We have a formula for Resistance of a wire
R=ρL/A………..implies R is propotional to length.
Where ρ, L and A are resistivity, length and the area of conductor respectively.
Then
R=ρL/πr2……………. ‘ r’ is the radius of cross section
Since L and r both are the units of length……one can be expressed in terms of the other
i.e. r = nL
Then
R=ρL/π(nL)2
R=KL/L2…………..k is the constant =ρ/πn2
i.e. R=K/L
ie R is inversely propotional to length…which violates the original formula…….

Any comments???

A nice insight to be brought up for dumb people like me. As we look at the derivation, it seems good. But it is restricted only to circle since πr2 is the are of a circle. It's always nice to come up with new ideas. Who knows, it might be Tregor's Law in the future.:)

Good luck!

never had any intrest in physics....

if you do a dimension testing you are eliminating a very important factor which is resistivity

here is how:

rho= omega*meter
length=meter
Area= meter^2

if you put in these values you get R = omega which is correct. eventhough rho is a constant (generally but it can change with temperature) you cannot eliminate it if you do a dimensional comparision

so the error is eliminatin rho from the dimensional analysis

good job you almost got me there!

a very good question. It has caught me on the wrong foot.

tho not a physics man, i like equations as long as it does not involve derivatives

i will have to go thru birbhadra's explanation to learn more.

Dimensional analysis can only indicate if there might be an error but it cannot be used to provide a robust argument why something is false.

Here is the problem with tegor's work.

Notice that he uses r=nL. Now, greater the value of L less is the value of n, right? I will use this fact later.

If you plug in K=rho/n^2 then according to his derivation you get,

rho/(n^2 L). Now, although nL is equal to a constant radius 'r', greater the value of L, less is the value of n and because of that you get a higher value of R in the derived 'formula'. So, R does increase when L increases. cool?

birbhadra:
we are assumming resistance for a given temperature at which resistivity is always the same.........so can't we just keep one single constant K for three constants..divided or multiplied.....in a dimensional analysis........elimination of rho was just to keep a new constant we can keep the rho as it is and get ....R=Rho*L/Pi*(nL)^2
which also implies R is inversely to L
tero dai
i'm still trying to follow you.........will be back

true but no matter what, any relationship MUST be dimensionally balanced regardlessly. it is not possible for a ralationship to be true and dimentionally unbalanced. but you are right only dimensional equality cannot gurantee a relationship.

for this case we already know that the given relationship is true. only the susequent simplification was incorrect which can be proven by D.A.

i do not know anything that is true and dimensionally imbalanced! if you do then please provide some insight. i want to make sure that i know if there is any

thanks

above comment was regarding tero dai's analysis

tregor

even if the temperature is constant, you will still have rho = omega*length

let me try one more time

rho = omega*length
lenght = length
area = length^2

so R = (omega*length)*(length)/(length^2) = omega
we still get

R = omega which is correct

regardless of rho being constant it still works. In the original post you eliminated rho and lumped it as a constant. can't do that.

tregor,

Your result is actually correct that R is inversely proportional to L but it does not contradict the law itself and this is because of scaling analysis. r=nL which is in 3D.
When you converted the scale here, it is scaled differently than in the 1D case (which would have been L^2), 2D case (just L) (the more familiar one) and the 3D case, which is the one you gave.

The DA is also correct since he or she has K as Rho by pi r^2.

can only lump constants if they dont have units or same units associated with them

birbhadra
let's do it without lumping the rho..
if the equation R=ρL/A………..implies R is propotional to length,...(where rho is taken as the constant at a given temperature)...
does R=ρ/Pi*n2*L not imply ...R is inversely to L?... cause Rho, Pi and n are constant..........
same assumption is made in the two equations that rho is a constant.......

the final equation is definitely wrong
just trying to figure out what can i not do to get the result.....
will appreciate more comments

tregor kasto yaar timi ta jhann tyesto ramro sanga expain gare. la hera.

let me try differently then. your only problem is that you somehow are stuck with the idea of rho being a constant haina?

ok here we go

since you assumed rho to be constant b/c temp is constant. similarly one can assume the area is constant and length is constant for a given uniform wire-like conductor, for e.g.

then lets assume our area is constant b/c we have a uniform area through out. we can do that can we not?

now,

R = rho*L/A

since rho and area are both constant we say K = rho/A

R = K*L
so it is now proportional to length. haina ta. you cannot ignore dimensions even if it is that os something assumed to be constant.

if you are still not convinced you really need to talk to a physics instructor.

have fun and bring more of this in future

finally i have a uniform conducting wire which is 1 meters in length, 0.002 meter^2 in area with rho being 5 ohm*meter

since nothing is changing over time, ie length, area and resistivity can be assumed as constants. can we not?

so now

R = K

however this case will hold true even wrt dimensional consistency because resistance is a constant if all the other variables are fixed.


ok i think i beat the problem to death. i won't comment on this any more. hope i was help

~Birbhadra

I think the problem lies in this assumption:
=========================
Since L and r both are the units of length……one can be expressed in terms of the other
i.e. r = nL

========================

I will show why it is flawed. See, for a wire of given cross-section, the area of cross-section is constant, hence its radius is also constant. Length has NOTHING to do with the cross-section. So no matter by how much you change the length, cross-section is same and hence the radius is constant. It cannot be related with the length. You cannot generalise a concept just by looking at a given piece of a wire.

I agree with Mr. Lonely just want to elaborate it.....

Putting r=nL you are restricting the sample as sth whose radius(of cross-section) is multiple of its length. Then in this case, you are right, resistance do decrease when length increase (inversely proportional). Because when length L increases, cross-section also increases (because of r=nL). Since increase in r has greater effect in R than L,(because of square term), resistance decreases but not due to increase in L but DUE TO INCREASE IN r

~
Tregor,

Could you kindly elaborate me how can you assume that r = nL in terms of Physics, not mathematics ?

Cheers,

tregor your argument is clearly wrong. Just tell me in this equation Power=I2R ( I square R). Does the power increase or decrease when the resistance R increases?