Injected current. So what I wrote here actually, is

Kirchoff's law. Saying that the current that I inject

splits into capacitive current, here. Charging the capacitance, and this, the

resistive current. Is the current that goes through the

resistance, and leaks out. Actually, current can only flow outside

through the resistance. And it can charge the capacitance.

If I inject a positive current, then the capacitance is charged, inside.

It's charged inside. Inside the cell, inside the cell, you

charge the capacitance with positive charges.

So the inside of the cell become positive, and that's what you see here.

The inside of the cell becomes positive, and that's our agreement.

If I inject positive current inside the cell, I get the positive voltage inside

the cell. Which I call depolarizing.

Okay, so now I have to take this equation.

And show you that when I solve this equation for V, I get such a behavior.

So let's see, so I'm trying now to solve this and this is actually very easy.

Because now you have a very simple linear, because this is constant, the

capacitance. This is constant, the resistance.

There is a constant current I. It's a linear, 1 dimensional partial

equation. So you only have dv, dt.

There is only time here, t is time. As I said before, this is now here, I

injected in time. So c, dv, dt plus vdivided by R is equal

I. And I need to solve this equation.

It's a very simple solution. So let me write to you the solution.

But before writing the solution for such I thing, I have to define initial

conditions. What do you start with?

So we already said that we start with v at t equals 0 equal lets call it 0.

So I start with v equal 0here. V equal 0.

And I get to some v with time. So I want to solve for v, I want to solve

this equation for v as a function of time.

So the solution looks like this. V as a function of time, is equal I which

is my current injection multiplied by R which is the resistance.

Multiplied by 1 minus e, exponent to the power of minus t time divided by R

multiplied by C. This is the general solution, for this

equation. Assuming that v at t equals 0 is indeed

0. So let's check it.