Chapter 23: Electric Potential

Welcome to Last Minute Lecture.

This free chapter overview is designed to help students review and understand key concepts.

These summaries supplement not replaced the original textbook and may not be redistributed or resold.

For complete coverage, always consult the official text.

All right, welcome back to the Deep Dive.

Today we're diving into a pretty fundamental topic when it comes to really understanding electricity.

We're talking about electric potential

and energy in electric fields.

Yeah, and it sounds a bit like, you know, physics textbook jargon, but bear with us, because really getting a grip on these ideas, it isn't just about memorizing some equations.

It's about unlocking a way simpler and I think a more intuitive way to figure out how all this electric stuff actually works.

It's kind of like, I don't know, getting the cheat codes to understanding circuits, charges, all of that.

Exactly.

So today we're setting out to explore this whole energy perspective on electricity.

And hopefully what you'll see is how it often lets us kind of sidestep all the complicated stuff that comes with dealing with forces and fields directly.

Right.

We're going to sort of unpack how these concepts you might vaguely remember from way back in like high school physics work, potential energy, which we'll just call you to make things easier.

They have these like really correct parallels in the world of electricity and they give us a really helpful place to start.

And then we'll build on that.

We'll get into electric potential or V, which is kind of like an energy map for electric fields.

Right.

And then the real key is how we're going to connect all this back to the electric field itself, which is usually called E, right?

Right.

And we'll see how these energy concepts make it way easier to analyze all these different arrangements of charges.

And we'll even get into why any of this matters for everyday things like, you know, the voltage of a battery, for example.

Absolutely.

Okay.

So to kick things off, let's do a quick flashback to some basic stuff.

You know, back in physics class, remember learn about work, potential energy, conservation of energy, like imagine lifting a book against gravity.

Yeah.

Turns out those core ideas, they have some really cool counterparts when it comes to electricity and they lay a super helpful foundation for what we're going to be digging into today.

Right.

Exactly.

I mean, think about it.

When you lift a book,

you're doing work against gravity, right?

And that work gets stored as gravitational potential energy.

Then you let go and boom, that potential energy turns back into motion.

That's kinetic energy.

Well, electrostatic forces, which are basically the forces between charges, they work in a pretty similar way.

They're what physicists call conservative forces.

Okay.

And what that means is the work done by the electric field, when it moves a charge from one spot to another, well, that work doesn't care about the specific path the charge took.

It just depends on where it started and where it ended up.

Huh.

So it doesn't matter how it gets there, just where it begins and where it ends up.

Okay.

Exactly.

So because these electrical forces are conservative, just like, you know, good old gravity, we can define something called electric potential energy, which we're using you to represent.

So in a nutshell, what exactly is electric potential energy?

What's the big deal?

Okay.

So electric potential energy, or you, is basically the energy that a system of charges has because of how they're arranged in space.

Like, are they close together?

Are they far apart?

That kind of thing.

It's basically stored energy that comes from the work you did to either bring those charges together or to move them apart.

Right.

Now, because that electrostatic force is conservative,

remember, start and end points matter, we get this super important relationship.

The work done by the electric field, when a charge goes from point A to point B, we'll call that Wabi, is equal to, get this, the negative of the change in potential energy.

Okay.

So the equation looks like this.

Wabi equals UA minus U, which also equals negative delta U.

Okay.

This is a big deal because it means we can just think about energy changes.

We don't have to constantly worry about those pesky forces.

Gotcha.

So it's all about the beginning and the end in terms of energy, which, yeah, that does seem way simpler than having to track the force at like every single tiny step of the way.

Right.

Now, what if we just have two -point charges hanging out?

We'll call them Q and Q0, separated by some distance, R.

Is there a way to actually calculate the electric potential energy between them?

Oh, absolutely.

For two -point charges like that, their electric potential energy, their U, is given by a pretty straightforward formula.

U equals one divided by four pi epsilon naught times Q times Q0, all divided by R.

Okay.

Now, epsilon naught, that's a special constant.

Physicists call it the permittivity of free space.

What's really interesting here is as that distance R between the charges gets bigger and bigger, like approaching infinity,

the potential energy, the U, gets closer and closer to zero.

Okay.

Actually, a lot of times we just define zero electric potential energy as when the charges are infinitely far apart.

So, this formula is basically telling us how much potential energy is packed into that two -charge setup at a certain distance.

So, I guess the further apart they are, the less they're like feeling each other, right?

Yeah, actually.

So, less energy stored in that interaction.

What happens if we have a whole bunch of charges, though?

Like, a party of charges all interacting.

How do we figure out the total electric potential energy for that whole system?

Okay.

So, in a system with multiple -point charges, the total potential energy is actually pretty straightforward to find.

You just calculate the electric potential energy for every single pair of charges in the system using that formula we just talked about.

Okay.

And then you add up all those individual energies.

So, it's like you're accounting for all the little two -charge interactions within the whole group.

So, we're not talking about some crazy complicated field calculation for the whole system, but instead we're looking at the energy stored in each little two -charge relationship and adding them up.

Exactly.

Okay, that makes sense.

All right, let's shift gears a little bit and talk about this idea of electric potential, or V.

How is that different from electric potential energy U?

What new angle does it give us?

So, electric potential or V, this is really useful because it basically tells us the potential energy per unit charge.

Okay.

So, in equation form, it's V equals U divided by Q0, where Q0 is a test charge.

Like, imagine we just plop a tiny charge into that electric field.

The beauty of electric potential is that it's a property of the space itself because of those other charges, regardless of how big our test charge is.

Okay.

It's like we're making a map of the electrical landscape, you know, and it tells us how much potential energy a single positive charge would have if we put it at any point.

And this map is way easier to work with than the electric field, which is all vectors.

I see.

So, potential's like painting a picture of what the electrical environment looks like, and then potential energy comes into play when we actually put a specific charge into that environment.

Exactly.

Okay.

Now, what about potential difference?

We often see this written as ViB, or delta V.

That seems like a pretty important term.

It is.

So, the potential difference that ViB between two points, A and B, it's the work done by the electric force per unit charge to move a charge from point A to point B.

Okay.

So, the equation's ViB equals WadaB over Q0, and it's also equal to, well, the difference in potential between those two points, Vi minus ViB.

And here's another thing.

It's also the negative of the work that we would have to do, again, per unit charge to move that charge from B to A, but this time against the electric force.

And guess what?

This potential difference, that's what we usually call voltage.

Ah, voltage.

That word's everywhere when we talk about batteries, circuits, all that good stuff.

Exactly.

So, when you see a battery that says like 12 volts, that's the potential difference between its plus and minus terminals.

That's the energy it has to push charge through a circuit.

That's it.

Now, thinking about this potential thing, let's imagine we've got a single point charge, Q, hanging out at a distance R.

Can we figure out a formula for the electric potential it creates based on what we've talked about?

Well, we know V equals U over Q0.

Right.

And we have that formula for U for two point charges.

So, it seems like we just divide that potential energy formula by Q0 to get the potential caused by just Q.

You got it.

The electric potential, V, caused by that single point charge, Q, at a distance R, is V equals 1 divided by 4 pi epsilon naught times Q divided by R.

We're assuming here, just like with potential energy, that the electric potential is zero super, super far away from that charge at infinity.

This formula basically gives us the electrical landscape created by that single charge.

Okay, so another handy formula.

Now, just like with potential energy, what happens if we have a bunch of point charges?

How do we find the total electric potential at a specific point, considering all those charges?

So, because potential is a scalar, the total electric potential at a point, thanks to all those charges, is just the sum of the potentials made by each charge.

Okay.

So, the equation looks like this.

V total equals the sum of all the V's, which is 1 over 4 pi epsilon naught times the sum of Q over R.

Here, Q is the charge of the ith charge, and R is how far that charge is from the point we're interested in.

Adding these up is way easier than dealing with vectors, which is what we'd have to do for the electric field.

Right, just simple addition instead of all those vector components.

Much nicer.

Okay, now, what if we're dealing with a whole smear of charge, not just individual charges, like if we have charge spread out on a plate or along a wire,

how do we figure out the potential then?

So, when we have a continuous charge distribution like that, we got to switch from adding to integrating.

Imagine chopping up that charge into super tiny bits, which we'll call dQ.

The potential dV made by each of these tiny bits at a point in space is dV equals 1 over 4 pi epsilon naught times dQ over R.

Okay.

To get the total potential V, we integrate dV over the whole charge distribution.

So, the formula is V equals 1 over 4 pi epsilon naught times the integral of dQ over R.

The trick here is to write dQ in a way that makes sense for the shape of that distribution.

Right, integrals.

They can look a little scary, but the idea behind them is pretty simple, like adding up the contributions from all those little bits of charge makes total sense.

You mentioned earlier a link between potential difference and the electric field E.

How are those two things connected?

Yeah, this is a really neat connection.

The potential difference between two points, A and B, is directly related to the electric field along any path connecting them.

Okay.

And we use something called a line integral for this.

It's basically van minus VB equals negative integral from B to A of E dot dl.

E is our electric field vector, and dl is a little tiny displacement vector along our path.

That dot product, E dot dl, just means we only care about the part of the electric field that's parallel to the direction we're moving it.

Now, here's the cool part.

For fields caused by charges, electrostatic fields, this line integral doesn't depend on the path you take.

So no matter what route we choose from B to A, the potential difference is the same.

Exactly.

And that, again, comes back to that conservative nature of the electrostatic force.

Okay.

So if we know the electric field somewhere, we can find the potential difference between any two points there by doing this integral.

That's super useful for connecting those two ideas.

Let's talk about actually calculating the electric potential for some common charge setups.

You mentioned some examples earlier.

Right.

There are two main strategies.

Either we directly add up or integrate the contributions from all the little charge pieces using that V equals one over four pi epsilon times the integral of dQ over R, or we integrate the electric field using that van minus VB equals a negative integral of E dot dl formula, which ways best often depends on the shape of the charge distribution and we already know what the electric field is.

Okay.

Let's take a charged conducting sphere as an example.

What does the electric potential look like inside and outside the sphere?

So for a charged sphere, all the extra charge hangs out on its surface.

What's really cool is that the electric potential outside the sphere at any distance R that's greater than the radius of the sphere, it's exactly the same as if all the charge was squeezed into a single point at the

Now inside the sphere is even more interesting.

The electric potential is constant everywhere.

So the potential is the same everywhere inside the sphere.

Yeah.

And it's equal to the potential at the surface, which is V equals one over four pi epsilon naught times Q over R.

And that constant potential inside is because there's no electric field inside a conductor when it's at equilibrium.

That's a key point.

Constant potential throughout a conductor at equilibrium.

What about those oppositely charged parallel plates like what we see in a capacitor?

What's the potential situation there?

So between those plates, the electric field is pretty much uniform pointing from the positive plate to the negative one.

Okay.

If the plates are a distance D apart and the electric field strength is E, then the potential difference between them is just V equals E times D.

Now, if we say the negative plate has zero potential, then the potential at any distance X from that where X is between zero and D is just V of X equals E times X.

So the potential changes linearly with distance, which makes sense because the electric field is uniform.

Right.

So a uniform field means the potential changes at a steady rate.

How about an infinitely long line of charge?

That seems a bit more abstract.

For an infinitely long line of charge where the charge is spread out evenly, we call that a constant linear charge density, lambda, the electric field gets weaker.

The U R from the line, it's proportional to one over R.

When we integrate that to find the potential difference, we see that the potential V at a distance R from the line changes logarithmically with R.

Logarithmically.

Yeah.

So the difference in potential between two points at distances R1 and R2 is proportional to the natural log of R2 over R1.

We usually set the potential of an infinite line charged to zero at random reference distance because it goes to infinity at infinity.

The key here is that logarithmic dependence on distance, which is different from what we see for a point charge.

Interesting how different charge setups lead to different potential landscapes.

You also mentioned a ring of charge and a finite line of charge earlier.

Any key takeaways for their electric potentials?

So for a ring of charge to find the potential at a point along the axis that goes through its center, we use the fact that every point on the ring is the same distance away.

That makes the math easier.

For a finite line of charge, finding the potential near it means adding up contributions from each tiny bit of charge, which gives us a more complicated equation, but we can still solve it.

The big takeaway is that for both of these cases and many others, calculating the electric potential, V, which is just a number, a scalar is often way easier than dealing with the electric field, E, which is a vector.

Once you have the potential, you can always find the electric field.

Right, because they're connected.

Okay, you also mentioned there's a limit to how high the potential of a conductor can get in air.

What's the physics behind that?

Yeah, so in air, if the electric field gets too strong near the surface of a conductor, it can cause the air molecules to ionize.

Basically, it rips electrons off the atoms, turning them into ions, and that causes something called corona discharge, where a charge starts leaking from the conductor into the air.

And this happens a lot more easily around sharp points or edges because the electric field is stronger there.

Ah, so that's why high -voltage equipment can't have sharp edges.

They could create those strong fields and cause sparks.

Exactly.

Okay, let's move on to equip potential surfaces.

What are these imaginary surfaces all about, and how do they help us visualize electric potentials?

Equip potential surfaces are 3D surfaces where the electric potential is the same everywhere.

Imagine you have some charges making an electric field.

If you could find all the points that have the same electric potential and connect them, you'd get an equip potential surface.

You can have a bunch of these surfaces, each for a different potential value.

So it's like taking a snapshot of the potential in 3D and then drawing surfaces that connect points with the same voltage.

You mentioned an analogy to something we see in real life.

Think of contour lines on a topographic map.

Each line connects points on the ground that are at the same elevation, which is basically gravitational potential energy per unit mass.

Equip potential surfaces are similar, but they connect points with the same electric potential, which is electric potential energy per unit charge.

Just like those contour lines show you how steep a hill is, equip potential surfaces show you the strength and direction of the electric field.

That's a really helpful analogy.

So what's the relationship between these equip potential surfaces and those electric field lines?

I'm guessing they're not running side by side.

You're right.

Electric field lines and equip potential surfaces are always perpendicular to each other.

Think about it.

If the electric field had a part that was parallel to an equip potential surface, moving a charge along that surface would take work, which goes against the whole idea of an equip potential surface where no work is done.

So the electric field has to be completely perpendicular.

The force would be perpendicular to the motion, meaning zero work.

Can two equip potential surfaces with different potentials ever cross?

Nope, they can't cross.

If they did, that point where they cross would have two different potential values, which isn't possible.

Every point in space has only one electric potential value.

Okay, that makes sense.

Yeah.

It'd be like one spot on a map having two different elevations at the same time.

Now, what about conductors at equilibrium?

How do equip potential surfaces relate to them?

This is key.

The surface of any conductor at equilibrium is always an equip potential surface.

And since the electric field is zero inside a conductor at equilibrium, the whole inside of the conductor is at the same constant potential too.

If there was a difference in potential, charges would move to even things out, which wouldn't be equilibrium anymore.

So a charged metal object at equilibrium is all at the same voltage.

What about a hollow space inside that conductor with no charges in it?

What's the electric field and potential like in that void?

Inside that empty space, the electric field is zero everywhere.

And if the field is zero, the potential has to be constant throughout.

Plus, that potential inside is the same as the potential of the conductor itself.

There's also no net charge on the inner surface of that space.

Any charge on the conductor stays on the outer surface.

This is the idea behind electrostatic shielding.

Wow.

So it's like a protected zone with constant potential and no electric field, even if the conductor itself is charged.

All right.

Let's talk about the potential gradient.

That sounds a bit math heavy.

It is a bit, but the idea is not too bad.

The potential gradient tells us how the electric potential changes as you move around in space.

It basically connects the electric field E and how the potential V changes.

In those XYZ coordinates, the parts of the electric field are given by the negative partial derivatives of the potential.

So X equals negative dv dx, E equals negative dv di, and Ez equals negative dv dz.

So how strong the electric field is in each direction tells us how quickly the potential is changing in that same direction.

And the negative sign shows us which way the field's pointing.

Exactly.

In vector form, it's E equals negative del V, where del V is the gradient operator working on the potential V.

The gradient itself is a vector that points in the direction where the potential is increasing the fastest.

The negative sign means that the electric field vector points in the direction where the potential is decreasing the fastest.

And that direction's always perpendicular to those equipotential surfaces, as we talked about.

Right.

So the electric field lines are like pointing downhill on the potential landscape, always moving from higher to lower potential.

And they do so along the steepest path perpendicular to those constant potential lines.

What about situations with radial symmetry, like the field around a point charge?

How does that gradient simplify?

When we have radial symmetry where the potential only depends on how far you are from the source, that radial distance, R, it's easier to use spherical coordinates.

Okay.

In that case, the electric field only has a radial part, E or R, and it's just R equals negative dv dr.

That partial derivative becomes an ordinary derivative, because V only depends on R.

So that connection between E and del V is pretty powerful.

It lets us go back and forth between the potential V and the electric field E.

If we know one, we can find the other.

And working with the potential is often a lot easier mathematically.

For sure.

It gives us two ways to think about electric fields, and we can choose whichever works best for the problem.

Sometimes it's easier to start with the electric field and then integrate to find the potential.

Other times, it's easier to start with the potential and then differentiate to find the field.

Okay.

Before we wrap up, let's do a quick review of those key terms.

We started with electric potential energy, U, the energy stored in a system of charges because of their arrangement.

Then we have electric potential, V, the potential energy per unit charge.

Potential difference, delta V or voltage is the work per unit charge to move between two points.

Equal potential surfaces are those surfaces where the potential is constant.

And finally, the potential gradient, del V, describes how the potential changes and is directly related to the electric field.

And the big takeaway is understanding these energy -based concepts gives you a really powerful and often much simpler way to figure out how electrical things work.

Instead of always dealing with sources and fields, you can often work with energies and

which can make tough problems a lot easier to handle.

Absolutely.

Whether you're trying to understand a simple circuit or getting into the nitty gritty of electromagnetism, these concepts of potential and energy are essential.

So as you're thinking about all this, think about how the idea of potential applies to other things beyond electricity.

Like we also have gravitational potential and other types of potential energy in physics.

What might be similar about these potentials and what could that tell us about the fundamental nature of forces and fields?

That's a great question to ponder the underlying unity of these potential concepts.

Thanks for walking us through this deep dive into electric potential and energy.

Until next time, keep those brains buzzing.