Locating Voltage Points

the voltage between A and C in Fig 2.13 a is zero because there is no connection between the two batteries.
the voltmeter is across two unrelated points.

potential (voltage) is measured between two points, ASSUMING the two points are related somehow.
the relationship can be from two parts of the same circuit,
or two positions in space where a charge distribution sets up an electric field and the corresponding potential field
in figure a, the two batteries are separate - each battery is its own circuit and the two are not related
while in figure b, the ground symbols connect the batteries, so they are part of the same circuit

looks like wwfeldman answered before me!
Edit: fix typo

Maxmichaels said:

Greetings. I’m working on a example on finding the voltages between the various points. Now if you look at figure 2.13 A) & B) I need to find the voltages between points A & C in both figure examples. Now in Fig.A) the voltages between A & C is zero, but in FigB) the voltage point is 3. Now in one example there is a ground symbol and the other their isn’t. I’m assuming that’s what gives the voltage change but I still don’t understand how the voltages in between A&C is 3 and the other is zero. Help.

Click to expand...

The answer given for (a) are wrong; the voltages are not zero, they are undefined. The two circuits are disconnected and the potential difference between the two cannot be measured.

I hope you have figured out the rest of the circuits. Whenever you see the ground symbol, you assume the circuits are connected at that point.

c_mitra is correct
the voltages are undefined - there is no analytical way to apply
the appropriate laws and calculate the voltage on should measure

but if you took a voltmeter and tried to measure it, it would likley read zero
it is almost equivalent to having nothing connected to a voltmeter - it will read zero

wwfeldman said:

it is almost equivalent to having nothing connected to a voltmeter - it will read zero

Click to expand...

That is true.

But that is because common voltmeters measure current and convert that to a voltage.

To measure the potential difference between two points, you need to carry one unit charge from one place to the other and measure the work done.

When you move a charge, you produce a current.

If the two circuits are not connected, then there is no way to carry the charge from one point to the other.

Moving a charge from one point to another is problematic. When you move a charge, it changes the local electric field and also creates another magnetic field and produces EM radiation which causes loss of energy.

Wise men say that you must move the charge so slowly that it is perfectly reversible.

There are voltmeters that have very low input current requirements (often called electrometer input voltmeters).

For example, there is an electric field (about 100V/m) close to the surface of the earth but that cannot be measured with a regular voltmeter.

Some historical electrometers can be seen at https://orau.org/health-physics-museum/collection/electrometers/index.html

Maxmichaels said:

Greetings. I’m working on a example on finding the voltages between the various points. Now if you look at figure 2.13 A) & B) I need to find the voltages between points A & C in both figure examples. Now in Fig.A) the voltages between A & C is zero, but in FigB) the voltage point is 3. Now in one example there is a ground symbol and the other their isn’t. I’m assuming that’s what gives the voltage change but I still don’t understand how the voltages in between A&C is 3 and the other is zero. Help.

Click to expand...

There is also some errors with the polarity. In b) V_bc should be -9V and not 9V.