Mutual energy

The concept of the mutual energy is consist of mutual energy formula, mutual energy theorem, mutual energy flow, Inductance.

What is the mutual energy

Today we begin to charge cell phone with transmitting antenna. The cell phone has a receiving antenna. It is clear there are energy transferred from the transmitting antenna to the receiving antenna.

Assume we have a transformator. There are energy transferred from primary coil to the secondary coil since there are mutual inductance. If we bring the secondary coil to another place which has a big distance to the primary coil, there still can receive some energy from the primary coil. The received energy is very weak. In this case, the primary coil becomes the transmitting antenna, the secondary coil becomes the receiving antenna. The energy sending from the transmitting antenna to the receiving antenna becomes the mutual energy. The energy flow from the transmitting antenna to the receiving antenna becomes the mutual energy flow.

Hence, mutual energy is the part of the energy received by the receiving antenna from the transmitting antenna. In the photon situation, the emitter is a small transmitting antenna stayed inside the atom. The absorber is a small receiving antenna inside the atom. The energy transferred from emitter to the absorber is the photon energy, this energy can be described as the mutual energy. The energy flow between the emitter to the absorber can be described as the mutual energy flow, hence the mutual energy flow is the photon's energy flow.

Difference between the mutual energy flow and the Poynting vector

When we speaking energy, energy flow we first we think the Poynting vector and Poynting theorem. However there are also mutual energy theorem which are used to describe the energy and energy flow between the two antenna. Comparing to the Poynting theorem, the mutual energy theorem is more accurately to describe the energy flow from transmitting antenna to the receiving antenna.

We all know for antenna, there are concept called "effective section area" and "section area". The section area is the section area of the wire of the antenna. If we use the section area times the Poynting vector to calculate the energy received by the receiving antenna, we will find we have get a energy which is too small to compare the real energy received by the receiving antenna. Hence we normally increase the section area a few times or even 100 times to the effective section area. If we use effective section area to calculate the energy, we can obtain a correct value. This tells us the Poynting vector and Poynting theorem is not really suitable to calculate the energy between the receiving antenna to the receiving antenna.

In other side, since there are is the mutual energy theorem to guarantee the energy received by the receiving antenna is exactly equal to the part of energy sends from the transmitting antenna to the receiving antenna. The mutual energy and mutual energy flow theorem is suitable to calculate the energy transferred from transmitting antenna to the receiving atenna.

Mutual energy flow is a part of energy flow corresponding to the Poynting energy flow of the superposed fields of the transmitting antenna and the receiving antenna.

Introduction of mutual energy

Poynting vector can be write as following,

$𝐒=𝐄\times 𝐇$

where $𝐄$ is electric field. $𝐇$ is magnetic H-field. Point vector are energy intensity of the energy flow of the electromagnetic field. Assume that,

$𝐄={𝐄}_1+{𝐄}_2$

$𝐇={𝐇}_1+{𝐇}_2$

Hence we have,

$𝐄\mathbf{\times }𝐇=({𝐄}_1+{𝐄}_2)\times ({𝐇}_1+{𝐇}_2)$

$={𝐄}_1\times {𝐇}_1+({𝐄}_1\times {𝐇}_2+{𝐄}_2\times {𝐇}_1)+{𝐄}_2\times {𝐇}_2$

Hence we can define

${𝐄}_1\times {𝐇}_1$

${𝐄}_2\times {𝐇}_2$

as self-energy items of the Poynting vector. And define

$({𝐄}_1\times {𝐇}_2+{𝐄}_2\times {𝐇}_1)$

as the mutual energy items of the Poynting vector.

Lorentz reciprocity theorem

$\underset{V1}{\int }({𝐄}_2(\omega )\cdot {𝐉}_1(\omega ))dV=\underset{V2}{\int }({𝐄}_1(\omega )\cdot {𝐉}_2(\omega ))dV$

Where $\omega$ is frequency.

Action-at-a-distance

The theory of action-at-a-distance are introduced by （K. Schwarzschild） （H. Tetrode） （A.D. Fokker）. According to this theory, an electric current will produce two electromagnetic potentials or two electromagnetic waves: one is the retarded wave, another is advanced wave. The emitter can send the retarded wave, but in the same time it also sends an advanced wave. The absorber can send the advanced wave, but in the same time it also sends a retarded wave. According to this theory, the sun cannot send the radiation wave out, if it stayed alone in the empty space. Infinite absorbers are the reason that the sun can radiate its light. The action can be written as following,

$S=-\sum _i{m}_{i}c\int (\frac{d{x}_{i\mu }}{d{\tau }_i}\frac{d{x}_i^{\mu }}{d{\tau }_i}{)}^{\frac{1}{2}}d{\tau }_i-\sum _i\sum _{j<i}\frac{e_i{e}_j}{c}\int \int \delta (s_{ij}^2)\frac{d{x}_{i\mu }}{d{\tau }_i}\frac{d{x}_j^{\mu }}{d{\tau }_j}d{\tau }_{i}d{\tau }_j=\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{u}\mathrm{m}$

where

$s_{ij}^2=(x_{i\mu }-x_{j\mu })(x_i^{\mu }-x_j^{\mu })$

$ds=c^{2}d{t}^2-d{x}_1^2-d{x}_2^2-d{x}_3^2$

Absorber theory

The absorber theory is introduced by Wheeler and Feynman （J. A. Wheeler） （J. A. Wheeler）. The absorber theory is build on the top of the above theory of the action-at-a-distance （A.D. Fokker） （K. Schwarzschild） （H. Tetrode） . A cording to this theory, electromagnetic field has no its own freedom. The electromagnetic field is adjective field. It is only a bookkeeper for the action or reaction between at least two charges. That means without a test charge or absorber, only the emitter alone can not produce the radiation. Absorber theory try to offer a better explanation to the recoil force of an accelerated or decelerated charge in empty space. The recoil force has been introduced by Dirac （P. A. M. Dirac）. But Wheeler and Feynman do not satisfy that Dirac did not offer a reasonable reason of that formula. Wheeler and Feynman try to use the absorbers stayed on the infinite big sphere to explain the formula given by Dirac. The absorber theory also emphases the importance of the absorber in the radiation process.

Transactional interpretation for quantum mechanics

The transactional interpretation of quantum mechanics introduced by John Cramer (John Cramer). The transactional interpretation is build on the top of Wheeler–Feynman absorber theory. In this theory, the emitter can send an offer wave to the absorber, when the absorber receive the offering wave, it can send a confirmation wave to the emitter. These two waves can have a handshake. This handshake process is the transactional process. In this process the photon or other particle is produced. The confirmation wave is advanced wave.

Mutual energy theorem

Welch's reciprocity theorem

$-\underset{t=-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{V2}{\int }{𝐄}_2(t)\cdot {𝐉}_1(t)dVdt=\underset{t=-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{V1}{\int }{𝐄}_1(t)\cdot {𝐉}_2(t)dVdt$

where $t$ is time. This theorem is introduced by W. J. Welch in 1960 （W. J. Welch）. In the Welch's reciprocity, the 2 fields one is retarded wave, the another is advanced wave.

Conjugate transform

It is not clear who first introduced the concept of the conjugate transform, but the details theory of the conjugate transform can be found in （Jin Au Kong）. It is important that if a field satisfies the Maxwell equations, after the conjugate transform, it still satisfies the Maxwell equations. If the original field is retarded wave, after the transform it becomes advanced wave. Vice Versa, if the original field is advanced wave, after the transform it becomes the retarded wave.

$ℂ(𝐄(t),𝐇(t),𝐉(t),𝐊(t),𝝐(t),𝝁(t))=(𝐄(-t),-𝐇(-t),-𝐉(-t),𝐊(-t),𝝐(-t),𝝁(-t))$

or

$ℂ(𝐄(\omega ),𝐇(\omega ),𝐉(\omega ),𝐊(\omega ),𝝐(\omega ),𝝁(\omega ))=({𝐄}^{*}(\omega ),-{𝐇}^{\mathbf{*}}(\omega ),-{𝐉}^{\mathbf{*}}(\omega ),{𝐊}^{*}(\omega ),{𝝐}^{\mathit{*}}(\omega ),{𝝁}^{\mathbf{*}}(\omega ))$

Where $ℂ$ is the conjugate transform. $𝐄$ is electric field. $𝐇$ Magnetic field. $𝐉$ current intensity. $𝐊$ magnetic current intensity. $𝝐$ is permittivity, $𝝁$ is permeability, $t$ is time, $\omega$ is frequency.

Rumsey's reciprocity theorem

V.H. Rumsey has introduced his summarize the Lorentz reciprocity theorem as "action and reaction". He has apply the complex conjugate transform to the his "action and reaction" theorem and obtained a new reciprocity theorem （V.H. Rumsey）,

$-\underset{V1}{\int }{𝐄}_2^{*}(\omega )\cdot {𝐉}_1(\omega )dVdt=\underset{V2}{\int }{𝐄}_1(\omega )\cdot {𝐉}_2^{*}(\omega )dV$

Rumsey's reciprocity theorem was derived by apply the conjugate transform to the Lorentz reciprocity theorem.

Inner product space for the electromagnetic fields on a closed Surface

Shuang-ren Zhao has defined the inner product for any two electromagnetic fields which are (Shuang-ren Zhao),

$({\xi }_1,{\xi }_2{)}_{{\Gamma }_s}=\underset{{\Gamma }_s}{\iint }({𝐄}_1(\omega )\times {𝐇}_2^{*}(\omega )+{𝐄}_2^{*}(\omega )\times {𝐇}_1(\omega ))\cdot \hat{n}d\Gamma$

where $\xi =[𝑬,𝑯]$, $\tau =[𝑱,𝑲]$, Shuang-ren Zhao has proved that the above inner products, satisfy the Inner product space 3 definitions. If ${\tau }_2$ is taken as a unit vector of ether current ${𝑱}_2$ or ${𝑲}_2$, the field ${\xi }_1$ can be calculated ether on the original source ${𝑱}_1$ or on the surface ${\Gamma }_s$. ${\Gamma }_s$ is any surface outside the two volumes $V_1$ and $V_2$.

$\hat{n}$ is a unit surface normal vector. Shuang-ren Zhao has proved that this kind of inner product satisfy inner product space 3 conditions,

• Conjugate symmetry:

$({\xi }_1,{\xi }_2)=({\xi }_2,{\xi }_1{)}^{*}$

• Linearity in the first argument:

$\begin{array}{cc}(a{\xi }_1,{\xi }_2)& =a({\xi }_1,{\xi }_2)\\ ({\xi }_1+{\xi }_2,{\xi }_3)& =({\xi }_1,{\xi }_3)+({\xi }_2,{\xi }_3)\end{array}$

• Positive-definiteness:

$\begin{array}{cc}(\xi ,\xi )& \ge 0\\ (\xi ,\xi )& =0\iff x=\mathrm{𝟎}.\end{array}$

According to this theory that the inner product of a retarded wave ${\xi }_1$ and an advanced wave ${\xi }_2$ vanish, if the sources of the two wave are inside the surface ${\Gamma }_s$, i.e.,

$({\xi }_1,{\xi }_2{)}_{{\Gamma }_s}=0$

where ${\tau }_1,{\tau }_2\in V$ are the source of ${\xi }_1,{\xi }_2$. ${\Gamma }_s$ is the boundary surface of the volume $V$

The inner product can also be defined in a close surface or open surface $\Gamma$ which stays between the volume $V_1$ and $V_2$. In this case the inner product of a retarded wave send from ${\tau }_1$ and the advanced wave ${\xi }_2$ send from ${\tau }_2$ are not zero.

$({\xi }_1,{\xi }_2{)}_{{\Gamma }_1}=({\xi }_1,{\xi }_2{)}_{\Gamma }=({\xi }_1,{\xi }_2{)}_{{\Gamma }_2}\ne 0$

Where ${\Gamma }_1$ and ${\Gamma }_2$ are similar surface like $\Gamma$.

The mutual energy theorem

Shuang-ren Zhao has introduced the mutual energy theorem (Shuang-ren Zhao) in early of 1987.

$-({\tau }_1,{\xi }_2{)}_{V_1}=({\xi }_1,{\tau }_2{)}_{V_2}$

$({\tau }_1,{\xi }_2{)}_{V_1}=\underset{V1}{\int }({𝑱}_1(\omega )\cdot {𝑬}_2^{*}(\omega )+{𝑲}_1(\omega )\cdot {𝑯}_2^{*}(\omega ))dV$

$({\xi }_1,{\tau }_2{)}_{V_2}=\underset{V2}{\int }({𝑬}_1(\omega )\cdot {𝑱}_2^{*}(\omega )+{𝑯}_1(\omega )\cdot {𝑲}_2^{*}(\omega ))dV$

The mutual energy theorem is similar to Rumasy's reciprocity theorem, but Shuang-ren Zhao thought this is an energy theorem instead some kind of reciprocity theorem. Since this theorem can be applied to a system with a transmitting antenna and receiving antenna. Shuang-ren Zhao believe this theorem tell us that the energy received by the receiving antenna is equal to the part of energy send from the transmitting antenna to the receiving antenna.

The time-domain cross-correlation reciprocity theorem

Adrianus T. de Hoop published the time-domain cross-correlation reciprocity theorem in the end of 1987 （Adrianus T. de Hoop） which can be seen as following,

$-\underset{t=-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{V2}{\int }{𝐄}_2(t+\tau )\cdot {𝐉}_1(t)dVdt=\underset{t=-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{V1}{\int }{𝐄}_1(t)\cdot {𝐉}_2(t+\tau )dVdt$

Huygens–Fresnel principle

Shuang-ren Zhao emphases that the mutual energy theorem is an energy theorem instead of some kind of reciprocity theorem. The theorem described an energy in the space. This theorem can be seen as Huygens–Fresnel principle (Shuang-ren Zhao), which can be written as,

$-({\tau }_1,{\xi }_2{)}_{V_1}=({\xi }_1,{\xi }_2{)}_{\Gamma }=({\xi }_1,{\tau }_2{)}_{V_2}$

where

$({\xi }_1,{\xi }_2{)}_{\Gamma }=\underset{\Gamma }{\iint }({𝐄}_1(\omega )\times {𝐇}_2^{*}(\omega )+{𝐄}_2^{*}(\omega )\times {𝐇}_1(\omega ))\cdot \hat{n}d\Gamma$

$\Gamma$ is any close surface or infinite big surface separating $V_1$ and $V_2$. We take the direction of $\hat{n}$ is from $V_1$ to $V_2$.

Assume ${𝐉}_2=\delta (x-x^{\prime })\widehat{𝐦}$

${𝐄}_1\cdot \widehat{𝐦}=({\xi }_1,{\xi }_2{)}_{\Gamma }$

Assume ${𝐊}_2=\delta (x-x^{\prime })\widehat{𝐦}$

${𝐇}_1\cdot \widehat{𝐦}=({\xi }_1,{\xi }_2{)}_{\Gamma }$

The forgot second Lorentz reciprocity theorem

I. V. Petrusenko introduced the forgot second Lorentz reciprocity theorem in 2009 （I. V. Petrusenko）. These theorems have been rediscovered for many times later, this shows they are very important.

The relation of the these theorems

It is not difficult to prove that the mutual energy theorem （Shuang-ren Zhao） and the time-domain cross-correlation reciprocity theorem（Adrianus T. de Hoop） are same theorem connected by Fourier transform, one is in the Fourier frequency-domain, another is in time-domain. The method of this mutual energy theorem is similar to （V.H. Rumsey） The reciprocity theory in arbitrary time-domain （W. J. Welch） is a special case where $\tau =0$ of the time-domain cross-correlation reciprocity theorem（Adrianus T. de Hoop） The forgot second Lorentz reciprocity theorem （I. V. Petrusenko） is also same to （V.H. Rumsey） Hence, All the above theorems can be see as one theorem. The same mathematical formula has two major applications (1) is used as reciprocity for example to find the directivity diagram of the receiving antenna from the directivity diagram of the transmitting antenna, in this case this formula can be referred as a reciprocity theorem; (2) to find the energy transfer between the transmitting antenna and the receiving antenna, then the same formula can be referred as the mutual energy theorem.

If this theorem is applied as the reciprocity theorem, doesn't mater there is advanced wave, since in the reciprocity theorem, it is allowed among the two fields one is a real field another is a virtual filed. It will be no problem that a virtual field to be an advanced field. However, when this theorem is applied as an energy theorem that need the two fields are all real in physics. The advanced field must be allowed in physics.

The difference between the Lorentz reciprocity theorem and the mutual energy theorem/time-domain cross-correlation reciprocity theorem

In Lorentz reciprocity theorem the two fields are all retarded fields. In the mutual energy theorem or cross-correlation time-domain reciprocity theorem, the two fields are one is retarded wave sending from the transmitting antenna, another is the advanced wave sending from the receiving antenna.

The mutual energy theorem or time-domain cross-correlation reciprocity theorem are energy theorem. It describe the energy relation of the two antenna. It is also can be applied as some kind of reciprocity theroem

The Lorentz reciprocity theorem reciprocity theorem is a mathematical theorem which transform of one filed in the mutual energy theorem/time-domain cross-correlation reciprocity theorem. Lorentz reciprocity theorem can be applied to find the directivity diagram actually is also because there is the mutual energy theorem stand on the back of it. However most electronic engineer used Lorentz reciprocity theorem more because there the advanced wave is exclude.

Mutual energy flow

The mutual energy flow theorem is same as the Huygens–Fresnel principle (Shuang-ren Zhao) in mathematics. However Shuang-ren Zhao realized there is an energy flow transferred from the transmitting antenna to the receiving antenna. Hence give this formula a new name, mutual energy flow theorem,

$-({\tau }_1,{\xi }_2{)}_{V_1}=({\xi }_1,{\xi }_2{)}_{\Gamma }=({\xi }_1,{\tau }_2{)}_{V_2}$

The inner product of the mutual energy flow theorem can also be defined in the time domain, hence there is,

$({\xi }_1,{\xi }_2{)}_{\Gamma }=\underset{t=-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{\Gamma }{\iint }({𝐄}_1(t)\times {𝐇}_2(t+\tau )+{𝐄}_2(t+\tau )\times {𝐇}_1(t))\cdot \hat{n}d\Gamma dt$

Which is the energy go through the surface $\Gamma$. $\Gamma$ is any surface between $V_1$ and $V_2$. $\Gamma$ can be close surface for example the outside surface of $V_1$, or a open surface for example is a infinite plane between $V_1$ and $V_2$.

$({\tau }_1,{\xi }_2{)}_{V_1}=\underset{t=-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{V1}{\int }({𝑱}_1(t)\cdot {𝑬}_2(t+\tau )+{𝑲}_1(t)\cdot {𝑯}_2(t+\tau ))dVdt$

$({\xi }_1,{\tau }_2{)}_{V_2}=\underset{t=-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{V2}{\int }({𝑬}_1(t)\cdot {𝑱}_2(t+\tau )+𝑯(t)\cdot {𝑲}_2(t+\tau ))dVdt$

$Q(\tau )=\underset{\Gamma }{\iint }({𝐄}_1(t)\times {𝐇}_2(t+\tau )+{𝐄}_2(t+\tau )\times {𝐇}_1(t))\cdot \hat{n}d\Gamma$

Normally there is $\tau =0$. $Q(\tau =0)$ can be defined as the mutual energy flow going through the surface $\Gamma$. $({\xi }_1,{\xi }_2{)}_{\Gamma }(\tau =0)$ is the whole energy go through the surface $\Gamma$ between the time $t=-\mathrm{\infty }$ and $t=+\mathrm{\infty }$.

The application of the mutual energy theorem

The wave expansion

Assume there are a complete set of the electromagnetic fields ${\xi }_0$,${\xi }_1,{\xi }_2,{\xi }_3,\mathrm{...}$. Assume these electromagnetic fields are normalized, i.e.,

$({\xi }_i,{\xi }_j)={\delta }_{ij}$

Then any electromagnetic field can be expanded as,

$\xi =\sum _{i=0}^N{a}_i{\xi }_i$

where

$(\xi ,{\xi }_j)=(\sum _{i=0}^N{a}_i{\xi }_i,{\xi }_j)=a_j$

Hence, we have

$\xi =\sum _{i=0}^N(\xi ,{\xi }_i){\xi }_i$

Spherical waves expansion

Shuang-ren Zhao proposed the spherical wave expansion method . The spherical wave expansion can be written as, ${\xi }_{nm}=[M_{nm},\frac{j}{\eta }{N}_{nm}]$ ${\eta }_{nm}=[N_{nm},\frac{j}{\eta }{M}_{nm}]$

Where the factor, $\frac{j}{\eta }$ make the second term is just a magnetic field, when the first term in the square brackets of the above formula is an electric field. $j=\sqrt{-1}$. where

$M_{nm}=\mathrm{\nabla }\times [g_{n}r{h}_n(kr)Y_n^m(\theta ,\varphi )]$

$N_{nm}=\frac{1}{k}\mathrm{\nabla }\times M_{nm}$

$h_n$ is class $n$ and second kind of the Hankel function， $k=\omega \sqrt{ϵ\mu }$

$\eta =\sqrt{\frac{\mu }{ϵ}}$

$r,\theta ,\varphi$ are coordinates of sphere.The corresponding unit vector are $r,\theta ,\varphi$

$Y_n^m(\theta ,\varphi )=[(2n+1)\frac{(n-m)!}{(n+m)!}{]}^{\frac{1}{2}}{P}_n^m(cos\theta )\exp (jm\varphi )$

$P_n^m$ is Legendre function， $n=0,1,\mathrm{...}$ $m=0,\pm 1,\mathrm{......}\pm n$

The formula of the spherical wave expansion can be written as,

$\xi =\sum _{nm}(a_{nm}{\xi }_{nm}+b_{nm}{\eta }_{nm})$

$a_{nm}=(\xi ,{\xi }_{nm}),b_{nm}=(\xi ,{\eta }_{nm})$

The mutual energy theorem is also a reciprocity theorem

The same theorem Zhao calls it the mutual energy theorem, but Welch and de Hoop does call it the time domain reciprocity theorem. In fact, this formula can be used as a reciprocity theorem. That is, the conjugate transformation is performed on all fields in the mutual energy theorem.

$-(ℂ{\tau }_1,ℂ{\xi }_2)=(ℂ{\xi }_1,ℂ{\xi }_2)=(ℂ{\tau }_1,ℂ{\xi }_2)$

we obtain，

$({\xi }_2,{\tau }_1)=({\xi }_1,{\xi }_2)=-({\xi }_2,{\tau }_1)$

Hence, in the original mutual energy theorem ${\tau }_1$ is emitter, ${\xi }_1$ is the retarded wave, ${\tau }_2$ is the absorber, ${\xi }_2$ is advanced wave. After the transform, ${\tau }_2$ becomes the emitter, ${\xi }_2$ becomes the retarded wave, ${\tau }_1$ becomes the absorber, ${\xi }_1$ becomes the advanced wave.

According to this reciprocity theorem, the directivity diagram of the receiving antenna is the same as that of the directivity diagram of the transmitting antenna. The pattern of the same electric charge where it is absorbed is the same as the pattern when it is radiated.

The calculation of the radiation and the absorption of the antenna

Assume the current element of the antenna 1 is ${𝐉}_1$, Which produce the retarded wave, $[{𝐄}_1,{𝐇}_1]$，

Assume the current element of the antenna 2 is ${𝐉}_2$ produce the advanced wave, $[{𝐄}_2,{𝐇}_2]$，The mutual energy theorem,

$-\underset{V2}{\int }{𝐄}_2^{*}\cdot {𝐉}_{1}dVdt=\underset{V1}{\int }{𝐄}_1\cdot {𝐉}_2^{*}dVdt$

tell us, the advanced wave of the receiving antenna ${𝐄}_2$ sucks the power form ${𝐉}_1$. And this power is same as the retarded wave ${𝐄}_1$ offers the power to the receiving antenna ${𝐉}_2$

We know that the directivity of the transmitter antenna is easier to be calculated but is more difficult to be measured. In contrast, the directivity diagram of the receiving antenna is not easier to be calculated, but is easier to be measured. Apply this theorem, we can apply the calculated directivity diagram of the radiation as the directivity diagram of the receiving antenna. And also can apply the measured directivity diagram of the receiving antenna as the directivity diagram of the radiation.

Theorems related to the mutual energy

Poynting theorem

$-\mathrm{\nabla }\cdot (𝐄\times 𝐇\mathbf{)}=𝐄\cdot \frac{\partial 𝐃}{\partial t}+𝐇\cdot \frac{\partial 𝐁}{\partial t}+𝐉\cdot 𝐄$

Where

$𝐒=𝐄\times 𝐇$

is the Poynting vector.

$u=\frac{1}{2}(𝐄\cdot 𝐃+𝐇\cdot 𝐁)$ is the energy of electric field.

Superposition principle

$𝐄=\sum _{i=1}^N{𝐄}_i$

$𝐇=\sum _{i=1}^N{𝐇}_i$

Assume ${𝐄}_i$ and ${𝐇}_i$ are the electromagnetic field of $i$-th charge. Assume that $i$-th charge accelerates or decelerates.

Poynting theorem of $N$ charges

Substitute superposition principle to Poynting theorem

$-\mathrm{\nabla }\cdot \sum _{i=1}^N\sum _{j=1}^N({𝐄}_i\times {𝐇}_j)=\sum _{i=1}^N\sum _{j=1}^N({𝐄}_i\cdot \frac{\partial {𝐃}_j}{\partial t}+{𝐇}_i\cdot \frac{\partial {𝐁}_j}{\partial t})+\sum _{i=1}^N\sum _{j=1}^N({𝐉}_i\cdot {𝐄}_j)$

Self energy items of the Poynting theorem

$-\mathrm{\nabla }\cdot \sum _{i=1}^N({𝐄}_i\times {𝐇}_i)=\sum _{i=1}^N({𝐄}_i\cdot \frac{\partial {𝐃}_i}{\partial t}+{𝐇}_i\cdot \frac{\partial {𝐁}_i}{\partial t})+\sum _{i=1}^N({𝐉}_i\cdot {𝐄}_i)$

Mutual energy items of the Poynting theorem for $N$ charges

Subtract the self-energy items from the Poynting theorem of $N$ charges, the mutual energy items of the Poynting theorem can be obtain

$-\mathrm{\nabla }\cdot \sum _{i=1}^N\sum _{j=1,j\ne i}^N({𝐄}_i\times {𝐇}_j)=\sum _{i=1}^N\sum _{j=1,j\ne i}^N({𝐄}_i\cdot \frac{\partial {𝐃}_j}{\partial t}+{𝐇}_i\cdot \frac{\partial {𝐁}_j}{\partial t})+\sum _{i=1}^N\sum _{j=1,j\ne i}^N({𝐉}_i\cdot {𝐄}_j)$

This can be referred as mutual energy formula.

Mutual energy items of the Poynting theorem for $2$ charges

Subtract the self-energy items from the Poynting theorem of $N$ charges, the mutual energy items of the Poynting theorem can be obtain

$-\mathrm{\nabla }\cdot \sum _{i=1}^2\sum _{j=1,j\ne i}^2({𝐄}_i\times {𝐇}_j)=\sum _{i=1}^2\sum _{j=1,j\ne i}^2({𝐄}_i\cdot \frac{\partial {𝐃}_j}{\partial t}+{𝐇}_i\cdot \frac{\partial {𝐁}_j}{\partial t})+\sum _{i=1}^2\sum _{j=1,j\ne i}^2({𝐉}_i\cdot {𝐄}_j)$

Or

$-\mathrm{\nabla }\cdot \sum _{i=1}^2\sum _{j=1}^{j<i}({𝐄}_i\times {𝐇}_j+{𝐄}_j\times {𝐇}_i)=\sum _{i=1}^2\sum _{j=1}^{j<i}({𝐄}_i\cdot \frac{\partial {𝐃}_j}{\partial t}+{𝐇}_i\cdot \frac{\partial {𝐁}_j}{\partial t}+{𝐄}_j\cdot \frac{\partial {𝐃}_i}{\partial t}+{𝐇}_j\cdot \frac{\partial {𝐁}_i}{\partial t})+\sum _{i=1}^2\sum _{j=1}^{j<i}({𝐉}_i\cdot {𝐄}_j+{𝐉}_j\cdot {𝐄}_i)$

Or

$-\mathrm{\nabla }\cdot ({𝐄}_1\times {𝐇}_2+{𝐄}_2\times {𝐇}_1)=({𝐄}_1\cdot \frac{\partial {𝐃}_2}{\partial t}+{𝐇}_2\cdot \frac{\partial {𝐁}_1}{\partial t}+{𝐄}_2\cdot \frac{\partial {𝐃}_1}{\partial t}+{𝐇}_2\cdot \frac{\partial {𝐁}_1}{\partial t})+({𝐉}_1\cdot {𝐄}_2+{𝐉}_2\cdot {𝐄}_1)$

References

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