Particle Simulation of Lower Hybrid Waves and Electron-ion ...
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![Page 1: Time rate of energy change of particle with v by a force F is: for a charge particle with added field E': Summing all the electron.](https://reader033.fdocuments.in/reader033/viewer/2022051516/56649d365503460f94a0e8dc/html5/thumbnails/1.jpg)
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![Page 3: Time rate of energy change of particle with v by a force F is: for a charge particle with added field E': Summing all the electron.](https://reader033.fdocuments.in/reader033/viewer/2022051516/56649d365503460f94a0e8dc/html5/thumbnails/3.jpg)
![Page 4: Time rate of energy change of particle with v by a force F is: for a charge particle with added field E': Summing all the electron.](https://reader033.fdocuments.in/reader033/viewer/2022051516/56649d365503460f94a0e8dc/html5/thumbnails/4.jpg)
![Page 5: Time rate of energy change of particle with v by a force F is: for a charge particle with added field E': Summing all the electron.](https://reader033.fdocuments.in/reader033/viewer/2022051516/56649d365503460f94a0e8dc/html5/thumbnails/5.jpg)
1Definition123 1 , , 1, 2,3i j k
2Definition123 1n
3We will start by defining it on R and follow up with
a general definition on n-D.
ricskewsymmettotally
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( ) ( ) ( ) ( )ijk abc ijk jki kij i j
( ) ai bj ckijk
ijk abkai bj bi aj
j k i
k i j ( )
ijk abcai bj ck a b c
a b c i j
![Page 7: Time rate of energy change of particle with v by a force F is: for a charge particle with added field E': Summing all the electron.](https://reader033.fdocuments.in/reader033/viewer/2022051516/56649d365503460f94a0e8dc/html5/thumbnails/7.jpg)
i iA B C D A B C D v v v vv v v v
ijk j k iab a bA B C D j k a b
ja kb jb ka A B C D
A C B D A D B C v v v vv v v v
6ijk ijk
3 ( )i i ijk j kA R A B A B ur ur
![Page 8: Time rate of energy change of particle with v by a force F is: for a charge particle with added field E': Summing all the electron.](https://reader033.fdocuments.in/reader033/viewer/2022051516/56649d365503460f94a0e8dc/html5/thumbnails/8.jpg)
i kj
kj
i j i j j
i j j ij j
i i
j j
(B v) (B v)
= ( B v )
= (B v )
= (B v )- (B v )
=v B -v ( B)
ijk
ijk ab a b
a ba b b a
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![Page 12: Time rate of energy change of particle with v by a force F is: for a charge particle with added field E': Summing all the electron.](https://reader033.fdocuments.in/reader033/viewer/2022051516/56649d365503460f94a0e8dc/html5/thumbnails/12.jpg)
![Page 13: Time rate of energy change of particle with v by a force F is: for a charge particle with added field E': Summing all the electron.](https://reader033.fdocuments.in/reader033/viewer/2022051516/56649d365503460f94a0e8dc/html5/thumbnails/13.jpg)
Time rate of energy change of particle with v by a force F is:
for a charge particle with added field E':
Summing all the electron in a circuit, we find that the sources do work to maintain the current at the rate
- sign is the Lenz's law. This is in addition to the Ohmic losses in thecircuit, which should be excluded from the magnetic energy content
![Page 14: Time rate of energy change of particle with v by a force F is: for a charge particle with added field E': Summing all the electron.](https://reader033.fdocuments.in/reader033/viewer/2022051516/56649d365503460f94a0e8dc/html5/thumbnails/14.jpg)
Thus, if the flux change is δF, the work done by sources is:
The problem of the work done in establishing a general steady-state distribution of currents and fields is shown in Fig.5.20
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The current distribution can be broken up into small current loops.A loop of current of cross-section area Δσ following a closed path C and spanned by a surface S with normal n as shown in Fig. 5.20.The work done against the induced EMF in terms of the change in magnetic induction through the loop is:
Express B in terms of the vector potential A, we have
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Since J∆σdl =Jd3x, the sum over all loops gives:
Stokes’s theorem implies that
Ampère’s law implies that:
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The identity
with P→A ; Q→H gives:
Assuming that the field distribution is localized, the second term (surface integral) vanishes. Hence we have
This is the magnetic equivalent of the electrostatic equation (5.147)
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Assuming that the medium is para- or diamagnetic , such that a linear relation exists between H and B, then
Hence the total magnetic energy will be
This is the magnetic analog of electrostatic equation
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If we assume that a linear relation exists btwn J and A, (5.144) implies that the total magnetic energy is:
This is magnetic analog of
If an object of permeability μ1 is placed in a magnetic field whose current source are fixed, the change in energy can be treated in close analogy with the electrostatic discussions of section 4.7 by replacing D→H; E→B.
H٠B-H0٠B
0=B٠H
0 - H٠B
0 + surface terms
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Hence we have:
This can also be written as
Both μ0 and μ1 can be functions of position, but they are assumed independent of the field strength.
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If the object is in otherwise free space , the change in energy can be written as:
This is equivalent to the electrostatic equation
(5.81)
(5.84)
pf/
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The force acting on a body can be derived from a generalized displacement and calculate
with respect to displacement.
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From (5.149) the total energy of N distinct circuits can be expressed as :
by converting (5.149) to
with the help of (5.32).
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Breaking up the integrals into sums of separate integrals over each circuit, we have:
Hence the coefficients L,M of inductance are given by
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Note that (5.32) reads:
The integral over d3x' (5.14) can be written as integral of A. If the cross-section of the ith circuit is negligible, then mutual inductance becomes:
Ai induced by J
j
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Since curl of A = B, the mutual inductance is:
Flux at i induced by Jj
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The self inductance is :
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If the current density is uniform throughout the interior, from Ampère’s law:
the magnetic induction, close to the circuit, is:
The inductance per unit length inside and outside the wire out to ρmax is:
2 πρdρ
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Because the expression BΦ fails at ρ>At distances large compared to A1/2 , the 1/ρ magnetic induction can be replaced by a dipole field pattern
Thus the magnetic induction can be estimated to be:
If we set
Hence
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Upon combining the different contributions, the inductance of the loop can be estimated to be:
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Consider quasi-static magnetic field in conducting media, the relevant equations are:
for uniform, frequency-independent permeable media.
Laplace equation gives Φ=0
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We can estimate the time τ for decay of an initial configuration with typical spatial variation defined by length L , then
(5.161) can be used to estimate the distance L over which fields exist in a conductor, subjected externally to fields with harmonic variation at frequency
hence
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For copper sphere of radius 1 cm, τ~5-10 m sec
molten iron core of the earth τ~105 years
Evidence: earth magnetic field reverse ~106 years ago, 0.5*104 years, B goes to 0
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Consider a semi-infinite conductor of uniform permeability and conductivity occupies the space z>0 :
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Because the diffusion equation (5.160) is second order in spatial derivatives and first order in time, the steady-state solution for Hx(z,t) can be written as the real part of
By eq(5.160) , h(z) satisfies :
A trial solution gives
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Dim[ k ]~ 1/length 1/δ.
This length is the skin depth δ:
Ex: Seawater Copper at room temperature
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With the boundary condition at z→∞, for z>0,
Since H varies in time, there is an electric field:
Hence the solution of Hx(z,t) is real part of
Taking the real part, together with (5.165),
,
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To compare the magnitude of electric field and magnetic induction, the dimensionless ratio is
by quasi-static assumption. The small tangential electric field is associated with a localized current density
The integral in z is an effective surface current:
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The time-averaged power input, for the resulting resistive heating (P=IV), per unit volume is
With (5.167),(5.168), we have
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A simple example:Two infinite uniform current sheets, parallel to each other and located a distance 2a apart, at z=±a. The current density J is in the y direction:
z=a
z=-a
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At time t=0, the current is suddenly turned off. The vector potential and magnetic field decay according to (5.160) , with variation only in z and t. Let, from Laplace transform,
z=a
z=-a
, we haveFrom (5.160),
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With a change of variable from p to k:
The initial condition can be used to determine h(k):
==> partial_z H_x = J_y
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The initial condition can be used to determine h(k):
Exploiting the symmetry in z, we can express cosine in terms of exponentials:
Inversion of the Fourier integral yields h(k),
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Hence the solution for the magnetic field at all t>0 is:
The integral can be expressed in terms of the error function:
κ=ka; ν=1/μσaa
Hence
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![Page 46: Time rate of energy change of particle with v by a force F is: for a charge particle with added field E': Summing all the electron.](https://reader033.fdocuments.in/reader033/viewer/2022051516/56649d365503460f94a0e8dc/html5/thumbnails/46.jpg)
Error function can be expanded in Taylor series, the result is: