Problem 22.7
From example 21.11, the electric field due to an infinite line charge is:
Radius of cylinder r=0.250 m
Length of cylinder l=0.400 m
Infinite positive line charge
λ=6.00 µC/m
(a) IDENTIFY:
Use Eq.(22.5) to calculate the flux through the surface of the cylinder.
SET UP: The line of charge and the cylinder are sketched in Figure 22.7.
g
Figure 22.7
EXECUTE: The area of the curved part of the cylinder is A  2 rl.
The electric field is parallel to the end caps of the cylinder, so E  A  0 for the ends and the flux through the
cylinder end caps is zero.
The electric field is normal to the curved surface of the cylinder and has the same magnitude at all points on this
surface.
Thus   0 and E
 E  EA cos  EA    / 2 P0r  2 rl  
l
P0

 6.00 10
6
C/m   0.400 m 
8.854 1012 C2 / N  m2
 2.71105 N  m 2 / C
(b) In the calculation in part (a) the radius r of the cylinder divided out, so the flux remains the same,
 E  2.71105 N  m2 / C.
(c)  E 
l
P0

 6.00 10
6
C/m   0.800 m 
8.854 1012 C2 / N  m2
 5.42  105 N  m2 / C (twice the flux calculated in parts (b) and (c)).
EVALUATE: The flux depends on the number of field lines that pass through the surface of the cylinder.
22.13. (a) IDENTIFY and SET UP: It is rather difficult to calculate the flux directly from   úE  dA since the
magnitude of E and its angle with dA varies over the surface of the cube. A much easier approach is to use
Gauss's law to calculate the total flux through the cube. Let the cube be the Gaussian surface. The charge enclosed
is the point charge.
EXECUTE:
 E  Qencl / P0 
9.60 106 C
 1.084  106 N  m2 / C.
8.854 1012 C2 / N  m2
By symmetry the flux is the same through each of the six faces, so the flux through one face is
1
6
1.084 10
6
N  m2 / C   1.81105 N  m2 / C.
(b) EVALUATE: In part (a) the size of the cube did not enter into the calculations. The flux through one face depends only
on the amount of charge at the center of the cube. So the answer to (a) would not change if the size of the cube were
changed