Vector form of ohm's law
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Gaurav Seth 3 years, 5 months ago
The simplest version of Ohm's law: V = IR
where V is the voltage drop across a resistor of resistance R when a current I flows through it.
Let us generalize this law so that it is expressed in terms of E and J, rather than V and I.
Consider a length l of a conductor of uniform cross-sectional area A with a current I flowing down it.
In general, we expect the electrical resistance of the conductor to be proportional to its length, and inversely proportional to its area (i.e., it is harder to push an electrical current down a long rather than a short wire, and it is easier to push a current down a wide rather than a narrow conducting channel.)
Thus, we can write
=>R=ηlA=>R=ηlA
The constant ηη is called the Resistivity, and is measured in units of ohm-meters. Ohm's law becomes
=>V=ηlAI=>V=ηlAI
=>Vl=ηIA=>Vl=ηIA
However, IA=JxIA=Jx (supposing that the conductor is aligned along the x-axis) and Vl=ExVl=Ex, so the above equation reduces to
Ex=η.JxEx=η.Jx
There is nothing special about the x-axis (in an isotropic conducting medium), so the previous formula immediately generalize
E=η.JE=η.J
This is the vector form of Ohm's law.
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