An electric current is flowing through a long cylindrical conductor with radius

Question

An electric current is flowing through a long cylindrical conductor with radius a = 0.45 m. The current density J =7.5 A/m^2 is uniform in the cylinder. In this problem we consider an imaginary cylinder with radius r around the axis AB. When r is less than a, express the current inside the imaginary cylinder in terms of r and J. Express the magnitude of the magnetic field B at r in terms of the current through the imaginary cylinder I and it's radius r. Express B in terms of J and r. For r = 0.5 a, calculate the numerical value of B in Tesla. When r is greater than a, express the current inside the imaginary cylinder in terms of r, a and J. Express the magnitude of the magnetic field. B, at r > a in terms of I and r. Express B in terms of J, a and r. For r = 2 a. calculate the numerical value of B in Tesla

Explanation / Answer

Here,

J = 7.5 A/m^2

a = 0.45 m

part a)

for the current inside the imaginary cyclinder

current inside the cyclinder = J * area^2

current inside the cyclinder = J * pi * r^2

b)

for the current inside , I
Using amphere circuital law

B * 2pi * r = u0 * I

B = u0 * I/(2pi * r)

the magnetic field is u0 * I/(2pi * r)

C)

Using Amphere circuital law

B * 2pi * r = u0 * Iin

B * 2pi * r = u0 *J * pi * r^2

B = u0 *r*J/(2)

part d)

at r = 0.5 a

B = u0 *r*J/(2)

B = 4pi *10^-7 * 0.45 * 0.5 * 7.5/(2)

B = 1.06 *10^-6 T

the magnetic field at 0.5 * a is 1.06 *10^-6 T

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