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This is my current understanding. Voltage is defined to be the potential difference between 2 points, hence it only makes sense for a voltmeter to be connected in parallel. But why must this voltmeter have such high resistance? Regardless of it's resistance the voltage will still be the same, so it will still have the same measurement. Say we take a theoretical voltmeter with 4 ohms resistance, the extra branch would draw more current in a way such that the voltage would still be the same. So what is the need for such high resistance?

Furthermore a real life voltmeter does not have infinite resistance, so does it calculate the voltage with the tiny bit of current that flows through?

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    $\begingroup$ If you're measuring the voltage across an impedance matched circuit with an internal resistance of 150 kOhms, then you've just effectively shorted it out and you're not measuring anything. $\endgroup$
    – Jiminy Cricket.
    Commented Aug 25, 2022 at 7:28

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Take for example this simple circuit:enter image description here

The volatage across $R_1$ is $U_1 =V \cdot \frac{R_1}{R_1+R_2}$ and the voltage across $R_2$ is $U_2 =V \cdot \frac{R_2}{R_1+R_2}$. Now suppose you want to measure these two voltages with a voltmeter, but your voltmeter has a really low resistance $R_v$. If you put the voltmeter parallel to $R_1$ to measure the voltage $U_1$, you will actually almost shortcut $R_1$ so the voltage that you will measure $ U_{meas}$ will be smaller than $U_1$.

More precisely, the resistor $R_1$ and the resistance of the voltmeter $R_v$ in parallel have the resistance $$R_{tot}=\frac{R_1 \cdot R_v}{R_1+R_v}$$ so the voltage that you will measure is actually $$U_{meas} = V \cdot \frac{R_{tot}}{R_{tot}+R_2} \neq U_1$$

But if $R_v$ gets really big $$R_{tot}=\lim_{R_v \to \infty} \frac{R_1 \cdot R_v}{R_1+R_v} = \lim_{R_v \to \infty} \frac{R_1 \cdot R_v}{R_v} = R_1$$

So the measured voltage in this case will be $$U_{meas} = V \cdot \frac{R_{tot}}{R_{tot}+R_2} = V \cdot \frac{R_{1}}{R_{1}+R_2} = U_1$$

For the second part of your question, yes a voltmeter use the tiny amount of current flowing trough to measure the voltage. You might know that a current induces a magnetic field, so the greater this tiny current is, the greater this magnetic field will be. We can then put a magnet next to it and measure the force on the wire and deduce the current from it, which then allows us to know the voltage using Ohms law.

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  • $\begingroup$ If we connected a low resistance voltmeter in parallel, would it not draw more current as well. And because voltage is always the same across parallel, then shouldn't the measured voltage be exactly the same despite the voltmeters low resistance. $\endgroup$
    – Quin Gardiner Bax
    Commented Aug 25, 2022 at 18:58
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    $\begingroup$ No. Analyze the circuit yourself. If we connect a low resistance voltmeter in parallel to R1, the total resistance in the circuit drops, and the current increases. With the larger current across R2, the voltage drop across R2 increases. Since the total voltage drop in the circuit remains equal to V, the voltage drop across R1 must decrease. Assign an R value to the voltmeter and you can see how much it drops with simple circuit analysis. $\endgroup$
    – Bill Watts
    Commented Aug 26, 2022 at 0:01
  • $\begingroup$ As you said, a voltmeter in parallel will increase the current in your circuit. Since the current is greater, the voltage drop will be even greater at other resistors, which implies that the measured voltage drop is smaller. Imagine also the case where you have a current source instead of a voltage source. In this case, if you connect a voltmeter parallel to a resistor with the same resistance as the voltmeter, you will actually only measure half of the voltage that you would have without the voltmeter. $\endgroup$
    – Nicolas Schmid
    Commented Aug 26, 2022 at 7:37
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The need is for the voltmeter resistance to be much greater than the impedance of the circuit between the two points being measured so that the parallel combination of the two has negligible effect on the impedance between the points, and thus negligible effect on the voltage being measured.

So there is no need for the voltmeter resistance to be infinite, just large enough compared to the impedance between the two points so as not to materially effect the voltage being measured.

Hope this helps.

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