How Do You Calculate the Maxwell Stress Tensor Between Cylindrical Shells?

[tomfrank]

Homework Statement

x and y are nonconducting cylindrical shells. Both cylindrical shells are surrounding long wires that are carrying current. the x shell out of the page and the y shell into the page.
x radius has a charge per unit length = to +$$\lambda$$
y radius has a charge per unit length = to -$$\lambda$$
I need to calculate the maxwell stress tensor at a midway between x and y. (from the picture)

Homework Equations

T^ij =$$\epsilon$$_o(E_iE_j-$$\delta$$_ijE²)+(1/$$\mu$$)(B_iB_j-$$\delta$$_ijB²)

The Attempt at a Solution

How do I exactly found the i and j component of the magnetic and electric field?

 

[tomfrank]

can anyone help me?

 

[kerimek]

To calculate the Maxwell stress tensor, you will need to first determine the electric and magnetic fields at the midpoint between the x and y cylindrical shells. This can be done using the Biot-Savart and Gauss's law equations, respectively. Once you have the electric and magnetic fields, you can plug them into the Maxwell stress tensor equation to calculate the stress tensor at the midway point.

To find the electric field, you can use Gauss's law, which states that the electric flux through a closed surface is equal to the charge enclosed by that surface divided by the permittivity of free space. In this case, the electric field will be radial and its magnitude will depend on the charge per unit length on the cylindrical shells and the distance from the midpoint.

To find the magnetic field, you can use the Biot-Savart law, which states that the magnetic field at a point due to a current-carrying wire is proportional to the current and inversely proportional to the distance from the wire. In this case, the magnetic field will be tangential and its magnitude will depend on the current in the wires and the distance from the midpoint.

Once you have the electric and magnetic fields at the midway point, you can plug them into the Maxwell stress tensor equation to calculate the stress tensor. The i and j components of the stress tensor will depend on the direction of the electric and magnetic fields at the midpoint. You can use the right-hand rule to determine the direction of the fields and thus the direction of the stress tensor components.