- #1

- 18

- 0

## Main Question or Discussion Point

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{equation}

\begin{matrix}

\text{i.e.}\: i \text{(biot)} = \phi \text{(biot) }

\end{matrix}

\tag{1}

\end{equation}

\begin{equation}

\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}

\end{equation}

Also:

\begin{equation}

\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}

\end{equation}

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

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

\begin{equation}

\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}

\end{equation}

Also:

\begin{equation}

\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}

\end{equation}

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

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

\begin{equation}

\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}

\end{equation}

\begin{equation}

\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}

\end{equation}

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.

if yes

\begin{equation}

\begin{matrix}

\text{i.e.}\: i \text{(biot)} = \phi \text{(biot) }

\end{matrix}

\tag{1}

\end{equation}

**(a) While converting to SI:**\begin{equation}

\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}

\end{equation}

Also:

\begin{equation}

\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}

\end{equation}

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

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

**(b) While converting to electrodynamic units:**\begin{equation}

\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}

\end{equation}

Also:

\begin{equation}

\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}

\end{equation}

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

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

**(c) While converting to electrostatic units:**\begin{equation}

\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}

\end{equation}

\begin{equation}

\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}

\end{equation}

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"?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"?