A very long conducting tube (hollow cylinder) has inner radius a and outer radius b. It carries charge per unit length +?, where a is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of charge has charge per unit length +?. (a) Calculate the electric field in terms of ? and the distance r from the axis of the tube for (i) r < a ; (ii) a < r < b; (iii) r > b. Show your results in a graph of E as a function of r. (b) What is the charge per unit length on (i) the inner surface of the tube and (ii) the outer surface of the tube?
Introduction We have to electric field due to a uniformly charged wire. Step 1 When , the Gaussian surface will contain the charge only due to the line in which passes through the axis of the cylinder. Hence the electric field will be Step 2 If , then the gaussian surface will pass through the conductor. Now, we know that the electric field inside the conductor is always zero. Hence in this region, the electric field is zero. Step 3 If we have , then the total charge per unit length is Hence the charge density of the Gaussian surface drawn with will be . Hence the electric field outside the cylinder will be