 = E
z
 = 0. Calculate  
 
y 
z 
x
 
 (a) the flux f
E
 through the cube 
 (b) the charge within the cube 
22. A very long uniformly charged wire oriented 
along the axis of a circle of radius R rests on 
its center with one of the ends (as shown in 
figure). The linear charge density on the wire 
is l. Evaluate the flux of vector E across the 
circle area. 
+ ?
R
 
23. Find the electric flux crossing the wire frame 
ABCD of length l width b and whose center 
is at a distance OP = d from an infinite line 
of charge with linear charge density ?. 
Consider that the plane of frame is 
perpendicular to the line OP. 
A 
B 
C 
D 
O 
b 
d 
?
P
 
24. Two point charges q and -q separated by the 
distance 2a. Evaluate the flux of electric  
field strength vector across a circle of radius 
R. 
a a
-q +q
R
 
25. A solid spherical region having a spherical 
cavity whose diameter ‘R’ is equal to the 
radius of the spherical region, has a total 
charge ‘Q’. Find the electric field at a point 
P as shown. 
P 
R 
x
 
26. An insulating sphere of radius R has a 
spherical hole of radius a located within its 
volume and centered a distance b from the 
center of the sphere, where a < b < R  (a 
cross section of the sphere is shown in 
figure). The solid part of the sphere has a 
uniform volume charge density ?. Find the 
magnitude and direction of the electric 
field E inside the hole, and show that E is 
uniform over the entire hole.