How Does Maxwell's Stress Tensor Apply to an Infinite Parallel Plate Capacitor?

[mike217]

Consider an infinite parallel plate capacitor with the lower plate (at z=-d/2) carrying the charge density$$- \sigma$$ and the upper plate (at z=d/2) carrying the charge density $$\sigma$$.

Determine all nine elements of the stress tensor in the region between the plates. Display your answer as a 3x3 matrix.

$$\left(\begin{array}{cc}Txx&Txy&Txz\\Tyx&Tyy&Tyz\\Tzx&Tzy&Tzz\end{array}\right)$$

To calculate the matrix I must calculate $$Tij = \epsilon(EiEj-0.5\delta ij E^2)+1/\mu (BiBj-0.5\delta ij B^2)$$

By calculating the E-field between the plates I get $$\sigma / \epsilon\ z$$. My question is how do I calculate the other EiEj and BiBj components.

Thank you.

 

[Nate810]

The other components of the electric and magnetic field can be determined using the equations for the electric and magnetic fields. For example, for the electric field, you can use Coulomb's law to determine E_x and E_y. Similarly, for the magnetic field, you can use the Biot-Savart law to calculate B_x and B_y. Once these components are known, you can calculate the stress tensor.

 

[wais]

To calculate the other components of the E-field, we can use the fact that the electric field is continuous at the interface between the two plates. This means that the E-field in the region between the plates is the same as the E-field on either side of the plates. Therefore, we can use the E-field of a point charge to calculate the other components.

For example, the E-field in the x-direction is given by Ex = \sigma / 2\epsilon_0. Using this value, we can calculate the other components as follows:

- Txx = \epsilon_0E_x^2 - 0.5\epsilon_0E^2 = 0.5\epsilon_0E_x^2 = 0.25\sigma^2/\epsilon_0
- Txy = Txx = 0.25\sigma^2/\epsilon_0
- Txz = Txy = 0.25\sigma^2/\epsilon_0
- Tyx = Txx = 0.25\sigma^2/\epsilon_0
- Tyy = \epsilon_0E_y^2 - 0.5\epsilon_0E^2 = 0.5\epsilon_0E_y^2 = 0
- Tyz = Tyx = 0
- Tzx = Txy = 0.25\sigma^2/\epsilon_0
- Tzy = Tyx = 0
- Tzz = \epsilon_0E_z^2 - 0.5\epsilon_0E^2 = 0.5\epsilon_0E_z^2 = 0.25\sigma^2/\epsilon_0

Therefore, the stress tensor in the region between the plates is:

\left(\begin{array}{cc}0.25\sigma^2/\epsilon_0&0.25\sigma^2/\epsilon_0&0.25\sigma^2/\epsilon_0\\0.25\sigma^2/\epsilon_0&0&0\\0.25\sigma^2/\epsilon_0&0&0.25\sigma^2/\epsilon_0\end{array}\right)