Solving Electric Field and Charge Distribution Problems Using Gauss's Law
Electric Field and Charge Distribution Problem
A long, straight wire is surrounded by a hollow metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of λ, and the cylinder has a net charge per unit length of 2λ. Let's use Gauss's law to find the following:
(a) Charge per Unit Length on Inner Surface of Cylinder
To find the charge per unit length on the inner surface of the cylinder, we use Gauss's law which states that the electric flux through a closed surface is proportional to the enclosed charge. The net charge enclosed by a cylindrical surface of radius r and length L is given by:
Q = λ*L + 2λ*L = 3λ*L
Since the charge is uniformly distributed, the charge per unit length on the inner surface of the cylinder is:
λinner = 3λ
(b) Charge per Unit Length on Outer Surface of Cylinder
Similar to part (a), the net charge enclosed by a cylindrical surface of radius r and length L is 3λ*L. Since the outer surface of the cylinder encloses all the charges, the charge per unit length on the outer surface is:
λouter = 3λ
(c) Electric Field Outside the Cylinder at Distance r from the Axis
The electric field outside the cylinder at a distance r from the axis can be calculated using Gauss's law. The electric field E at a distance r from the axis is given by:
E = 3λ / (4πε0r2)
A long, straight wire is surrounded by a hollow metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of λ, and the cylinder has a net charge per unit length of 2λ. From this information, use Gauss's law to find the following: (Use any variable or symbol stated above along with the following as necessary: ε0 and π.) (a) the charge per unit length on the inner surface of the cylinder. λinner = (b) the charge per unit length on the outer surface of the cylinder. λouter = (c) the electric field outside the cylinder a distance r from the axis. magnitude
A.- λinner = 3λ B.- λouter = 3λ C.- E = 3λ / (4πε0r2)