Law of Ohm

The law of Ohm is a physical Loi making it possible to connect the intensity of the electric current crossing a dipole to the tension on its terminals.

Macroscopic point of view

In D.C. current

The potential difference or tension U (in Volt S) at the boundaries of a consumer of resistance R (in Ohm S) is proportional to the intensity of the electric current I (in amp S) which crosses it.

$U\; =\; R.I\; \backslash ,$

One can deduce some:

• $I\; =\; \backslash \; frac\; U\; R$ if R is nonnull

• $R\; =\; \backslash \; frac\; U\; I$ if I is nonnull

Resistance is expressed in Ohm S (Symbole: Ω).

This law bears the name of Georg Ohm which worked on the conducting behavior of the S metal. It satisfactorily applies to the metal drivers thermostated , i.e. maintained with a constant Température. When the temperature changes, the value of resistance also changes in a more or less simple way, which forces to introduce corrective terms. By convention, one preserves the law and one introduces the corrective terms into the value of the resistance of the driver.

In Alternative course

The preceding law spreads with the case of the sinusoidal currents by using the complex notations. One notes $\backslash \; underline\; \{U\},\; \backslash \; underline\; \{I\}$ the tension and the current complexes. The law of Ohm is written then:

$\backslash \; underline\; \{U\}\; =\; \backslash \; underline\; \{Z\}.\; \backslash \; underline\; \{I\}$

With $\backslash \; underline\; \{Z\}\; \backslash ,$: complex impedance of the dipole considered, which can be made up of linear dipoles (resistances, condensing S and Inductance S).

Local point of view (mesoscopic)

Statement of the local law of Ohm

From a local point of view, i.e. mesoscopic, the law (local) of Ohm is stated by saying that the Mobilité of the charge carriers is independent of $||\backslash \; vec\; \{E\}||$.

If one notes $\backslash \; driven\; \backslash ,$ the mobility of the charge carriers, their speed is written then $\backslash \; vec\; \{v\}\; =\; \backslash \; pm\; \backslash \; driven\; \backslash \; vec\; \{E\}$ (the direction of the movement depends on the sign of the carriers); the Density of current $\backslash \; vec\; \{J\}$ associated with a density of carriers $n\; \backslash ,$ is worth as for it:

$\backslash \; vec\; \{J\}\; =qn\; \backslash \; vec\; \{v\}\; =qn\; \backslash \; driven\; \backslash \; vec\; \{E\}$, where $q\; \backslash ,$ is the electric charge of the carrier (in absolute value).

One notes $\backslash \; sigma\; =\; qn\; \backslash \; driven\; \backslash ,$ the electric Conductivité of material (for only one type of carrier).

There is then the local law of Ohm for only one type of carrier:

$\backslash \; vec\; \{J\}\; =\; \backslash \; sigma\; \backslash \; vec\; \{E\}$.

If one has several types of carriers, such as for example the electron S and the holes in a semiconductor, the density of current becomes:

$\backslash \; vec\; \{J\}\; =\; \backslash \; sum\_k\; n\_k\; q\_k\; \backslash \; vec\; \{v\}\; \_k$,

with $\backslash \; vec\; \{v\}\; \_k=\; \backslash \; mu\_k\; \backslash \; vec\; \{E\}$,

thus $\backslash \; vec\; \{J\}\; =\; \backslash \; left\; n\_k\; q\_k\; \backslash \; mu\_k\; \backslash \; right\; \backslash \; vec\; \{E\}$.

There is then total conductivity:

$\backslash \; sigma=\; \backslash \; sum\_k\; n\_k\; q\_k\; \backslash \; mu\_k\; \backslash ,$

See also Law of Nernst-Einstein.

Relationship with the macroscopic law of Ohm: definition of resistance

Let us consider a portion of driver of a point has at a point B and of cross-section S , one then has the potential difference which is worth:

$V\_A-V\_B\; =\; \backslash \; int\_\; \{has\}\; ^\; \{B\}\; \backslash \; vec\; \{E\}\; .d\; \backslash \; vec\; \{L\}$

and intensity:

$i=\; \backslash \; int\; \backslash \; int\_S\; \backslash \; vec\; \{J\}\; .d\; \backslash \; vec\; \{S\}\; =\; \backslash \; int\; \backslash \; int\_S\; \backslash \; sigma\; \backslash \; vec\; \{E\}\; .d\; \backslash \; vec\; \{S\}\; =\; \backslash \; sigma\; \backslash \; int\; \backslash \; int\_S\; \backslash \; vec\; \{E\}\; .d\; \backslash \; vec\; \{S\}$

Let us multiply by a constant the potential difference $V\_A-V\_B\; \backslash ,$, then the boundary conditions are unchanged as well as the lines of field of $\backslash \; vec\; \{E\}$, and the expression $\backslash \; int\; \backslash \; int\_S\; \backslash \; vec\; \{E\}\; .d\; \backslash \; vec\; \{S\}$ is multiplied consequently constant, consequently the report/ratio:

$\backslash \; frac\; \{V\_A-V\_B\}\; \{I\}$ is independent of this constant, it is a " constante" (it nevertheless depends on various parameters the such temperature) called electrical resistance and noted $R\; \backslash ,$.

$R=\; \backslash \; frac\; \{V\_A-V\_B\}\; \{I\}\; =\; \backslash \; frac\; \{\backslash \; int\_\; \{has\}\; ^\; \{B\}\; \backslash \; vec\; \{E\}\; .d\; \backslash \; vec\; \{L\}\}\; \{\backslash \; sigma\; \backslash \; int\; \backslash \; int\_S\; \backslash \; vec\; \{E\}\; .d\; \backslash \; vec\; \{S\}\}$

This formula makes it possible to calculate the resistance of various material geometries (thread-like, cylindrical, spherical,…).

See too

• Impedance
• Resistivity
• Resistance
• Electricity
• Laws of Kirchhoff (law of the meshs and law of the nodes)
• ohmic Material
• Principle of superposition
• Theorem of reciprocity
• Theorem of Thévenin
• Theorem of Norton
• Theorem of Millman

External bonds

• Exercises law of Ohm

Simple: Ohm' S law

Random links:

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