User:Vilnius/Maxwell-Lodge effect

The Maxwell-Lodge effect is a phenomenon of elctromagnetic induction in which an electric charge, near a solenoid which field change slowly, feels an electromotive force (e.m.f.) even if the magnetic field is practically static inside and null outside. It can be consider an analogous of the classic Aharonov-Bohm effect, where instead the field is exactly static inside and null outside.

It appears in literature with this name in an article of 2008, referring to a previous writing of the 1889 by the physicist Oliver Lodge.

Description
Considering a infinite solenoid (ideal solenoid) with n coils per length unit, over which a current $$I(t)$$ flows, the magnetic field inside it is


 * (1) $$\mathbf B = \mu n I(t)$$

while outside is null.

From the second and the third Maxwell's equations


 * $$\begin{cases}

\nabla \times \mathbf{E} &= -\dfrac{\partial\mathbf B}{\partial t} \\ \nabla \cdot \mathbf{B} &= 0 \end{cases}$$

and from definitions of magnetic potential and electric potential stems:


 * $$\mathbf E = - \mathbf \nabla \phi - \frac{\partial \mathbf A}{\partial t}$$

that without electric charges reduces to


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

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 $$\mathbf B$$ through the surface having the above curve as its edge, i.e.


 * $$\int_{S} \mathbf B \cdot d \mathbf S = \int_S \nabla \times \mathbf A \cdot d \mathbf S = \oint_{l} \mathbf A \cdot d\mathbf l$$,

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


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

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

On the other hand, if the current changes, also $$\mathbf B$$must 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 $$\mathbf B$$, the only possible cause of the e.m.f. is $$\mathbf A(r)$$.

Interpretation
Bearing in mind that the concept of field has been introduced into physics to ensure that actions on objects are always local, i.e. by contact (direct and mediated by a field) and not by remote action, as Albert Einstein feared in the EPR paradox, the result of the Maxwell-Lodge effect, like the Aharonov-Bohom effect, seems contradictory. In fact, even though the magnetic field is zero outside the solenoid and the electromegnetic radiation is negligible, a test charge experiments with the presence of an electric field.

The question arises as to what the information on the presence of the magnetic field from inside the solenoid leads to the electric charge.

From the calculations it seems evident that the source is the potential vector $$\mathbf A$$, traditionally considered as a simple support to the calculation, to the disappointment of the physicist Richard Feynmann.