2015-01-31
Where does the energy
The famous free energy researcher Don Smith believed that Einstein's famous formula E=mc2 describes a way to store energy. This approach is the best visual representation of what we will talk about next.
Let's approach this issue from the position of limit values. What can be the maximum potential energy of an isolated ball? If we look at the classical formula: \[ W_{c}=\frac {Q^{2}} {2C} \tag{2.1}\] then it turns out that the smaller \(C\), i.e. the capacity, the higher the potential energy. Then what can be the minimum capacity? To do this, let's recall the formula for the capacity of a solitary sphere: \( C=4\pi \varepsilon \varepsilon _{0}r\), where: \(r\) is the radius of the sphere, and \(\varepsilon , \varepsilon _{0}\) are the relative and absolute permeabilities. What is the minimum radius that a sphere with a charge can have? Yes, that's right, it's the radius of an electron \(r_{e}\) [1]. Where do we find its own capacity: \[ C_{e}=4\pi \varepsilon \varepsilon _{0}r_{e} \tag{2.2}\] It is clear that the charge of such a sphere will be exactly equal to the charge of an electron -- \(e\). We take the relative permittivity \(\varepsilon\) to be equal to one (as for a vacuum) and obtain the maximum energy for the minimum capacity -- the potential energy of the electron charge: \[ W_{ce}=\frac {e^{2}} {8\pi \varepsilon _{0}r_{e}} \tag{2.3}\]
Recall that the electron charge is \( e=1.6\cdotp 10^{-19}\) (C), and the classical electron radius is: \( r_{e}=2.82\cdotp 10^{-15}\) (m).
But the resulting formula is exactly equal to half of Einstein's mass-energy: \[ W_{ce}=\frac {e^{2}} {8\pi \varepsilon _{0}r_{e}}\;=\;\frac {m_{e}c^{2}} {2}, \tag{2.4}\] where: \(m_{e}\) is the mass of an electron equal to \(9.1\cdotp 10^{-31}\) (kg), \(c\) is the speed of light equal to \(3\cdotp 10^{8}\) (m/s). Thus, we obtained the connection: charge-mass-energy and answered the question — where does the energy come from.
The potential energy of the electron system will be maximum if the capacitance of the capacitor in which they are located tends to zero. Apparently, such a limiting state of the electron system is an electron gas or electron plasma in a vacuum.
This is also the answer to the original question - where does the energy go if the capacitance of the electron system is increased? The electrons are simply bound by the capacitance and cease to be free, and the larger the capacitance, the more bound they are.
Where is the second half of the mass-energy?
Apparently, it is contained in the magnetic field inside the electron. It is known that the electron has its own angular momentum, or spin, the properties of which cannot be explained from the point of view of ordinary mechanics [4]. But this means that the electric charge in the electron is mobile, albeit not in the classical sense, and if so, it must also represent a magnetic charge. Thus, we obtain a electron model suitable for our further calculations and research: outside it is an electric charge, and inside it is a magnetic charge. We will further support the second statement by the presence of inductance in the electron.
In this paper, we will not delve into the depths of quantum physics, but will consider free charges from the point of view of electrical engineering and radio electronics.
From this point of view, our readers will be interested in looking at this idea in a more expanded form, where a model of an electron will be presented taking into account the second magnetic field and the data obtained here.
Electron -- an ideal oscillatory circuit?
Since the electron is a kind of elementary capacitance, then why can't it also be an elementary inductance? And indeed, we find such a justification in the work [2], from where we take the formula for the inductance of the electron: \[L_{e} = \frac{m_{e}} {n\, e^{2}} = {m_e r_e^2 \over e^2} = \frac {\mu_{0}r_{e}} {4\pi} \tag{2.5}\] where: \(n\) -- specific charge density, \(\mu_{0}\) — magnetic constant equal to \(1.26\cdotp 10^{-6}\) (H/m). Formula (2.5) is obtained if we take only one charge as \(n\). By the way, in [3] it is proposed to consider such inductance as an analogue of the mass of matter.
For a complete picture, we have to make the last assumption that the electron is an ideal oscillatory circuit, with its own resonant frequency, wave resistance and infinite quality factor. As is known, energy in an ideal oscillatory circuit can circulate forever or until a radiating antenna is connected to the circuit, for example.
Another interesting conclusion can be this: since the electron is an oscillatory circuit,tour, so while it is a particle, all its potential energy is reactive. It becomes active when the electron becomes a wave, and we can feel the manifestations of this energy in the form of light, heat, etc.
If all our assumptions are correct, then the task of extracting energy from an electron comes down to one simple rule: we must create conditions for the electron under which its reactive energy can be transformed into active energy. In the next section, we will consider such conditions, and now we will give some parameters of the electron, which are not always found in classical literature.
Some parameters of the electron
We will obtain one more value, and then we will bring together all the data on the electron. From formula (2.2) we take the electron's own capacitance and calculate the potential on its conditional surface: \(\varphi_e = e / C_e = 5.11 \cdotp 10^5\) (V). This raises a natural question: where does this potential go when we charge, for example, a metal ball with negative charges? To answer this, we can draw an analogy with a capacitor of small capacitance, but charged with a high voltage, which is connected to a discharged capacitor of large capacitance; the entire potential is distributed over the large capacitance, and on the total capacitance, we get only a small increase in voltage.
Knowing that half of the electron's potential energy is in the magnetic charge, we can calculate the current inside it: \(I_e = c \sqrt{m_e / L_e} = c\, e / r_e = 1.7 \cdotp 10^4\) (A). What's interesting here is that if we divide the potential on the electron's surface by this current, we get a resistance of 30 Ohms. But since, according to our model, the electric charge is outside and the magnetic charge is inside the electron, they do not intersect in the reactive state, but begin to interact only when they transition to an active state, for example, into radiation. Then this resistance begins to work as a wave resistance. The model of a spherical dielectric resonator of the H-type is the best fit for this approach.
Let's present reference data on the electron, which we will need later. In this work you can see a detailed derivation and physical meaning of these quantities.
- classical radius: \(r_{e} = 2.82\cdotp 10^{-15}\) (m);
- mass: \(m_{e} = 9.1\cdotp 10^{-31}\) (kg);
- charge: \(e = 1.6\cdotp 10^{-19}\) (C);
- surface potential: \(\varphi_e = 5.11 \cdotp 10^5\) (V);
- self-capacitance: \(C_{e} = 3.14\cdotp 10^{-25}\) (F);
- self-inductance: \(L_{e} = 2.82\cdotp 10^{-22}\) (H);
- characteristic impedance: \(Z_{e} = \sqrt {L_{e} / C_{e}} = 30\) (Ohm);
- Thompson resonant frequency: \(\nu_{e} = \frac {1}{2\pi\sqrt {L_{e}C_{e}}} = 1.69\cdotp 10^{22}\) (Hz);
- Wavelength: \(\lambda_{e} = c / \nu_{e} = 1.77\cdotp 10^{-14}\) (m).
Closing the problem, we can get this length in another way -- from the classical radius of the electron, simply multiplying it by \(2\pi\): \[\lambda_{e} = 2 \pi r_e \tag{2.6}\] Formula (2.6) is a check for all the reference data given above. Here we mean the wavelength of an electron from the point of view of radio electronics, since, for example, the Compton wavelength is equal to \(2.43 \cdotp 10^{-12}\) m (which differs from the one we obtained by the value of the fine structure constant [5]), and according to de Broglie, it generally depends on its speed.
It is interesting that if we calculate the internal energy of an electron using Planck's constant \(h\) and the resonant frequency \(\nu_{e}\) according to Thompson, then it will coincide with Einstein's, with an accuracy determined by the fine structure constant \(\alpha\): \[W_e = m_e c^2 = h\, \nu_{e} \alpha \tag{2.7}\] In the same way, we can find the magnetic moment of an electron, imagining that its entire charge rotates in a circle: \[\mu_e = {e\, \omega_e r_e^2 \over 2 \alpha} = {e\, r_e c \over 2 \alpha} \tag{2.8}\] This gives us the magnetic moment electron, equal to \(9.27\cdotp 10^{-24}\) J/T, which corresponds to the experimentally measured value.
From this perspective, it may be interesting to look at the coefficient-free laws of Coulomb and Ampere, as well as at the detailed derivation of some of these quantities.