What Happens When You Connect Different Voltage Batteries in Parallel?

[Amaterasu21]

Hi all,
I've been thinking about a hypothetical circuit with (say) a 6V cell of negligible internal resistance, a 4V cell of negligible internal resistance, and a resistor in parallel with each other, and I can't figure out what the potential difference across the resistor would be. I've tried to apply Kirchoff's voltage rule about the emfs and p.d.s around a closed loop, but I can't see how to apply it without contradictory answers. Any help?

 

[ergospherical]

The circuit theory assumptions have to break down for this circuit, for if the wires are truly resistance-free then KVL applied around the upper loop implies $6-4 = 0$. There must be some impedance lumped with the cells (be it in the wires or the cells themselves).

 

[Dale]

I can't see how to apply it without contradictory answers.

That is correct. It is a self contradictory scenario and therefore all you can get is contradictory answers.

 

[anorlunda]

You can conceive of several other contradictions leading to seeming paradoxes.

For example, a short circuit across an ideal voltage source. Or an open circuited ideal current source.

They can all be resolved by remembering that valid circuit analysis should never result in infinite voltage or infinite currents. If your scenario leads to such an infinity, then circuit analysis rules do not apply.

 

[Baluncore]

... a 6V cell of negligible internal resistance, a 4V cell of negligible internal resistance, and a resistor in parallel with each other, ...

That circuit definition contains a KVL contradiction.
You must specify the internal resistance of both cells before solving the circuit.

 

[DaveE]

An important point to remember here is that Kirchoff's Voltage and Current laws are only applicable for circuit analysis. Circuit analysis is a highly abstract method of modelling the real world using ideal components. These are generally restricted to resistors, capacitors, inductors, voltage sources, and current sources (there are a few others that you rarely see). These aren't the same as real voltage sources, inductors, etc. Think of it as a graphical version of simple mathematical equations. One consequence of this is that you can construct circuits that don't make sense. Like a 4V battery in parallel with a 6V battery. This is equivalent to the algebraic problem x=4 and x=6, find the value of x.

So, for real world electrical components we will use these simple ideal circuit elements to model the real component as a small network. For example, batteries may be shown as a voltage source with one (or more) resistors. How you choose to model your components allows you to decide how accurate (and difficult) your solutions are.

There are a lot of really well educated people that don't understand this. Including a rather famous MIT physics professor. It's not that they don't know physics, it's that they don't know how EEs talk about network analysis with "lumped element" models (also, they may not know about the problems with voltage and/or current probes in the lab).

If you study network theory, you'll learn of two subtle, but impossible scenarios*. The first is any circuit loop that includes only voltage sources and/or capacitors. The second is any circuit node that only has current sources and/or inductors connected to it. Your network has the former.

* There is a trivial solution where all of the initial conditions match perfectly.