Question

Find the magnetic vector potential, A, and magnetic flux density, B, caused by a length of 2ℓ of current I=I0​a^z​ at the midpoint of the line.

Answer 1

For both the magnetic vector potential
`(\(A\))` and the magnetic flux density
`(\(B\)):`

1. **Magnetic Vector Potential
`(\(A\)):`**

`\[ A_z(\rho, \phi, z=0) = (\mu_0 I_0)/(2\pi) (1)/(\rho) \]`

2. **Magnetic Flux Density
`(\(B\)):`**

`\[ B_\rho = -(\mu_0 I_0)/(2\pi) (1)/(\rho^2) \]`

`\[ B = B_\rho a^\rho \]`

Step-by-step explanation:

To find the magnetic vector potential
`(\(A\))` and magnetic flux density
`(\(B\))` caused by a length of
`\(2\ell\)` of current
`\(I=I_0 a^z\)` at the midpoint of the line, we can use the Biot-Savart law and the relationship between
`\(A\)` and
`\(B\)` in the context of magnetostatics.

1. **Magnetic Vector Potential
`(\(A\)):`**

The magnetic vector potential
`(\(A\))` is given by the line integral of the magnetic field
`(\(B\))` along a closed loop. For an infinitely long straight current, the magnetic vector potential can be expressed in cylindrical coordinates as follows:

`\[ A(\rho, \phi, z) = (\mu_0 I)/(4\pi) \int (d\phi)/(\rho) \]`

Since the current is along the
`\(z\)-axis.`, the contribution to
`\(A\)` will be along the
`\(z\)-axis.`

`\[ A_z(\rho, \phi, z) = (\mu_0 I_0)/(4\pi) \int_0^(2\pi) (d\phi)/(\rho) \]`

Evaluating the integral will give you the expression for
`\(A_z\).`

2. **Magnetic Flux Density
`(\(B\)):`**

The magnetic flux density
`(\(B\))` is related to the magnetic vector potential
`(\(A\))` by the curl operation.

`\[ B = \\abla * A \]`

In cylindrical coordinates, this simplifies to:

`\[ B_\rho = (1)/(\rho) (\partial A_z)/(\partial \phi) \]`

You can find the expression for
`\(B_\rho\)` using the derived expression for
`\(A_z\).`

The magnetic flux density
`\(B\)` is then given by
`\(B = B_\rho a^\rho\).`

Note: The result will depend on the specific geometry of your problem, and in this case, the length of the current
`(\(2\ell\))` and the location where you are evaluating the magnetic field are crucial.