# Which lead of a voltmeter is subtracted from which to get the potential difference?

I'm working on a problem (on paper, not in a lab) in which I'm using the Hall Effect to determine the sign of the charge carriers in a current. In order to get this right, I need to know which end of the wire is at a higher voltage.

Say I have a voltmeter with a red wire plugged into the + terminal and a black wire plugged into the - terminal of the voltmeter.

I touch the red wire to point A in a circuit, and I touch the black wire to point B.

$$\\Delta V\$$ gets displayed on the voltmeter display.

Is the voltmeter reporting $$\V_A - V_B\$$ or is it reporting $$\V_B - V_A\$$?

Let's say the voltmeter is functioning properly and that it behaves in the "standard" way, if there is a standard way, and further let's assume I don't have a physical voltmeter to test this out.

EDIT: Since there was interest in the original textbook question, here it is. Given the image below, what sign are the charge carriers? • Va - Vb . This is extremely easily solved experimentally using a battery of known potential points if you DID have a meter to test it out - consider getting one to further your understanding if you want to. Apr 14, 2020 at 21:25
• @QuickishFM Thanks. That's what I thought, since black (negative) is often called "ground". This means my answer key is incorrect! I was initially doubting myself and not the answer key but later suspected the answer key has this one wrong... Apr 14, 2020 at 21:29
• might be a trick question ... what is the exact wording of the question? ... what is the wording of the answer key? Apr 14, 2020 at 21:32
• Some of my textbooks had some bad answer keys, thats not too uncommon unfortunately. Having said that, you could post a picture of the question and the answer key for us to verify if you understood the question right as well, since answer keys are largely correct to begin with. Apr 14, 2020 at 21:33
• OK, I've updated the original post with the original textbook problem. I've since realized that the answer key is right after all... funny how that works! The problem with the answer key is it gave the answer only, and not the process that leads to the answer. Apr 14, 2020 at 21:55

Your multimeter's COM terminal is the one from which measurements are referenced. We usually push the black lead in there.

The 'V' terminal takes the red lead and the meter displays the voltage with reference to the COM socket.

So yes, the meter is displaying $$\ V_A - V_B \$$ or $$\ V_V - V_{COM} \$$.

For those that are curious about the textbook problem I posted as an edit to my original post, here is my solution...

First, let's forget about the magnetic field entirely. What causes current to flow along the slab? There is an electric potential difference between one end of the slab and the other, and therefore there's an electric force acting on charged particles. This is the potential difference that is measured by the voltmeter along the length of the slab.

In which direction is the current? Let's look at the figure. What is happening to the voltage along the length of the slab? The voltmeter is telling us that the point that is farther away is 0.73 V higher than the point that is nearer to us (this is the part I had to be sure of, since everything depends on it... and hence the reason for my original post). And conventional current flows from high voltage to low voltage. So that means the current is flowing from the far end of the slab to the near end of the slab.

OK now let's re-introduce the magnetic field. I have a current flowing towards me along the slab. Using the right-hand rule for cross-products, which way does the magnetic field cause the charge carriers to move? It depends on whether the charge carriers are positive or negative.

Let's say they're positive. Using the right-hand rule, I point my fingers out of the page / towards myself (in direction of current) and rotate my palm so that it points up (in direction of B), then the force is to the left (the direction my thumb points). This is for positive charge carriers, so the left edge of the slab will become positively charged (since the charge carriers move over to the left side) and the right edge of the slab will become negatively charged (because of a deficit of positive charges that have been moved away by the B field).

Now suppose the charge carriers are negative, but B is still the same. Now we have electrons flowing away from us (because electron current is in the opposite direction of conventional current... this is the part I did incorrectly originally and the reason why I got a different answer than the answer key). Using the right-hand rule, I point my fingers away from me, rotate my palm until it faces up, and then flip my result (since the charge is negative). So the force on the negative charge carriers would be to the left, so the left edge becomes negative and the right edge becomes positive.

This movement of charges to one edge or the other (perpendicular to the flow of the current) is called the Hall Effect. The charge separation created by the Hall Effect is the cause of the voltage we read along the width.

So which is correct, positive or negative charge carriers? We can get that by looking at the voltage difference across the width of the slab. The voltmeter at the top of the figure is telling us that the right edge is at a lower voltage than the left edge (since the voltage difference is negative... this is another place where the answer to my original post is crucial). Therefore the right edge must be negative and the left edge positive. And now looking back at our two right-hand rule applications, we see that the right edge being negative must agree with positive charge carriers.

So the charge carriers are positive.

The previous sentence is the only info that the answer key gave. Now, based on my logic above, I believe the answer key was right after all.

• well, technically the charge carriers could be some other mobile positive-charge thing. Maybe the entire apparatus is made of antimatter and the charge carriers are positrons. Maybe the slab is actually a fluid and the charge carriers are predominantly cations. Apr 15, 2020 at 2:18
• @Hearth That's true. I was implicitly assuming the slab is metal... an unspecified metal (could be copper, could be doped silicon, etc) with charge carriers of unknown sign. Apr 15, 2020 at 18:57