A long, straight wire with a circular cross section of radius R carries a current I. Assume that the current density is not constant across the cross section of the wire, but rather varies as J=αrJ=αr, where αα is a constant. (a) By the requirement that J integrated over the cross section of the wire gives the total current I, calculate the constant αα in terms of I and R.
(b) Use Ampere’s law to calculate the magnetic field B(r) for (i) r≤Rr≤R and (ii) r≥Rr≥R. Express your answers in terms of I.

(a) α =

(b) For r≤R: B(r) = μ_0.

For r≥R: B(r) = μ_0.

Explanation:

(a) The current I enclosed in a straight wire with current density not constant is calculated by:

where:

dA is the cross section.

In this case, a circular cross section of radius R, so it translates as:

For these circunstances, α =

(b) Ampere's Law to calculate magnetic field B is given by:

μ_0.

(i) First, first find for r ≤ R:

Calculating B(r), using Ampere's Law:

μ_0.

.μ_0

B(r) = .μ_0

B(r) = .μ_0

For r ≤ R, magnetic field is B(r) = .μ_0

(ii) For r ≥ R:

So, as calculated before:

I

Using Ampere:

B.2.π.r = μ_0.I

B(r) = .μ_0

For r ≥ R, magnetic field is; B(r) = .μ_0.

explanation:

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