# Question regarding inverse relation of resistance with area of cross section

It is said that resistance is inversely proportional to area of cross section. But greater area will have greater electric flux, and greater electric flux will have greater magnetic flux, and greater the magnetic flux will have greater eddy current, which is opposite to the current flowing. So this will oppose the current hence resistance increases. Am I right?

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Even if the claim were correct, you would not be true; you would be right while the statement you are making would be true. ;-)

But the explanation you are offering isn't right. The reason why the resistance goes like $1/A$ is simply that for a constant current $I$, the current per unit area – current density – is $j=I/A$ and according to the microscopic Ohm's law $J=\sigma E$, it's the current density that dictates the voltage per unit length of the wire (also known as the electric field): $$\frac {V}{\ell}\equiv |\vec E| = \frac{1}{\sigma} \cdot \frac{I}{A}$$ Comparing this equation with the usual Ohm's law $V=RI$, we see that $$R = \frac{\ell}{\sigma A}$$ Because you're only changing $A$ while $\ell,\sigma$ are kept fixed, you see that $V\sim 1/A$. Magnetic fields don't play any role here at all while electric fields do play role but we never calculate any "electric flux" to construct the right justification.

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well the resistance you are talking about is the resistance which the flow of current experiences because of collision of electrons with fixed kernels. the value of this resistance is inversely proportional to area of conductor

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Let's analyse each segment of your post:

It is said that resistance is inversely proportional to area of cross section.

Yes, that is true for an conductor when other physical parameters are fixed (length, resistivity, temperature)

But greater area will have greater electric flux...

Yes.

...and greater electric flux will have greater magnetic flux...

The Electric current in the conductor will cause a magnetic field that wraps around the conductor, the field (or flux density) will be proportional to the current.

...and greater the magnetic flux will have greater eddy current, which is opposite to the current flowing.

That would be the case only if the magnetic flux changes, which in turn means that the current and voltage has to change with time. In other words: it does not matter for DC. I will ignore the various forms of fluctuating current and only focus on sinusoidal AC.

The eddy currents in high frequency AC cause a variety of effects, the most relevant of which, in this case of a single conductor is 'Skin effect'. But even in that case, the resistance is inversely proportional to diameter (decreases with increase in area):

where δ is called the skin depth. The skin depth is thus defined as the depth below the surface of the conductor at which the current density has fallen to 1/e (about 0.37) of current density at surface.

So this will oppose the current hence resistance increases. Am I right?

Yes, resistance does increase when you take into account the effect of eddy current in certain situations (fluctuating voltage like high freq AC) compared to just DC without eddy currents. But if you implied that larger area does not decrease resistance, you are wrong in this case.

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