typical differential equation
DESCRIPTION
Telegraphist equation, Helmholtz equation and Shcrodinger equationTRANSCRIPT
8. Typical Differential Equations of
Electrodynamics or Mathematical Physics
Monalisa Malelak2321505
Telegrapher's equations
• The telegraphist's equations (or just telegraph equations) are a pair of linear differential equations which describe the voltage and current on an electrical transmission line with distance and time
• They were developed by Oliver Heaviside who created the transmission line model, and are based on Maxwell's Equations.
Generalized Telegraphist’s Equation
Time Domain – F(z,t)
Frequency Domain – F(z,jω)
We can write the Generalized Telegraphist Equation
Generalized Telegraphist’s Equation
Wave Equation
RGIt
IRCLG
t
ILC
z
I
RGVt
VRCLG
t
VLC
z
V
)(
)(
2
2
2
2
2
2
2
2
Generalized Telegraphist’s Equation
b
a
cFt
Fb
t
Fa
x
F2
2
2
2
: permitivity, conductivity and permeability.
,,
If we subtitute a=Єμ and b=σμ F for the fields E and H
𝜕2𝐹𝜕 𝑧 2
=a 𝜕2𝐹𝜕𝑡 2
+b 𝜕𝐹𝜕𝑡
𝜕2𝐹𝜕 𝑧 2
=a ( 𝑗 𝜔 )2𝐹 +b 𝑗 𝜔𝐹or
Telegraphist’s Equation with a,b > 0; c=0
𝜕2𝐸𝜕𝑧 2
=ϵμ 𝜕2𝐸𝜕𝑡 2
+σμ 𝜕𝐸𝜕𝑡
It will be Wave equation for lossy dielectrics
If we subtitute a=Єμ and b=σμ F for the fields A or φ
𝜕2 𝐴𝜕𝑧 2
=ϵμ 𝜕2 𝐴𝜕𝑡 2
+σμ 𝜕 𝐴𝜕𝑡
It will be Potential equation for magnetic vector potential and the electric scalar potential in conductors and lossy dielectric
If we subtitute a=Єμ and F for the fields E and H
or
Telegraphist’s Equation with a > 0; b=0, c=0
It will be Wave equation for electromagnetic waves in loss-free dielectric
If we subtitute a=L’C’ andF for the fields v or i
It will be wave equation for propagation of voltage and current traveling waves along loss-free transmission line
𝜕2𝐹𝜕 𝑧 2
=a 𝜕2𝐹𝜕𝑡 2
𝜕2𝐹𝜕 𝑧 2
=a ( 𝑗 𝜔 )2𝐹
𝜕2𝐸𝜕𝑧 2
=ϵμ 𝜕2𝐸𝜕𝑡 2
𝜕2𝑣𝜕𝑧 2
=L ′C ′ 𝜕2𝑣𝜕𝑡 2
If we subtitute b=σμ and F for the fields E and H
or
Telegraphist’s Equation with b > 0; a=0, c=0
It will be equation for conduction field in conductors with skin effect
𝜕2𝐹𝜕 𝑧 2
=b 𝜕𝐹𝜕𝑡
𝜕2𝐹𝜕 𝑧 2
=b 𝑗 𝜔𝐹
𝜕2𝐸𝜕𝑧 2
=σμ 𝜕𝐸𝜕𝑡
𝜕2𝐻𝜕𝑧 2
=σμ 𝜕𝐻𝜕𝑡
or
Helmholtz Equation
• In mathematics, Helmholtz Equation is the partial differential equation which represents the time-independent form of the original equation, result from applying the technique of separation of variables to reduce the complexity of the analysis.
• The Helmholtz equation often arises in the study of physical problems involving partial differntial equation (PDEs) in both space and time
• The Helmholtz equation can also represent other wisely used equations, such as the group diffusion equation for the neutron flux in a reactor or the renowned Schroedinger equation.
This is Helmholtz Equation
00sin kce
0)( 200
2 EE
022 EkE
Schroedinger Equation
In quantum mechanics, the Schroedinger equation
is a partial differential equation that describes
how the quantum state of a physical system changes
with time.
Schroedinger Equation
ajort
a 222
22
22
)(2
potWWm
a
m : electron massW : total electron energy in the nucleus fieldWpot : potensial energy of an electron in the nucleus field ħ : Planck’s quantum constant divided by 2πѠ : angular frequency of the material wave
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