# Current through a resistor

My question refers to the image below: From what I have previously thought, in order to calculate current through a resistor you would use Ohm's Law, meaning you get the voltage across the resistor divided by the resistance of the resistor.

Usually, you can work out the voltage across a resistor using the potential divider rule, but in the cases presented in the images, you are given voltages on either side of the resistor.

So my question is, why is the lower voltage subtracted from the higher voltage? I understand that adding the two voltages would go against the conservation of energy, but I can't think of a logical or a mathematical reason for the subtraction.

• Re, "Adding the two voltages would go against the conservation of energy" Voltage is not energy. Voltage is a potential: The difference between the voltage at two different points in a circuit is proportional to the amount of energy that an electron gains from the electric field or loses to the electric field in moving from the one point to the other. It's just like how the difference in height between two points is proportional to the amount of energy that a massive object gains from or loses to the gravity field when you lift it or lower it between the two points. – Solomon Slow Jan 14 '20 at 15:13
• Ok that makes sense, thanks – Albert Jan 14 '20 at 15:27

$$V$$ in ohms law is the difference between potentials, by definition. Now, why would the law require subtraction instead of addition? Imagine than instead of voltage your law is proportional to the length of some object. If in you system of coordinates, one end of the object has a coordinate $$x_1$$ and the other end a coordinate $$x_2$$, the length would be the difference between them, adding them does not have any meaning. The same happens with the voltage. A specific voltage is an absolute measure relative to some origin of coordinates. You want the difference, which is independent of this system of coordinates.

• I see, your length analogy made it easy to understand, thank you. – Albert Jan 14 '20 at 15:19

So my question is, why is the lower voltage subtracted from the higher voltage?

$$V_A$$ and $$V_B$$ in your diagram represent electrical potentials with respect to some reference point in the circuit (which is not shown). In applying Ohms law $$V$$ is the difference in potential between the two terminals of the resistor.

In your first diagram the electrical potential $$V_A$$ is apparently greater than $$V_B$$ with respect to that reference. The difference in potential across $$R_1$$ is therefore $$V_{A}-V_B$$, and current flows in the direction shown.

In the second diagram the situation is reversed.

Hope this helps.

When I was at school I was constantly told that the flow of electricity was like the flow of water in a river. Voltage, or Potential Difference was like the difference in height of two parts of the river. So the tallest waterfall in the world (called the Angel Falls) would have a large potential because the difference in height (Htop - Hbottom) is large. You do the subtraction in order to find the potential difference.

This means that by the time the water reaches the bottom of the falls it has picked up a lot of kinetic energy. If a river only drops slightly, the potential difference is less, and less kinetic energy is gained per unit of water. So a large sluggish river may have only a small potential difference. This is just like the voltage difference between the two sides of a resistor.

Using the same analogy, the amount of water flowing past a point in the river is a measure of the current. The Angel Falls is tall but not a lot of water flows down it so the current is small. The Thames through London is wide and deep. Although the potential difference is not much from one point to another, a heck of a lot of water slowly flows. The current is high but the potential difference is low.

Hope that is coherent and makes some sort of sense!