A long, straight, solid cylinder, oriented with its axis in the z−direction, carries a current whose current density is J⃗. The current density, although symmetrical about the cylinder axis, is not constant but varies according to the relationship J⃗=2*I0/πa^2 *[1−(r/a)^2]k for r≤a=0 for r≥a where a is the radius of the cylinder, r is the radial distance from the cylinder axis, and I0 is a constant having units of amperes. Part A Using Ampere's law, derive an expression for the magnitude of the magnetic field B⃗ in the region r≥a. Part B Obtain an expression for the current I contained in a circular cross section of radius r≤a and centered at the cylinder axis. Part C Using Ampere's law, derive an expression for the magnitude of the magnetic field B⃗ in the region r≤a.
A long, straight, solid cylinder, oriented with its axis in the z−direction, carries a current whose current density is J⃗. The current density, although symmetrical about the cylinder axis, is not constant but varies according to the relationship J⃗=2*I0/πa^2 *[1−(r/a)^2]k for r≤a=0 for r≥a where a is the radius of the cylinder, r is the radial distance from the cylinder axis, and I0 is a constant having units of amperes.
Part A
Using Ampere's law, derive an expression for the magnitude of the magnetic field B⃗ in the region r≥a.
Part B
Obtain an expression for the current I contained in a circular cross section of radius r≤a and centered at the cylinder axis.
Part C
Using Ampere's law, derive an expression for the magnitude of the magnetic field B⃗ in the region r≤a.
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