# Analogy of electrical and magnetic quantities

The analogy of electrical and magnetic quantities is a consequence of the strong symmetry in Maxwell's equations between the electrical and magnetic quantities that occur. These analogies are helpful for understanding electromagnetic and electrotechnical relationships and phenomena and are often given in textbooks.

The sizes of the stationary flow field have a strong analogy to fluid mechanics and thermodynamics and can be explained quite clearly (see also electro-hydraulic analogy ). The magnitudes of the electrostatic and magnetic fields are rather abstract, but can be well understood using the analogy. In addition, the difference between electric and magnetic fields (e.g. electric and magnetic monopoles , Lenz's rule ) becomes very clear in the analogies.

Electrical quantities Magnetic sizes
Electrostatic field Stationary flow field Magnetic field
Particulate swell size Electric charge

${\ displaystyle Q}$

no source size known

(Fictional magnetic monopole )

Field strength Electric field strength

${\ displaystyle {\ vec {E}} = {\ frac {\ vec {F}} {Q}}}$

Magnetic field strength

${\ displaystyle {\ vec {H}}}$

Material parameters Permittivity

${\ displaystyle \ varepsilon}$

Specific conductance / resistance

${\ displaystyle \ sigma = {\ frac {1} {\ rho}}}$

permeability

${\ displaystyle \ mu}$

Complex permittivity

${\ displaystyle \ varepsilon - {\ text {j}} {\ frac {\ sigma} {\ omega}}}$

permeability

${\ displaystyle \ mu}$

Flux density quantity Electric flux density

${\ displaystyle {\ vec {D}} = \ varepsilon {\ vec {E}}}$

Current density

${\ displaystyle {\ vec {J}} = \ sigma {\ vec {E}}}$

Magnetic flux density

${\ displaystyle {\ vec {B}} = \ mu {\ vec {H}}}$

Flow size Electric flow

${\ displaystyle {\ mathit {\ Psi}} = \ int \ limits _ {A} {{\ vec {D}} \ cdot {\ text {d}} {\ vec {A}}}}$

Current (charge flow)

${\ displaystyle i = \ int \ limits _ {A} {{\ vec {J}} \ cdot {\ text {d}} {\ vec {A}}}}$

Magnetic river

${\ displaystyle \ Phi = \ int \ limits _ {A} {{\ vec {B}} \ cdot {\ text {d}} {\ vec {A}}}}$

Flow through volume Contained cargo

${\ displaystyle \ oint \ limits _ {A} {{\ vec {D}} \ cdot {\ text {d}} {\ vec {A}}} = {{Q} _ {includes}}}$

Integral set of nodes

${\ displaystyle \ oint \ limits _ {A} {{\ vec {J}} \ cdot {\ text {d}} {\ vec {A}}} = 0}$

Integral magnetic knot set

${\ displaystyle \ oint \ limits _ {A} {{\ vec {B}} \ cdot {\ text {d}} {\ vec {A}}} = 0}$

Node set 1. Kirchhoff's law

${\ displaystyle \ sum \ limits _ {nodes} {i} = 0, \ \ \ \ i = \ int \ limits _ {A} {\ left ({\ vec {J}} + {\ frac {{\ text {d}} {\ vec {D}}} {{\ text {d}} t}} \ right)} \ cdot {\ text {d}} {\ vec {A}}}$

Magnetic node set

${\ displaystyle \ sum \ limits _ {nodes} {\ Phi} = 0}$

Integrals

Field strength quantities

Electric voltage

${\ displaystyle u = \ int \ limits _ {s} {{\ vec {E}} \ cdot {\ text {d}} {\ vec {s}}}}$

Magnetic tension

${\ displaystyle {{V} _ {m}} = \ int \ limits _ {s} {{\ vec {H}} \ cdot {\ text {d}} {\ vec {s}}}}$

potential Electrical potential

${\ displaystyle {{\ varphi} _ {P}} = \ int \ limits _ {P} ^ {\ operatorname {O}} {{\ vec {E}} \ cdot {\ text {d}} {\ vec {s}}}}$

Magnetic potential

${\ displaystyle {{V} _ {P}} = \ int \ limits _ {P} ^ {\ operatorname {O}} {{\ vec {H}} \ cdot {\ text {d}} {\ vec { s}}}}$

Integral swell size Electrical source voltage ( law of induction )

${\ displaystyle {{u} _ {0}} = \ oint \ limits _ {(A)} {{\ vec {E}} \ cdot {\ text {d}} {\ vec {s}}} = - \ int \ limits _ {A} {{\ frac {{\ text {d}} {\ vec {B}}} {{\ text {d}} t}} \ cdot {\ text {d}} {\ vec {A}}}}$

Magnetic source voltage (flooding)

${\ displaystyle \ Theta = \ oint \ limits _ {(A)} {{\ vec {H}} \ cdot {\ text {d}} {\ vec {s}}} = \ int \ limits _ {A} {\ left ({\ vec {J}} + {\ frac {{\ text {d}} {\ vec {D}}} {{\ text {d}} t}} \ right) \ cdot {\ text {d}} {\ vec {A}}}}$

Mesh sets 2. Kirchhoff's law

${\ displaystyle \ sum \ limits _ {mesh} {u} = \ sum \ limits _ {mesh} {{u} _ {0}}}$

Magnetic mesh set

${\ displaystyle \ sum \ limits _ {mesh} {{V} _ {m}} = \ sum \ limits _ {mesh} {\ Theta}}$

Energy density Electrical energy density

${\ displaystyle w = {\ frac {DE} {2}}}$

Power dissipation density

${\ displaystyle {{p} _ {V}} = SE}$

Magnetic energy density

${\ displaystyle w = {\ frac {BH} {2}}}$

Field energy Electric field energy

${\ displaystyle W = {\ frac {Qu} {2}}}$

Magnetic field energy

${\ displaystyle W = {\ frac {\ Psi i} {2}}}$

Electrotechnical component capacitor resistance Inductance / coil
property capacity resistance Inductance
Definition equation capacity

${\ displaystyle C = {\ frac {Q} {u}}}$

Conductance / resistance

${\ displaystyle G = {\ frac {i} {u}} = {\ frac {1} {R}}}$

Inductance

${\ displaystyle L = {\ frac {\ Psi} {i}} = {\ frac {N \ Phi} {i}}}$

Design equation

from field sizes

${\ displaystyle C = {\ frac {\ int \ limits _ {A} {{\ vec {D}} \ cdot {\ text {d}} {\ vec {A}}}} {\ int \ limits _ { s} {{\ vec {E}} \ cdot {\ text {d}} {\ vec {s}}}}}}$ ${\ displaystyle G = {\ frac {\ int \ limits _ {A} {{\ vec {J}} \ cdot {\ text {d}} {\ vec {A}}}} {\ int \ limits _ { s} {{\ vec {E}} \ cdot {\ text {d}} {\ vec {s}}}}}}$ ${\ displaystyle L = {\ frac {\ int \ limits _ {A} {{\ vec {B}} \ cdot {\ text {d}} {\ vec {A}}}} {\ int \ limits _ { s} {{\ vec {H}} \ cdot {\ text {d}} {\ vec {s}}}}}}$
Design equations

for a homogeneous field

capacity

${\ displaystyle C = \ varepsilon {\ frac {A} {l}}}$

Electrical conductance / resistance

${\ displaystyle G = \ sigma {\ frac {A} {l}} = {\ frac {1} {R}}}$

Inductance / Magnetic Resistance

${\ displaystyle L = {{N} ^ {2}} \ mu {\ frac {A} {l}} = {\ frac {{N} ^ {2}} {{R} _ {m}}}}$

resistance

${\ displaystyle R = \ rho {\ frac {l} {A}} = {\ frac {u} {i}} = {\ frac {1} {G}}}$

Magnetic resistance

${\ displaystyle {{R} _ {m}} = {\ frac {1} {\ mu}} {\ frac {l} {A}} = {\ frac {{V} _ {m}} {\ Phi }} = {\ frac {{N} ^ {2}} {L}}}$

Current-voltage relationship ${\ displaystyle i = C {\ frac {{\ text {d}} u} {{\ text {d}} t}}}$ Ohm's law

${\ displaystyle i = Gu}$

${\ displaystyle u = L {\ frac {{\ text {d}} i} {{\ text {d}} t}}}$

## annotation

1. a b c d The magnetic flux results from the integration of the flux density over an area. In the case of a coil with one turn, this is precisely the area enclosed by the turn. The surface in coils with several turns is actually a screw or helical surface . Since these windings are usually permeated by one and the same magnetic flux, they are viewed as individual windings and the linked magnetic flux is defined in electrical engineering . This results in the additional parameters or , in accordance with most textbooks . For a turn or if the three-dimensional conductor geometry in a coil is actually correctly taken into account, the linked flux can be replaced by the magnetic flux and (see also magnetic flux ).${\ displaystyle N}$${\ displaystyle N}$${\ displaystyle {{N} ^ {2}}}$${\ displaystyle \ Psi}$${\ displaystyle \ Phi}$${\ displaystyle N = 1}$

## Individual evidence

1. K. Lunze: Introduction to electrical engineering - textbook . Verlag Technik, 1988, ISBN 3-341-00504-8 .
2. ^ E. Philippow: Fundamentals of electrical engineering . Verlag Technik, 2000, ISBN 978-3-341-01241-3 .