Question

Ampere's law on magnetic field of steady current stated in an integral form is: $\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu I$ or $\oint_C \mathbf{H} \cdot d\mathbf{l} = I$ In a differential form: $\nabla \times \mathbf{B} = \mu \mathbf{J}$ or $\nabla \times \mathbf{H} = \mathbf{J}$ In free space $\mu = \mu_0 = 4\pi \times 10^{-7} H/m$. Problem 1 An exercise of Ampere's law. A long solid cylindrical conductor of radius a is placed along z axis. A magnetic field is induced when a current flows through the conductor. Suppose a current through it is characterized by a current density $\mathbf{J} = \hat{z} J_0/r$ where $J_0$ is a constant and r is the radial distance from the z axis. 1. Find the magnetic fields \mathbf{H} and \mathbf{B} for $0 \le r \le a$. 2. Find the magnetic fields \mathbf{H} and \mathbf{B} for $r > a$. 3. Compute the curls $\nabla \times \mathbf{H}$ and $\nabla \times \mathbf{B}$. Check to see if Ampere's law in differential form is satisfied.

          Ampere's law on magnetic field of steady current stated in an integral form
is:
$\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu I$
or
$\oint_C \mathbf{H} \cdot d\mathbf{l} = I$
In a differential form:
$\nabla \times \mathbf{B} = \mu \mathbf{J}$
or
$\nabla \times \mathbf{H} = \mathbf{J}$
In free space $\mu = \mu_0 = 4\pi \times 10^{-7} H/m$.
Problem 1 An exercise of Ampere's law.
A long solid cylindrical conductor of radius a is placed along z axis. A
magnetic field is induced when a current flows through the conductor. Suppose
a current through it is characterized by a current density $\mathbf{J} = \hat{z} J_0/r$ where $J_0$
is a constant and r is the radial distance from the z axis.
1. Find the magnetic fields \mathbf{H} and \mathbf{B} for $0 \le r \le a$.
2. Find the magnetic fields \mathbf{H} and \mathbf{B} for $r > a$.
3. Compute the curls $\nabla \times \mathbf{H}$ and $\nabla \times \mathbf{B}$. Check to see if Ampere's law in
differential form is satisfied.

Ampere's law on magnetic field of steady current stated in an integral form
is:
𝐁· d𝐥 = μ I
or
𝐇· d𝐥 = I
In a differential form:
∇×𝐁 = μ𝐉
or
∇×𝐇 = 𝐉
In free space μ = μ0 = 4π× 10^-7 H/m.
Problem 1 An exercise of Ampere's law.
A long solid cylindrical conductor of radius a is placed along z axis. A
magnetic field is induced when a current flows through the conductor. Suppose
a current through it is characterized by a current density 𝐉 = ẑ J0/r where J0
is a constant and r is the radial distance from the z axis.
1. Find the magnetic fields 𝐇 and 𝐁 for 0 ≤ r ≤ a.
2. Find the magnetic fields 𝐇 and 𝐁 for r > a.
3. Compute the curls ∇×𝐇 and ∇×𝐁. Check to see if Ampere's law in
differential form is satisfied.

Added by Richard H.

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