# Please clear my electromagnetism doubt about units

It is a well known fact that, in electromagnetic units, strength of a shell and strength of current flowing through its boundary are same. See here.

\begin{matrix}
\text{i.e.}\: i \text{(biot)} = \phi \text{(biot) }
\end{matrix}
\tag{1}

(a) While converting to SI:

\begin{matrix}
i\: \text{biot} = i\: (10 \text{Amp}) = 10i\: \text{Amp} = I \text{Amp}\\
\text{where biot} = 10 \text{Amp and}\: 10i = I
\tag{a}
\end{matrix}

Also:

\begin{matrix}
\phi\: \text{biot} = \phi\: (10 \text{Amp}) = 10\phi\: \text{Amp} = \Phi \text{Amp}\\
\text{where biot} = 10 \text{Amp and}\: 10\phi = \Phi
\tag{b}
\end{matrix}

Therefore in SI (by comparing with equation (1)):

##I## (Amp) = ##\Phi## (Amp)

(b) While converting to electrodynamic units:

\begin{matrix}
i\: \text{biot} = i\: (\sqrt{2}\: \text{ed}) = \sqrt{2}i\: \text{ed} = j\: \text{ed}\\
\text{where biot} = \sqrt{2}\: \text{ed and}\: \sqrt{2}\: i = j
\tag{c}
\end{matrix}

Also:

\begin{matrix}
\phi\: \text{biot} = \phi\: (\sqrt{2}\: \text{ed}) = \sqrt{2}\phi\: \text{ed} = \Phi_j\: \text{ed}\\
\text{where biot} = \sqrt{2}\: \text{ed and}\: \sqrt{2}\: \phi = \Phi_j
\tag{d}
\end{matrix}

Therefore in electrodynamic (by comparing with equation (1)):

##j## (ed) = ##\Phi_j## (ed)

(c) While converting to electrostatic units:

\begin{matrix}
i\: \text{biot} = i\: (3\times10^{10}\: \text{StatAmp}) = 3\times10^{10}i\: \text{StatAmp} = k\: \text{StatAmp}\\
\text{where biot} = 3\times10^{10}\: \text{StatAmp and}\: 3\times10^{10}\: i = k
\tag{e}
\end{matrix}

\begin{matrix}
\phi\: \text{biot} = \phi\: (3\times10^{10}\: \text{StatAmp}) = 3\times10^{10}\phi\: \text{StatAmp} = \Phi_k\: \text{StatAmp}\\
\text{where biot} = 3\times10^{10}\: \text{StatAmp and}\: 3\times10^{10}\: \phi = \Phi_k
\tag{f}
\end{matrix}

Therefore in electrostatic units (by comparing with equation (1)):

##k## (StatAmp) = ##\Phi_k## (StatAmp)

i.e. in all units, strength of the shell and strength of current flowing through its boundary are same.

Question 1: Am I correct?

if yes

Question 2: Then why is it said that "strength of the shell and strength of current flowing through its boundary are same only in certain units like electromagnetic units"?

## Answers and Replies

Dale
Mentor
2020 Award
Well, 1 biot is not really equal to 10 A. They are equivalent in some equations but not in other equations. The issue is that Maxwell's equations and the other equations of electromagnetism are different when written in different units. So the straight substitution that you are trying to do is not correct.

1 biot^2 equals 1 dyne in emu, but 1 A^2 does not equal 1 N in SI