Ohm's law and resistivity, again

[Marbles]

Sorry everyone, I forgot to post my efforts in solving the problem, so here I go again:

The question says: Suppose you wish to fabricate a uniform wire out of 1.00 g of copper. If the wire is to have a resistance of R=0.500 Ohm, and a density of d=8920 kg/m3, and if all the copper is to be used, what will be the:

a)length of the wire? >>ans. is 1.82 m

b) diameter of the wire? >>ans. is 0.28 mm


The equation is: R=p(rho)*l(length)/A(cross sectional area)

I found the area only, I'm lost!
But at least, based on the area, my diameter is supposed to be correct, but it wasn't!
I got 0.001>>lol

Anyways, I will show you what I did to find the area and the diameter:

First I found the volume of the wire: v=m(mass)/d(density)=0.001 kg/(8920 kg/m3)
and I got 1.12e(-7) m3

Then I found height, guess how!

R=p*l/A right?? ok, I canceled the length on top (numerator) with the length in the area equation: L*h and I was left with R=p/h, which means h=p/R=1.72e(-80) (rho or resistivity of copper, got it from our textbook)/0.5= 3.44e(-8) m

Now I substituted both values in the formula: A=v/h=1.12e(-7)/3.44e(-8)=3.25 m2

Now I used it to find the diameter: d=A/pi=3.25/3.14(pi)= 0.001 mm

I know that there are several mistakes, so I'd like you all to help me improve my solution and also help me find the length. I'd really appreciate that!

 

[JaredJames]

R= Resistance
p = Resistivity
A = Area
L = Length
V = Volume
m = Mass
D = Density

R = pL/A

Rearrange

R/p = L/A

V = m/D = LA

Now, multiply each side by the volume

V(R/p) = LA(L/A)

VR/p = LAL/A = L²

Sqrt(VR/p) = L

That's a true beauty and I'm proud of it. I'm sure you can work it from here.

_{*I normally wouldn't give such a direct answer, but this one baffled me and took a lot of effort.}

 

[Marbles]

wow I get everything, except that, why did we multiply both sides by v??

 

[JaredJames]

wow I get everything, except that, why did we multiply both sides by v??

Well we know volume.

So, by multiplying each side by it we can cancel out the area value, leaving only length.

This means instead of having two unknowns (A and L) we only have one (L).

It's like adding a constant to each side.

 

[Marbles]

I know that area is going to be canceled...but in what sense are we doing this? why? what makes u know that this is the way it's going to work? I'm not getting it!

 

[JaredJames]

We're doing it so that instead of there being 2 unknowns, we only have 1.

So long as A and L are unknown, it is impossible to solve. So by introducing volume on each side, all it does is remove the A value so we don't have to worry about it.

It's purely to reduce the equation so we can solve it.

Perhaps you should plug in the values and see if it works?

 

[Marbles]

aha I see...so that's how it works! wow! I'd never be able to think this way!
But wait! what about the diameter?

 

[JaredJames]

What about the diameter?

I'd assume the wire is round:

A=pi*r².

Now you know volume and you know length. Leaving only r to find.

V = A*L

 

[Marbles]

isn't it V/L=pi*r^2??
we find r^2 then square root? then why am I getting different answers?

 

[JaredJames]

Right hang on, I'll work it out myself.

 

[JaredJames]

OK

a) answer I got was 1.825m - note resistivity is 1.68E-8 not 1.72E-80 as you have above.

b) answer I got was 2.8E-4m or 0.28mm - remember, the number you get at the end is the radius, you double it for diameter.

Both agree with the answers you gave in the OP.

 

[Marbles]

omg how did u solve for r?? I'm not able to get it right no matter what I do!

 

[JaredJames]

V = AL

V/L = A

A=pi(r²)

A/pi = r²

Sqrt(r²) = r

2r = Diameter

 

[JaredJames]

omg how did u solve for r?? I'm not able to get it right no matter what I do!

Are you getting the right answer for the length?

 

[Marbles]

yes I am getting the right answer for the length!

 

[Marbles]

Oh yay! I finally got it right! I'm really glad! Thank you so much for you help and support...and above all, your patience!