Lab: Plot graph of resistance, R (in Ohms) versus 1/d^2

[mirs]

Homework Statement

We were given five samples of nichrome wire, each with a different diameter but the same length. A micrometer was used to measure the diameters. We then measured and recorded the electrical resistance, R in Ohms of each nichrome sample with a DMM. I will try to summarize all of this as clearly as possible below:

Sample #, Diameter (± 0.01 mm), Resistance
20, 0.80 mm, 23.6 ohms
22, 0.62 mm, 36.5 ohms
24, 0.56 mm, 57.4 ohms
28, 0.31 mm, 159.0 ohms
30, 0.26 mm, 226.8 ohms

Length of wire: 10.00 ± 0.01 mWe are required to plot a graph of resistance, R (in Ohms) versus the inverse square of the diameter (in m^-2), i.e.; "R" vs. 1/d^2. We then have to calculate the slope of the graph, and from the slope determine the resistivity p of nichrome wire. We then compare our value of p with the published value of p = 1.18 x 10^-6 ohm*m.

Homework Equations

The electrical resistance R (Ohm) of a particular cylindrical sample of material is related to the resistivity by:

R = pL/A = 4pL/pi^2 = (4pL/pi)(1/d^2)
R = 1/d^2

where L is the length of the wire, A = (pi*d^2)4 is the cross-sectional area, p is the resistivity, and d is the diameter of the wire.

The Attempt at a Solution

My problem lies in calculating the inverse squared diameter itself, and plotting the values. My first instinct was to convert the diameters to m, so instead of 0.80 mm we have 0.80 x 10^-3 m. I then square the value, then invert it to get 1562500 m^-2. HOW does this make any sense? It also looks off when i try to graph the values on excel; with 1/d^2 on the x-axis and R on the y-axis, the slope should be 4pL/pi, right? I've written out all of the equations relevant to this lab right out of the lab manual -- so why are the 1/d^2 values so large? Did I miss something, what could I be doing wrong?

Any help would be immensely appreciated!

 

[TSny]

My first instinct was to convert the diameters to m, so instead of 0.80 mm we have 0.80 x 10^-3 m. I then square the value, then invert it to get 1562500 m^-2. HOW does this make any sense?

Why do you say that this looks wrong?

with 1/d^2 on the x-axis and R on the y-axis, the slope should be 4pL/pi, right?

Yes.

 

[Student100]

First, why would the cross sectional area of the wires be $A=(\pi*d^2)4$? I'm assuming you meant: $A=\frac{(\pi*d^2)}{4}$.

For the inverse diameter equals R, what would the slope of the best fit be? $y=mx+b$ so $R=m(1/d^2)$ I don't explicitly see anything wrong with your attempt, and the values should be quite large when converted to meters. So I think you're on the right track.

I don't know why they have you do this way, seems kind of roundabout to me, unless I'm reading the post wrong.

Whoops, didn't see that TSny had already replied.

 

[mirs]

Why do you say that this looks wrong?
Yes.

Basically my graph is throwing me off since it's not linear. Would it be incorrect if I take the most linear portion of the graph and calculate the slope manually from there, instead of adding a trendline and having excel do it?

 

[mirs]

First, why would the cross sectional area of the wires be $A=(\pi*d^2)4$? I'm assuming you meant: $A=\frac{(\pi*d^2)}{4}$.

For the inverse diameter equals R, what would the slope of the best fit be? $y=mx+b$ so $R=m(1/d^2)$ I don't explicitly see anything wrong with your attempt, and the values should be quite large when converted to meters. So I think you're on the right track.

I don't know why they have you do this way, seems kind of roundabout to me, unless I'm reading the post wrong.

Whoops, didn't see that TSny had already replied.

Sorry, you're right, it is over 4, not *4

 

[Student100]

Basically my graph is throwing me off since it's not linear. Would it be incorrect if I take the most linear portion of the graph and calculate the slope manually from there, instead of adding a trendline and having excel do it?

Check calculations and add the zero point.

Unless I mucked it up myself, should be nice and linearish. Also, adjust your x scale, and do a best fit, instead of whatever that is. You should be okay, I think you just need to check the scale really and one number might off.

 

[TSny]

I agree with Student100. Your data looks good, but it appears that you didn't plot the data correctly.
Check the 4th data point in particular.

 

[mirs]

Check calculations and add the zero point. View attachment 109069

Unless I mucked it up myself, should be nice and linearish. Also, adjust your x scale, and do a best fit, instead of whatever that is. You should be okay, I think you just need to check the scale really and one number might off.

That looks more like it... thank you so much for your help!

 

[Student100]

View attachment 109072
That looks more like it... thank you so much for your help!

Now it's looking good. Remember to label your axis and give a title. Also, not sure if you're required to plot the error bars or not, but if so don't forget them!

 

[mirs]

I agree with Student100. Your data looks good, but it appears that you didn't plot the data correctly.
Check the 4th data point in particular.

Perfect. Thank you for all your help!

 

[mirs]

Now it's looking good. Remember to label your axis and give a title. Also, not sure if you're required to plot the error bars or not, but if so don't forget them!

Does this look ok?

 

[Student100]

Does this look ok?

View attachment 109074

Looks good to me.

 

[mirs]

Looks good to me.

I don't think we are required to add error bars but is it better to include them? Thanks again!

 

[Student100]

I don't think we are required to add error bars but is it better to include them? Thanks again!

I would remove them then, I didn't check if they were right, and no sense to do so if not required.

 

[mirs]

I would remove them then, I didn't check if they were right, and no sense to do so if not required.

OK, will do, thank you. I am now comparing my experimental value with the theoretical value. The slope is 2x10^-5, so I worked out the following:

|[(1.18 x 10^-6 ohm*m) - (2 x 10^-5 ohm*m)]/(1.18 x 10^-6 ohm*m)| x 100 = 1595 %

That is huge! I worked it out a couple of times and I can't seem to figure it out. Is it possible that my slope is off?

 

[mirs]

OK, will do, thank you. I am now comparing my experimental value with the theoretical value. The slope is 2x10^-5, so I worked out the following:

|[(1.18 x 10^-6 ohm*m) - (2 x 10^-5 ohm*m)]/(1.18 x 10^-6 ohm*m)| x 100 = 1595 %

That is huge! I worked it out a couple of times and I can't seem to figure it out. Is it possible that my slope is off?

EDIT -- Wait, the slope does not give me p! I need to solve for p, since that is 4pL/pi and sub in the given value for L. Oops.

 

[TSny]

Your graph is giving the slope to only one significant figure. How many sig figs for the slope do you think you should have?

 

[mirs]

Your graph is giving the slope to only one significant figure. How many sig figs for the slope do you think you should have?

Considering the diameter is measured to 2 sig figs, the slope should have 2 as well, right?

 

[TSny]

Considering the diameter is measured to 2 sig figs, the slope should have 2 as well, right?

Sounds good to me.

 

[mirs]

Sounds good to me.

Thank you! One last question -- if the uncertainty for measuring the diameter is ± 0.00001 m, how would that change in the inverse square of the diameter? I would think to convert the absolute uncertainty to a relative uncertainty; i.e. 0.0008 ± 0.00001 m would be 0.0008 m ± 1.25 %, and then multiply that by 2 since it is being squared, and then happens if I take the inverse? Do I just divide 1 by 0.0008 and leave the relative uncertainty alone?

 

[TSny]

if the uncertainty for measuring the diameter is ± 0.00001 m, how would that change in the inverse square of the diameter? I would think to convert the absolute uncertainty to a relative uncertainty; i.e. 0.0008 ± 0.00001 m would be 0.0008 m ± 1.25 %, and then multiply that by 2 since it is being squared,

I believe that's right. The effect of squaring is to double the relative uncertainty if the relative uncertainty is small.

and then happens if I take the inverse? Do I just divide 1 by 0.0008 and leave the relative uncertainty alone?

I'm not sure I understand what you're asking here. The relative uncertainty in 1/d² should essentially be the same as the relative uncertainty in d² as long as the relative uncertainty in d is small.

 

[mirs]

I believe that's right. The effect of squaring is to double the relative uncertainty if the relative uncertainty is small.

I'm not sure I understand what you're asking here. The relative uncertainty in 1/d² should essentially be the same as the relative uncertainty in d² as long as the relative uncertainty in d is small.

That answers my question actually, sorry I was unclear. Thank you for your help!