Electric field model using a cylindrical capacitor

[kesaluj]

Homework Statement

In a lab experiment we measured the potential at different points within a cylindrical capacitor electric field modeling plate thing (apparently that's the best I could do to translate that into English). The positive electrode was connected in the middle and the negative one was on the outside. The measured values decreased as we got further away from the center.

Homework Equations

3. The Attempt at a Solution [/B]

The part I'm unsure about (other than what to even call this experiment) is the theoretical values. We were given these two formulas:

V(r) = \frac {U \cdot \ln(\frac {r}{r_o}) }{\ln(\frac {r_o}{r_i})},
E(r) = - \frac {U }{r \cdot \ln(\frac {r_o}{r_i})},

where r_o is the outer radius, r_i is the inner radius, and U is the voltage used. Both of them result in negative values, since r_o is clearly larger than r in the logarithm in first equation and there's a minus sign in the second one. The absolute values, though, look about right.

I've tried to research this, but my mediocre physics knowledge is failing me. I get that this is a Gaussian surface thing and I know that we're looking at the \frac {V_{{r_o-}{r}}} {V_{r_i-r_o}} proportion, but I'm having a hard time making sense of the negative values, especially since we were straight-up given these formulas in the lab report instructions.

Does anyone have any pointers?

 

[BvU]

Hello kesaluj,

Look at it from the maths side: you can check that your relations satisfy $E = - {dU\over dr}$

And: at $r=r_0$ you have $V= 0$ when at $r=r_i$ you get $V = -U$
So U increases going outwards, meaning E points inward -- so it's negative.

within a cylindrical capacitor electric field modeling plate thing

Something like an electrolytic trough or https://ocw.mit.edu/resources/res-6-001-electromagnetic-fields-and-energy-spring-2008/chapter-7/07.pdf ?

 

[kesaluj]

Thank you for responding!

And: at $r=r_0$ you have $V= 0$ when at $r=r_i$ you get $V = -U$
So U increases going outwards, meaning E points inward -- so it's negative.

Ah. So the values we measured are supposed to be negative? Looks like I'll have to read up on this, because I clearly understand less than I hoped.

Something like an electrolytic trough or https://ocw.mit.edu/resources/res-6-001-electromagnetic-fields-and-energy-spring-2008/chapter-7/07.pdf ?

Yeah. I think it was a sheet of conducting paper with evenly spaced out electrodes, covered with some insulating foil that had holes over the electrodes where you're supposed to make the measurements.

 

[kuruman]

So U increases going outwards, meaning E points inward -- so it's negative.

Except that

The positive electrode was connected in the middle and the negative one was on the outside.

Doesn't this mean that the middle is at a higher electric potential than the outer electrode? If that's the case, doesn't the electric field point from a region of high potential towards a region of low potential, in this case outward? I think that the equations posted by OP do not match the experimental procedure.

 

[BvU]

Good point. I missed that -- only looked at the expressions, which were OK if the battery was connected with + outer and - on inner.
Measurements positive and decreasing value outward indicate the opposite. But then E points outward.

 

[kesaluj]

The instructions said that these equations were correct for a setup with the outer electrode's potential equal to zero and the measured values definitely decreased outwards (unless they we supposed to be negative and we missed that). Are the equations wrong or did we mess something up?

 

[kuruman]

What did the instructions say about quantity U in the equations? How was that defined?

 

[kesaluj]

What did the instructions say about quantity U in the equations? How was that defined?

It was set to 10 V.

 

[kuruman]

You did not answer my question. Usually, when equations are introduced in instructions, the meaning of the symbols is explained. Is there such an explanation for U in your instructions? If so, what is it? Also, when you connected your voltmeter to your electrodes, where did its + terminal go? I'm trying to establish whether you actually measured U or whether you measured -U and thought you were measuring U.

 

[kesaluj]

You did not answer my question. Usually, when equations are introduced in instructions, the meaning of the symbols is explained. Is there such an explanation for U in your instructions? If so, what is it?

Sorry! It was defined as the voltage of the battery.

Also, when you connected your voltmeter to your electrodes, where did its + terminal go? I'm trying to establish whether you actually measured U or whether you measured -U and thought you were measuring U.

My lab partner connected those and unfortunately I didn't check. If the voltometer had been connected the "wrong" way round and the actual measurements were negative, would the rest of this make sense ? The battery plugged in with the "+" in the middle and the absolute value of the measurements decreasing outwards? The equations they'd given us clearly produce negative results (I've seen another group's lab report and it seems like they just ignored the sign here, but I'd like to try to do this justice).

 

[kuruman]

I think you may safely assume that you reversed the voltmeter connections and ignore the sign. Since you didn't check the connections yourself one can only speculate. Yes, the magnitude of the electric field will decrease as you move outwards regardless of how you connect the battery and the voltmeter. That's because, in cylindrical geometry, the electric field lines are along radial lines (inward or outward) between a smaller circle and a larger one and the field is stronger where the lines are closer together nearer the center.

Next experiment please be doubly careful, record everything and keep an eye on your partner.

 

[kesaluj]

I think you may safely assume that you reversed the voltmeter connections and ignore the sign.

Thank you! To sum up: the formulas they gave us are both correct in this case, but our measurements were reversed and so the results (both theoretical and experimental) should all be negative?

Next experiment please be doubly careful, record everything and keep an eye on your partner.

I did try to be careful this time, but it's easy to miss something and I'm not very experienced or knowledgeable when it comes to physics. To justify: the cylindrical capacitor was just a portion of the whole experiment, so there were lots of other things to keep track of too.

 

[kesaluj]

I apologize for not letting this go, but... would anyone be able to explain to me the reason behind all the measured/calculated potentials being negative if the battery was connected from the positive side?

 

[kuruman]

Look at the drawing. It shows the battery connected as you said and a possible connection for the voltmeter. The convention is that "red" means "higher potential" than "black." The red lead is connected to the "plus" terminal of the voltmeter, sometimes labeled "Input" or "V"; the black terminal is connected to the "minus" terminal of the voltmeter sometimes labeled "COM" or "GND". When the voltmeter is connected as shown, it displays positive values because the plus terminal is at higher potential than the minus terminal. If your connections, either the battery or the voltmeter, are such that the plus terminal of the voltmeter is at lower potential than the minus terminal, then the voltmeter will read negative values. I hope this helps.

 

[kesaluj]

Look at the drawing. It shows the battery connected as you said and a possible connection for the voltmeter. The convention is that "red" means "higher potential" than "black." The red lead is connected to the "plus" terminal of the voltmeter, sometimes labeled "Input" or "V"; the black terminal is connected to the "minus" terminal of the voltmeter sometimes labeled "COM" or "GND". When the voltmeter is connected as shown, it displays positive values because the plus terminal is at higher potential than the minus terminal. If your connections, either the battery or the voltmeter, are such that the plus terminal of the voltmeter is at lower potential than the minus terminal, then the voltmeter will read negative values. I hope this helps.

Thank you. I do understand what could have happened with the measurements, but I still can't really make sense of the theoretical values being negative in the first place. The formulas we're supposed to use clearly give out negative results and I don't understand where that comes from.

 

[kuruman]

The formulas we're supposed to use clearly give out negative results and I don't understand where that comes from.

The formulas that you posted do not match the experiment that you performed. The formula you posted for the electric field is appropriate for an electric field that points radially in towards the center. That's what the negative sign up front means. It comes from the result that, for the electric field to point radially inward, you need the outer electrode to be at higher potential than the central electrode. As I mentioned earlier, electric field lines point from a region of high electric potential to a region of lower electric potential. That's not how you hooked the battery, so you should expect the electric field lines to be radially outward and you should use the formula for the electric field without the negative sign.

 

[kesaluj]

The formulas that you posted do not match the experiment that you performed. The formula you posted for the electric field is appropriate for an electric field that points radially in towards the center.

Excellent! I thought that might be the case, but the instructions state that the two formulas I posted are for "when the outer electrode's potential is zero" and that's why I was confused. They don't mention any other scenario. By any chance, would swapping r_i and r_o in the logarithm in the denominator do the trick?

 

[kuruman]

By any chance, would swapping $r_i$ and $r_o$ in the logarithm in the denominator do the trick?

Perhaps you should try to derive these expressions yourself. Then you will understand better what's going on. Assume that the center is negatively charged and that the potential is zero at $r = r_o$. You should get the equations that were given to you.