# Why is the total potential difference in a circuit the sum of the individual potential differences?

I am having difficulty in applying my understanding of electrical potential to circuits. Why is the total potential difference in a circuit the sum of the individual potential differences among capacitor plates that are in series? I tried finding a solution before but I didn't understand the references to voltage drops. As far as I know, the electrical potential is the energy required per unit charge to take a point charge and move it from one point to another, with infinity usually set as the initial position (for convenience?). Please correct me if I am confused.

• "Voltage drop" just means the difference in potential across a circuit component. – Keith McClary Sep 19 '15 at 3:45
• I would have to know what is your current "understanding" of electrical potential in circuits, even though you are clear about it for free charges. For this reason, could you please tell how would you prove what is the voltage drop across a single capacitor connected to a battery? – Rol Sep 20 '15 at 15:38
• Because the electric fields are additive and so are their integrals over paths. – gented Dec 9 '16 at 15:28

To understand the answer, you need to be aware of the concept of electric potential. Electric potential is a scalar quantity. In any circuit, there is a potential at any given point on the wire. The difference in potential between any two points in this circuit is sometimes called potential difference or voltage. You can understand the difference between potential and potential difference better here.

Imagine you had a circuit as shown below:

Assume that the potentials at $A,B,C$ are $V_A,V_B, V_C$. The potential difference between A and B is $V_A - V_B$. The potential difference between B and C is $V_B - V_C$. Adding the two, we get the expression $V_A - V_C$ which is the correct term for potential difference across AC.

I hope that this makes it clear that potential differences between two sets of points can be added to obtain the potential difference between the first and last point.

You have essentially answered your own question. You are correct that the potential as we call it is a measure of the amount of work that would be done per unit of charge as it is moved through that potential.

Electric fields are conservative in that the amount of work that must be put into moving a particle from one potential to another is exactly equal to the amount of work that would be done BY the particle were it to make a return trip. Energy in equals energy out in such a round trip.

This makes the amount of work done path independent, and thus the rule you stated (a trivial rule actually that a person by the name of Kirchoff gets credit for stating as a LAW) is true: The sum of the potentials across each of the components must equal the total potential.

There is no infinity of course, in the circuit. It's all relative to an arbitrary zero. There is no telling what the absolute potential actually is and it doesn't matter. The motivation for electrons to move is the potential DIFFERENCE between two ends of a wire, so that is what we are interested in quantifying.

The potential difference or voltage is actually the difference in the level/numbers of electrons or positive charges on the two terminals of battery or across the terminals of passive elements. Law of conservation of energy states that the total energy of an isolated system remains constant, that is why the sum of the potential differences across the passive elements in the circuit is always equal to the potential difference provided by the source/active elements.

Because potential exactly means that you have its potential at every point and to get from start to finish you need to make a number of steps. That is one leap can be broken into a series of steps, every step is closer to the target. As you move closer to the target, your potential becomes closer to the target potential. That is, you have some potential difference at every point. But, it should be not surprise that their sum is equal to the initial difference of potentials. I wounder how the field would look like to make this to fail.

Look at this like you are on a surface of a landscape, with all its ups and downs. Now, draw some contour, that you are willing to move along. Yes, there is a lift at the voltage source point

So, why you slide down from the hill, you will need to arrive at the battery bottom level. Why the hell is difference of individual heights amounts to the total height of the hill? Why is the sum of increments is equal to the total distance?