❓How is the Maxwell stress tensor expressed solely in terms of the magnetic field tensor by using the

How is the Maxwell stress tensor expressed solely in terms of the magnetic field tensor by using the results from parts (a) and (b)?

Take the standard expression for the magnetic Maxwell stress tensor $\mathbf{T}{ik}$ (in terms of $\mathbf{B}$) and convert it entirely into an expression involving the magnetic field tensor $\mathbf{F}{ij}$.

The process involves two main substitutions, using the identity derived in part (b) and the trace of that identity.

📐 Expressing $\mathbf{T}{ik}$ in Terms of $\mathbf{F}{ij}$

1. Start with the Definition of the Maxwell Stress Tensor

The magnetic part of the Maxwell stress tensor is:

$\mathbf{T}_{ik} = \frac{1}{\mu_0} \left( \mathbf{B}_i \mathbf{B}k - \frac{1}{2} \mathbf{B}^2 \delta{ik} \right)$

2. Use the Result from Part (b)

The result derived in part (b) is:

$\mathbf{F}{ij} \mathbf{F}{jk} = \mathbf{B}^2 \delta_{ik} - \mathbf{B}_i \mathbf{B}_k$

Solving this for the dyadic term $\mathbf{B}_i \mathbf{B}_k$:

$\mathbf{B}i \mathbf{B}k = \mathbf{B}^2 \delta{ik} - \mathbf{F}{ij} \mathbf{F}_{jk}$

Brief audio

Proof and Derivation related to FAQ

How is the Maxwell stress tensor expressed solely in terms of the magnetic field tensor by using the results from parts (a) and (b)? | FAQsviadean on Notion