Abstract
The magnetic susceptibility of the electron gas in a metal is normally very small. Only if the effective electron mass is low, the susceptibility will be in magnitudes higher (because of the strong Landau diamagnetism). Therefore in a static inhomogeneous magnet field free electrons in such a metal are repulsed in the direction of the decreasing magnet field - though they gain NO energy of the magnet field.
So there is an overall velocity of the electrons in one direction and thus they are collected in one area of the material and build up an electric potential and electric field. The electrons are moving against this electric field and so they are changing kinetic in potential energy. This can be used to convert heat energy into electric energy.
Most importantly, the magnet field is only used to change the movement of the free electrons in one direction, but not to gain energy. However exactly this movement in one direction builds up an electric potential and electric field, and this can be used.
Introduction: Susceptibility of the electron gas
In a classical system, there can be no magnetization of electrons in a solid (Bohr-van Leeuwen theorem). However in quantum physics there are two effects because of the spin of electrons and the Landau levels. Thus, the susceptibility of the electron gas in a metal is given by the Pauli paramagnetism and the Landau diamagnetism.
If the effective electron mass in a solid is equal to the free electron mass, then the Pauli paramagnetism is three times higher than the Landau diamagnetism. The overall susceptibility is very small.
If the electron mass is quite small then the Landau diamagnetism will dominate, since only the latter depends strongly on the effective electron mass. In this case the susceptibility the will be in magnitudes higher (like in Bismuth with m_e=0.05m*)
Description of the idea
The next figure shows in the lower part a permanent magnet which produces a static, inhomogeneous magnetic field in the upper part. Above, there is medium with a low effective electron mass like Bismuth in a fixed position.
. +-----V------+ Voltmeter . | | . | | . +---+---+ | The susceptibility of the wires are . Bismuth | - - -| | (nearly) zero - no repulsion. . |- - - -| | . | | | The susceptibility of Bismuth is . | /|\ | | much higher. . | | | | Thus the electrons are repulsed in . | | | | one direction and produces an . | | | electric potential. . | + + +| | . |+ + + +| | . +---+---+ | . | | . +------------+ . . \ \ | / / Magnetic field . \ | | | / . +-------------------+ . |........ N ........| Permanent magnet . |...................| . |...................|
Although there should be no movements of the materials, externals electric fields (or temperature differences), the electrons will be repulsed in the Bismuth because of the Landau diamagnetism.
So, the electrons are collected in the upper part. Therefore the electrons produce an electric potential and an electric field. This electric field will enhance and then prevents a further movement of the electrons.
With a magnetic field of B=2T, dB=17T/m there is a force of every free electron in Bismuth F=-1.2 10^-22 N (see calculation below). The electric potential for a 1 cm part Bismuth is U_Bi= -7.5 10^-6 V
A circuit
If we want to have a circuit, there must be also another wire in the non-uniform magnet field, e.g. with a Voltmeter. Normally the force of free electrons in a metal is about 10^5 times smaller compared to the force in Bismuth. So approx. the U_Bi should be measured at the Voltmeter.
The effect of -7.5 10^-6 V is not very high. However, this effect can be enhanced, if more of this parts are switched together. With 130 of this Bismuth-copper pairs, the electric potential would be 0.001 V.
The main points are
The static magnetic field does not do work on the electrons, but it changes the direction of the motion of the electrons (normal thermal motion) because of the repulsion due to the Landau diamagnetism.
The electrons don't gain energy of the magnet field.
However the electrons have a uniform overall direction to the decreased magnetic field, this means they move to one area of the conductor. These additional charges in this area mean an electric potential and electric field. This electric field will enhance and then will prevent a further movement in this direction.
If this potential is building up (in the beginning) or in the case of a
circuit, there is another important fact:
The charges will flow against the electric field (because the overall
direction of the electrons is always in direction of the decreasing magnetic
field effect).
Moving against an electric field means the electrons have to change kinetic
energy in electric potential energy. And here the conversion of heat into
electric energy happens.
A slow electron is "colder" than the ions around. At the next collision of the electron with a ion of the conductor, it gains energy from this ion. The electron has then the "normal" temperature and therefore kinetic energy, then again moves against the field, will be slower and collide with a ion and so on. So the electrons absorb kinetic energy of the ions, thus the semiconductor will cool down.
This effect is limited to only a few materials with very small effective electron masses and to high magnetic fields.
The possible electric potential and current strongly depend on the materials. If there are materials with different electron masses, which are switched to together, an electric current must flow trying to equalize this difference. However the magnetic field will further produce the different electric potential. Therefore a current can be produced which flows without intermission. Since the electrons have to flow against the electric potential, kinetic energy will be converted in electric energy. In other words, heat (without a temperature difference) could be converted in electric energy.
This effect does not contradict the 1st law of thermodynamics, since the whole energy will be conserved, only the distribution will be changed. However it contradicts the 2nd law of thermodynamics, since heat energy (and not only heat difference) will be converted into electric energy.
To be more specific about the 2nd law:
It just says that the entropy will increase, this means a state goes only in
a state with a higher probability.
But there is no PROVE that this means ALWAYS that temperature difference inside observed area will be smaller. This is just an observed knowledge. Or the other way round: Until now, there is no system found where the most probabilistic state is a state where one part is warmer than the other.
Estimation of the effect
The total susceptibility of the electron gas in a metal is
(e.g. Magnetism in Condensed Matter, Stephen Blundell, pp. 152)
3 n µ_0 µ_B^2
X = ------------- * L
2 E_F
n = number of electrons per unit volume
E_F = Fermi Energy
µ_0 = magnetic permeability of free space
µ_B = Bohr magneton
L = factor of Landau diamagnetism = 1 – (1/3) * (m_e/m*)^2
With m_e is effective electron mass and
m* is the free electron mass
The average susceptibility for one free electron is simply X / n
The magnetic moment M is
M ~ (B X) / µ_0 with B = magnetic field
The magnetic moment of a free electron is therefore
3 µ_B^2 B
M = --------- * L
2 E_F
The force acting on a magnetic moment of a sample in an
inhomogeneous magnetic field is
F= M dB
dB = gradient of the magnetic field
I use the following numbers:
B= 2T
dB= 17T/m
Bismuth: E_F=0.03 eV m_e=0.05
=> L=-132.3 M=-7.1 10^-24 F=-1.2 10^-22 N
Copper: E_F=7 eV m_e=1.3
=> L=0.80 M=1.8 10^-28 F=3.6 10^-27 N
The length of the medium Bismuth should be quite small, L = 0.01m
otherwise the decrease of the magnet field is too dominant.
The electric potential is
L F FL
U = Integral --- dl = -- with e ... electronic charge
0 e e
The electric potential in Bismuth is therefore:
U_Bi = -7.5 10^-6 V
The electric potential of copper is 10^5 times smaller!
Gerhard Kainz
*********************************