A long hollow cylindrical conductor inner radius 23 mm outer

A long, hollow, cylindrical conductor (inner radius 2.3 mm, outer radius 6.3 mm) carries a current of 24 A distributed uniformly across its cross section. A long thin wire that is coaxial with the cylinder carries a current of 43 A in the opposite direction. What is the magnitude of the magnetic field (a) 1.1 mm, (b) 2.4 mm, and (c) 7.1 mm from the central axis of the wire and cylinder?

Solution

As we need to determine the magnetic field due to the current carrying conductors, we will use Ampere\'s law to determine the values. Ampere\'s law, as we know, states:  B.dl = oI(enclosed). Where dl is a small distance along the loop selected to apply ampere\'s law. We will use the same to solve the given problem. For each of the distances mentioned, we will assume a loop of radius same as that distance and coaxial with the cylinder.

Part a.) For distance from the central axis, say r, to be 1.1 mm, we have only the thin wire lying inside it.

Hence we get: B(2R) = 43o

or, B = 43 x 2 x 10^-7 / 1.1 x 10^-3 = 78.182 x 10^-4 T

Part b.) For 2.4 mm, the distance lies inside the outer cylindrical shell conductor. We would first determine the current density for the outer cylindrical shell then find the current enclosed in the loop of radius 2.4 mm

Current density = J = 24 / (6.3*6.3 - 2.3*2.3)x10^-6 = 2.2219 x 10^5 A/m^2

Now the point under consideration is 2.4 mm from the axis. Hence the current flowing through the cylindrical shell that would lie inside the loop of radius 2.4 mm would be: 2.2219 x 10^5 x 3.14 (2.4*2.4 - 2.3*2.3) x10^-6 = 0.3279 amperes

Therefore the net current lying inside the loop would be 43 - 0.3279 = 42.6721

Hence, by ampere\'s law, we get: B(2 * 3.14 * 2.4 x 10^-3) = 42.6721 x o

B = 35.56 x 10^3 T is the required magnetic field.

Part c.) For distance of 7.1 mm, the net current enlosed would be: 43 - 24 = 19A

By ampere\'s law, we get: B(2 x 3.14 x 7.1 x 10^-3) = 19 x o

B = 19 x 2 / 7.1 x 10^-3 = 5.352 x 10^3 T is the required magnetic field