Typical Differential Equation

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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