Question

A long, straight, solid cylinder, oriented with its axis in the z− direction, carries a current whose current density is J . The current density, although symmetrical about the cylinder axis, is not constant but varies according to the relationship J =2Iπa2[1−(ra)2]k^for≤a= 0 for≥a where a is the radius of the cylinder, r is the radial distance from the cylinder axis, and I is a constant having units of amperes. Part A Using Ampere's law, derive an expression for the magnitude of the magnetic field B in the region r≥a . Express your answer in terms of some or all of the variables I , a , r , and magnetic constant μ0 B) Obtain an expression for the current I contained in a circular cross section of radius r≤a and centered at the cylinder axis. Express your answer in terms of some or all of the variables I, a, r, and magnetic constant μ0 C) Using Ampere's law, derive an expression for the magnitude of the magnetic field B in the region r≤a . Express your answer in terms of some or all of the variables I, a, r , and magnetic constant μ0 . .

Answer 1

Use Ampère's Law to derive the magnetic field B outside and inside a cylindrical conductor with varying current density. Outside the conductor, B is proportional to the total current and inversely proportional to the radius. Inside the conductor, B depends on the current enclosed by a circle with radius r.

Step-by-step explanation:

To calculate the magnetic field B inside and outside a cylindrical conductor with non-uniform current density using Ampère's Law, we first note that for r ≥ a (outside the cylinder), the enclosed current equals the total current I.

Applying Ampère's law in integral form ∫ B ⋅ dl = μ_0 I_enclosed, and using symmetry, we see that B is constant along a circle of radius r.

Therefore, the integral simplifies to B(2r) = μ_0 I, giving us B = μ_0 I / (2r) for the region outside the conductor.

For r ≤ a (inside the cylinder), the current I' within a radius r from the cylinder's axis is computed by integrating the provided current density function over the cross-sectional area, yielding I' = ∫∫ J ⋅ dA.

Inserting the given J into the integral and solving provides I' as a function of r.

We then apply Ampère's Law again inside the wire to find B by considering the symmetrical path within the cylinder, with the result being B = μ_0 I' / (2r).