A charge of uniform linear density 2.89 nC/m is distributed along a long, thin, nonconducting rod. The rod is coaxial with a long conducting cylindrical shell with an inner radius of 6.23 cm and an outer radius of 15.5 cm.
If the net charge on the shell is zero, what is the surface charge density on the inner surface of the shell?
What is the surface charge density on the outer surface of the shell?
total charge in rod:
2.89 nC/m
total charge on inner shell: same
charge density=charge/area
consider one meter of length
= 2.89nC/m*1m / (Pi*.0623^2*1m)
=237nC/m^2
outer surface
has to be same charge
consider one meter of length
= 2.89nC/m*1m / (Pi*.15^2*1m)
= 39.6nC/m^2 check the math with a calculator.
To find the surface charge density on the inner and outer surfaces of the conducting cylindrical shell, we can use Gauss's law.
Gauss's law states that the electric flux through a closed surface is proportional to the charge enclosed by that surface. In this case, the surfaces we are interested in are the inner and outer surfaces of the cylindrical shell, and we need to find the charge enclosed by each surface.
To find the charge enclosed by the inner surface of the shell, we can consider a Gaussian cylinder whose axis coincides with the axis of the cylindrical shell. The length of the Gaussian cylinder should be the same as the length of the rod.
Since there is no charge enclosed by the Gaussian cylinder (the net charge on the shell is zero), the electric flux through the Gaussian surface is also zero. Therefore, the electric field inside the Gaussian cylinder is zero.
Now, we can apply Gauss's law to a cylindrical surface with radius r (where r varies from the inner radius of the shell to the outer radius). The electric flux through this cylindrical surface is given by:
Φ = E * A = (E * 2πr) * L,
where E is the electric field, A is the area of the cylindrical surface (circumference * length), and L is the length of the rod.
Since the electric field is zero inside the Gaussian cylinder, the flux through the cylindrical surface is also zero. Therefore, the electric field outside the Gaussian cylinder is also zero. This implies that the charge on the inner surface of the shell is equal in magnitude and opposite in sign to the charge on the rod within the Gaussian cylinder.
Now, we can find the charge on the inner surface of the shell:
Charge on inner surface of shell = Charge on rod within Gaussian cylinder
The charge on the rod within the Gaussian cylinder can be found by integrating the charge density along the length of the rod (which coincides with the length of the Gaussian cylinder). The charge density is given as 2.89 nC/m, which means the charge per unit length.
To find the surface charge density on the inner surface of the shell, divide the charge on the inner surface by the circumference of the inner surface:
Surface charge density on inner surface = (Charge on inner surface of shell) / (2π * inner radius of shell)
Next, we need to find the charge on the outer surface of the shell. Similar to before, the charge on the outer surface of the shell is equal in magnitude and opposite in sign to the charge on the rod outside the Gaussian cylinder.
The charge on the rod outside the Gaussian cylinder can be found by integrating the charge density along the length of the rod beyond the length of the Gaussian cylinder.
To find the surface charge density on the outer surface of the shell, divide the charge on the outer surface by the circumference of the outer surface:
Surface charge density on outer surface = (Charge on outer surface of shell) / (2π * outer radius of shell)
By following these steps, you can find the surface charge densities on both the inner and outer surfaces of the conducting cylindrical shell.