# Can we apply the equivalence principle in electromagnetics to multiple regions simultaneously?

Note: I posted the same question on the electrical engineering stack exchange with no luck so far, possibly because it's quite theoretical, so I'm posting it here again. I'm not sure if it's considered bad practice to this; if so, I apologize.

Setup 1

An object $$(\epsilon, \mu)$$ occupies volume $$V_1$$ and is enclosed by surface $$S_1$$. It sits in vacuum $$(\epsilon_0, \mu_0)$$, denoted by $$V_0$$. In $$V_0$$ there is an electric field $$\vec{E}_\mathrm{inc}$$ which impinges on said object. As a result, a set of fields $$(\vec{E}_1, \vec{H}_1)$$ is established inside $$V_1$$.

We know that we can introduce equivalent current sources on $$S_1$$, which generate the correct fields $$(\vec{E}_1, \vec{H}_1)$$ inside $$V_1$$, but exactly cancel ("extinct") the field outside, so that the total field in $$V_0 \backslash V_1$$ is $$0$$.

Setup 2

Now suppose that in addition to the first object, we introduce another object which occupies volume $$V_2$$ and is bounded by surface $$S_2$$, and has the same material properties as the first object $$(\epsilon, \mu)$$.

I know that the equivalence principle can be applied individually to each object one at a time, to study the fields generated inside them.

However, does anything prevent me from treating these objects as one, i.e. $$V_1 \cup V_2$$ and $$S_1 \cup S_2$$? In other words, can I simultaneously introduce equivalent currents which exist on $$S_1$$ and equivalent currents that exist on $$S_2$$, which collectively generate the correct fields inside both objects, and collectively generate a $$0$$ field in $$V_0 \backslash (V_1 \cup V_2)$$?

I couldn't find any clear references to such a situation online; any insight would be appreciated. Thanks!

• If you use the equivalent currents simultaneously generated by objects V1 and V2 then they will work as you described. What is wrong is to use the current on V1 when V1 is by itself and use the current on V2 when V2 is by itself, and then apply those currents together because there is an interaction between them that are ignored. – hyportnex Aug 23 '19 at 21:02
• That's helpful, thanks, @hyportnex . Do you know of any references / papers where such a situation is encountered or used somehow? All textbooks I've looked in just treat the usual case of a single object. – EM_IE Aug 23 '19 at 22:53
• I have only studied the case of scattering off metal objects, i.e., external reflections. In this context you may see any good book on radar scattering (say, a radar is looking at two airplanes flying next to each other, or two cars in neighboring lanes, etc.) An equally important and difficult problem is scattering off a phased array antenna where the interaction between neighboring elements is taken into account. – hyportnex Aug 23 '19 at 23:31
• Thanks. You bring up a good point that maybe I should study the PEC case first and then extend the arguments to penetrable objects. In the PEC case with setup 2, the "simultaneous" and "individual" situations should probably lead to the same equivalent currents, I suspect. – EM_IE Aug 23 '19 at 23:57