Understanding Electromagnetic Duality: From B_i to E_i Using Tensor Notation

[gentsagree]

So, I'm trying to show that by duality $B_{i}\rightarrow E_{i}$, using tensor notation. I've done it in a different way, and it works (starting from $\overline{F}_{ij}$, the dual of F_ij), but I would like to get it from B_i directly. Where am I going wrong?

This is what I did:

$$B_{i}=\frac{1}{2}\epsilon_{ijk}F^{jk}\rightarrow B'_{i}=\frac{1}{2}\epsilon_{ijk}(i\overline{F}^{jk})=\frac{1}{2}i\epsilon_{ijk}(-\frac{1}{2}i\epsilon^{jk\rho\sigma}F_{\rho\sigma})$$
$$=\frac{1}{4}\epsilon_{ijk}\epsilon^{jk\rho\sigma}F_{\rho\sigma}=\frac{1}{4}\epsilon_{0ijk}\epsilon^{jk0i}F_{0i}$$

where in the last line I have inserted an extra 0-index in the Levi-Civita symbol (although I am not sure I know how to deal with zeros with epsilon), and made the substitution $(\rho,\sigma)\rightarrow(0,i)$.

However I calculate this to be $-\frac{3}{2}F_{i0}$ when it should be just $F_{i0}=E_{i}$

Any advice?

 

[Meir Achuz]

I don't understand your first equation for B'. B_i in the dual F is the 0i row, not what you write.