Maxwell–Lodge effect

The term appeared in the scientific literature in a 2008 article, referring to an article of 1889 by physicist Oliver Lodge.

Description

Consider an infinite solenoid (ideal solenoid) with n turns per length unit, through which a current ${\displaystyle I(t)}$ flows. The magnetic field inside the solenoid is,

${\displaystyle \mathbf{B}=\mu nI(t)}$ (1)

while the field outside the solenoid is null.

From the second and third Maxwell's equations,

${\displaystyle \{\begin{array}{ll}\mathrm{\nabla }\times \mathbf{E}& =-{\displaystyle \frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}}\\ \mathrm{\nabla }\cdot \mathbf{B}& =0\end{array}}$

and from definitions of magnetic potential and electric potential stems:

${\displaystyle \mathbf{E}=-\mathrm{\nabla }\varphi -\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}}$

that without electric charges reduces to

${\displaystyle \mathbf{E}=-\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}}$ (2)

Resuming the original definition of Maxwell on the potential vector, according to which is a vector that its circuitation along a closed curve is equal to the flow of ${\displaystyle \mathbf{B}}$ through the surface having the above curve as its edge, i.e.

${\displaystyle {\int }_S\mathbf{B}\cdot d\mathbf{S}={\int }_S\mathrm{\nabla }\times \mathbf{A}\cdot d\mathbf{S}={\oint }_l\mathbf{A}\cdot d\mathbf{l}}$,

we can calculate the induced e.m.f., as Lodge did in his 1889 article, considering ${\displaystyle l}$ the closed line around the solenoid, or convenience a circumference, and ${\displaystyle S}$ the surface having ${\displaystyle l}$ as border. Assuming ${\displaystyle a}$ the radius of the solenoid and ${\displaystyle r>a}$ the radius of ${\displaystyle l}$, the surface crossing it is subjected to a magnetic flux ${\displaystyle \pi a^2\mathbf{B}}$ which is equal to circuitation ${\displaystyle l}$: ${\displaystyle C(l)=2\pi r\mathbf{A}(r)}$. From that stems

${\displaystyle \mathbf{A}(r)=\frac{1}{2}{a}^2\frac{\mathbf{B}}{r}}$.

From (2) we have that the e.m.f. is null for ${\displaystyle \mathbf{B}}$ constant, which means, due to (1), at constant current.

On the other hand, if the current changes, ${\displaystyle \mathbf{B}}$ must also change, producing electromagnetic waves in the surrounding space that can induce an e.m.f. outside the solenoid.

But if the current changes very slowly, one finds oneself in an almost stationary situation in which the radiative effects are negligible and therefore, excluding ${\displaystyle \mathbf{B}}$, the only possible cause of the e.m.f. is ${\displaystyle \mathbf{A}(r)}$.

It is possible to make calculations without referring to the field ${\displaystyle \mathbf{A}(r)}$. Indeed, in the framework of Maxwell equations as written above:

${\displaystyle {\int }_l\mathbf{E}\cdot dl=-\frac{d}{dt}{\int }_S\mathbf{B}\cdot dS=-\frac{d}{dt}{\int }_{\pi a^2}\mathbf{B}\cdot dS}$,

being ${\displaystyle \mathbf{B}}$ negligible outside the solenoid. Thus

${\displaystyle \mathbf{E}2\pi r=-\pi a^2\frac{d\mathbf{B}}{dt}\Rightarrow \mathbf{E}=-\frac{a^2}{2r}\frac{d\mathbf{B}}{dt}}$.

This doesn't avoid the problem that ${\displaystyle \mathbf{B}}$ is practically null in places where the e.m.f. manifests itself.