Vector density of current

One noting I the Electric current in a portion of driver, and is $\backslash \; mathrm\; D\; \backslash \; vec\; S$ a vector element of surface of a cross-section of this driver, one poses:

$\backslash \; vec\; j$ the vector density of current

such as

$\backslash \; mathrm\; di\; =\; \backslash \; vec\; J\; \backslash \; cdot\; \backslash \; mathrm\; D\; \backslash \; vec\; S$.

There is then

$i\; =\; \backslash \; iint\_S\; \backslash \; vec\; \{J\}\; \backslash \; cdot\; \backslash \; mathrm\; D\; \backslash \; vec\; \{S\}$.

The sign of I is then related to the orientation surface S .

Expression of J

It is shown, owing to the fact that $i=\; \backslash \; frac\; \{\backslash \; mathrm\; dq\}\; \{\backslash \; mathrm\; dt\}$, that if there is one type of carrier (electron for example) one a:

$\backslash \; vec\; \{J\}\; =\; Q\; \backslash \; cdot\; N\; \backslash \; cdot\; \backslash \; langle\; \backslash \; vec\; v\; \backslash \; rangle$

where

• Q is the load of a carrier,
• N the density volume of the carriers (many carriers per unit of volume) and
• $\backslash \; langle\; \backslash \; vec\; v\; \backslash \; rangle$ the Flight Path Vector means of the carriers.

If there are several types of carriers (electrolytic Solution for example) one will have:

$\backslash \; vec\; j=\; \backslash \; sum\_k\; q\_k\; \backslash \; cdot\; n\_k\; \backslash \; cdot\; \backslash \; langle\; \backslash \; vec\; \{v\}\; \backslash \; rangle\_k$

It should be noted that $\backslash \; langle\; \backslash \; vec\; \{v\}\; \backslash \; rangle\_k$ depends on the other charge carriers.

Vector density of current surface j_S

Let us suppose that a dimension of the driver is low in front of the others, one will then have a “sheet” thickness E negligible, then:

$i\; =\; \backslash \; iint\_S\; \backslash \; vec\; \{J\}\; \backslash \; cdot\; \backslash \; mathrm\; D\; \backslash \; vec\; \{S\}\; =\; \backslash \; int\; \backslash \; left\; (\backslash \; int\_0^e\; \backslash \; vec\; J\; \backslash \; \backslash \; mathrm\; Dy\; \backslash \; right)\; \backslash \; \backslash \; mathrm\; dx\; \backslash \; cdot\; \backslash \; vec\; u$

one poses then:

$\backslash \; vec\; \{j\_S\}\; =\; \backslash \; int\_0^e\; \backslash \; vec\; J\; \backslash \; \backslash \; mathrm\; Dy$

what gives

$i\; =\; \backslash \; int\; \backslash \; vec\; \{j\_S\}\; \backslash \; \backslash \; mathrm\; dx\; \backslash \; cdot\; \backslash \; vec\; \{U\}$

See too

• Law of Ohm

• Conservation of the electric charge

Random links:

Raion de Orhei | The Vault-Stick (Two-Sevres) | Rich Jules François | The Community of communes of the valley of Hogneau | The Community of Chamousset communes as Lyonese | Affenpinscher

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