Question
Fri February 18, 2011 By: Sneha Ghodke
 

Mam, is there vector form of Ohm's Law?

Expert Reply
Sat February 19, 2011
Dears student
We all know the simplest version of Ohm's law:
 

 

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 and , 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
 

 

The constant $\eta$ is called the resistivity, and is measured in units of ohm-meters. Ohm's law becomes
 

 

However, (supposing that the conductor is aligned along the $z$-axis) and , so the above equation reduces to
 

 

There is nothing special about the $z$-axis (in an isotropic conducting medium), so the previous formula immediately generalizes to
 

 

This is the vector form of Ohm's law.
 
Hope this helps.
Regards
Team
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