# What does applying a potential difference mean?

I have been studying the physics of accelerating electrons through a vacuum using an electron gun. It says that a potential difference (p.d.) is applied between the filament lamp and an anode. The electrical energy of the electrons is converted to the K.E as the electrons are attracted to the anode.

What does applying a p.d actually mean? I know a p.d is a measure of energy transferred by electrons per unit charge so does it just mean creating a difference in charge to cause an electric current?

How does increasing this difference in charge increase the potential difference?

• The difference in electric charge causes the electromagnetic force to pull electrons towards the plus end. Increasing the difference in charge increases the potential difference since $F \propto \frac{qQ}{r^2}$ which means it scales linearly with the net charge present in the area that is attracting the electrons. Jan 20, 2021 at 13:27
• Thanks! @Martin van IJcken
– M G
Jan 20, 2021 at 13:28
• @MartinvanIJcken That should really be an aswer.
– noah
Jan 20, 2021 at 13:32

Applying a potential difference means modifying a system such that the potential energy at point A is higher than at point B. In the case of an electric potential, this potential difference is caused by the electromagnetic force.

$$F_{el} = -\frac{1}{4\pi \epsilon_0}\frac{qQ}{r^2}$$

The magnitude of a potential difference is equal to the work done on a particle if it moves between the two points, so an electron at distance R from a point with electric charge is

$$U = \int_0^R F_{el}(r)\cdot dr = \int_0^R -\frac{1}{4\pi \epsilon_0}\frac{-eQ}{r^2}\cdot dr = -\frac{eQ}{4\pi \epsilon_0} \frac{1}{R}$$

Thus, when we increase the charge $$Q$$, we see that the potential energy at the point A decreases, and the difference between point A and B increased.

Keep in mind that in an electric circuit the potential is not caused by a single point source but rather a positive electric charge at the anode, and an equal negative charge at the cathode.

a p.d is a measure of energy transferred by electrons per unit charge

This is actually not fully correct. You can have a potential difference even when there's no energy transfer. You can have a potential difference in an open circuit. E.g. there is a potential difference across the two terminals of a battery all the time, but current only flows and energy is only being transfered when the terminals are connected.

• Electric potential energy $$U_e$$ is the potential energy related with a charge's location in its surroundings. Just like gravitational potential energy is related to an object's location high up.

• (Electric) potential is the same, but per charge, $$\frac{U_e}q$$. Often it is implied electric, so you'll often just hear the term potential being used.

• An (electric) potential difference between two points is called voltage, $$V=\frac{U_{e2} }q-\frac{U_{e1}} q$$.

The (electric) potential difference, or voltage, is a measure of "how strongly" a charge "wants" to move away from its current location. Just like an object tends to fall from the shelf if possible towards lower locations - a tendency of moving towards lower potential energy - an electric charge has a tendency of moving towards points of lower potential. The potential difference tells us "how strong" this tendency is.

But just because this tendency is established - just because the charge really, really "wants" to move - it still might not be able to. Plenty of negative charges are located on one battery terminal and they repel each other hugely and really, really "want" to move to another spot with lower repulsion. Such other spot is the other battery terminal. So when you add a path (a conductive material) from this terminal to the other, then the charges immediately move. But not before you add the path. You can have potential difference between two points without current or energy transfer. Then it is just stored and "waiting" to be released - which is the essense of a battery (and capacitor and similar storage components).