Now, let's expand this discussion a little bit.

So, work done along the path comprised of arcs and radial parts.

If we think of a more complicated case as depicted on the right side,

instead of just one arc and one linear part,

what if we have multiple arcs and multiple linear parts?

Then again, there is no work done along an arc,

and only work done along the radial parts.

If you do the math,

because the starting point become the ending point in between,

they will be cancelled out.

So, only the real starting point and the real ending point will matter.

So, no matter how you design the path,

that is comprised of multiple arcs and linear parts,

if you have the same starting point and ending point,

the path along those are different ways will be the same, right?

So, what about the smooth path?

So, like in the case of spherical coordinates,

we can think of doing more chopping,

more smoothing out, and still the principle holds through.

So, therefore, the work done in carrying a unit charge from

a to b is independent of path, right?

So, in this picture,

we see we have another point called p naught,

instead of a and b, and we just learned

the work done in carrying a unit charge from a to b is independent of path.

So, it means it's like state function,

where electric potential difference between a and b

will be constant no matter what path you take.

Now, why do we need another point p naught that we call reference point?

So, before we were talking about work,

and work is actually only from a to b.

However, if we have another reference point,

then we can give some quantitative amount.

So, if you think of the analogy before with a rock,

it's the difference between using

sea level as your height for judging the potential energy of the rock.

Exactly. So, we can give absolute number of electric potential to each point,

and that is very convenient in everyday life.

Okay. So, W depends,

W unit means the unit charge work,

depends only on the end points,

and it can be represented as the difference between the two numbers, right?

So, in fact, it is a state function of let us start

with potential Phi B and Phi A, the difference,

and let's choose a reference point P naught and agree to evaluate the

integral by using a path that always go by way of point P naught.

Then the W unit will remain the same.

However, we can determine Phi for any point in space.

So, Phi becomes a scalar field,

which we call electrostatic field.

So, for convenience, we will that P naught be the point at infinity y.

If you make r infinite,

then this becomes zero.

So, naturally, the electrostatic potential at

infinity will be zero and zero is a good reference point, right?

So, for a single charge at the origin,

the potential Phi is given for any point x,

y, z as follows.

Phi of x, y, z is equal to minus q over four Pi Epsilon naught

times one over r. Then we can assign numbers to any point of interest.

Now, I just mentioned that we can put zero volt at infinity.

But in practice it's hard to connect infinity, right?

So, where can we find zero volt in our everyday life.

The ground.

Yes. As Melody points out we can find zero volt from the ground.

Then you may ask,

why does ground has the same potential as the point at infinity?

Melody, can you explain to us?

Yeah. Can I have this?

Sure.

So, if you think of the Earth capacitance

of the surrounding environment,

and at infinite, there's zero volts, right?

So, the change in voltage for

a capacitor is equal to the change in charge over the capacitance.

However, since this is a very large system,

the capacitance is very large essentially infinite.

So, if your Delta V is equal to zero,

then on the other side of this capacitor,

which is the earth will also be zero.

Very good.

In addition, if you connect anything to this large source of charges,

then everything will balance out.

So, if you connect something here, has a positive,

there's a lot of negative charges that can make this zero as well.

Perfect. That's a very good explanation.

Okay. So, now we're going to think about some mathematics again,

with a focus on why we introduced this electrostatic potential or electric potential,

and why we add this concept in addition to electric field.

Okay? In order to do that,

let's first check if the superposition principle

can be applied to electrostatic potential as well.

So, we will start from the definition.

The electrostatic potential of arbitrary point P is

equal to minus p naught is the reference point,

P naught to P,

E_subtotal dot ds all right?

E_subtotal dot ds.

Now, we know the true electric field acting on that position of interest can

be decomposed into components because we can apply superposition to electric fields.

So, then we can replace this E_subtotal by Sigma of each component j, okay?

Then, we also know we can interchange the Sigma and integral here.

So, if we pull this out of the equation,

and put Sigma to the right side,

then what happens is,

this will become the potential

contributed to by each component of charge in the world, right?

So, we will call that Phi j.