1. I-3 Electric Potential

2. Main Topics Conservative Fields.
The Existence of the Electric Potential.
Work done on Charge in Electrostatic Field.
Relations of the Potential and Intensity.

3. Conservative Fields
There are special fields in the Nature in which the total work done when moving a particle on along any closed path is zero. We call them conservative.
Such fields are for instance:
Gravitational - we move a massive particle
Electrostatic - we move a charged particle

4. The Existence of the Electric Potential
From the definition of a conservative field it can be shown that work done by moving a charged particle from some point A to some other point B doesn’t depend on the path but only on the difference of some scalar quality in both points. This quality is called the electric potential .

5. Work Done on Charge in Electrostatic Field by an External Agent I
If we (as an external agent) move a charge q from some point A to some point B then we do work by definition:
W(A->B)=q[(B)-(A)]

6. Work Done on Charge in Electrostatic Field II
Since doing positive work means increasing of the energy, we can define a potential energy U
U=q
Then:
W(A->B)=q[(B)-(A)] =U(B)-U(A)

7. Work Done on Charge in Electrostatic Field III
In almost all situations we are interested in the difference of two potentials. We define this difference as the voltage V
VBA= (B)-(A)
Then:
W(A->B)=q VBA

8. Work Done on Charge in Electrostatic Field IV
So we come to the general formula:
W=q[(B)-(A)]=U(B)-U(A)=qVBA
Try to understand well the difference
between the potential, the potential energy and the voltage!
between the work done by the field and an external agent!

9. The Impact of the Potential
Since the potential exists, we can describe fully the electrostatic field using the scalar potential field (r) instead of the vector intensity field E(r).
We need only one third of information
Superposition means just adding numbers
Some terms converge better

10. Relations Between Potential and Intensity I
It is convenient to describe this in terms of potential energy and force so we don’t have to care about the polarity of the charge and use examples from the gravitation field.
Let’s have a charged particle and a force F acting on it.
If the particle moves by dl the field does work dW=F.dl .

11. Relations  versus E II
The sign of this work depends on the projection of the path vector dl into the force F.
If the field does a positive work it must be at the cost of lowering the potential energy of the particle i.e. dW>0 means dU<0.
So: dW = F.dl = -dU
To get work for some finite path A->B we just integrate.

12. Uniform  Homogeneous Field I
Let us move a unit charge some distance d in the direction of the intensity E. Since both vectors are parallel the work is positive and the integral is a simple multiplication:
Ed=-[(B)-(A)]>0
This means (B)=(A)-Ed so along the field lines the potential  is decreasing!

13. E=-[(B)-(A)]/d=[(A)-(B)]/d
Uniform Field II
We can also get E from the potentials
E=-[(B)-(A)]/d=[(A)-(B)]/d
So the magnitude E is the slope of the potential and the vector points in the direction of decreasing potential.
Roughly E depends on the change (derivative) of  and  on sum (integral) of E.

14. Uniform Field III
If we want to know the work done on a non-unity charge by the field or an external agent, we have to take into account the charge and deal with the potential energy. Bigger charge “sees” bigger slope in potential energy and negative charge sees an opposite slope!
Roughly: E depends on the change (derivative) of  and  depends on sum (integral) of E.

15. The Units The unit of  or V is 1 Volt. [ ] = [U/q] => V = J/C
[E] = [/d] = V/m
[] = [kq/r] = V => [k] = Vm/C =>
[0]=CV-1m-1

16. Homework 2
The homework is selected for “problem” sections that are in the end of each chapter.
21 – 17, 19
22 – 1, 2, 4, 6, 12, 26
23 – 7, 10
You are free to try to answer to the questions in the questions sections!

17. Things to read
Chapter 23-1, 23-2

18. Work Done by the Field A->B
Now we can divide it by the “test” charge q
This is generally ho to get  from E.