Electrostatica Potencial

1
CHAPTER 2
ELECTROSTATIC POTENTIAL
2.1 Introduction
Imagine that some region of space, such as the room you are sitting in, is permeated by
an electric field. (Perhaps there are all sorts of electrically charged bodies outside the
room.) If you place a small positive test charge somewhere in the room, it will
experience a force F = QE. If you try to move the charge from point A to point Bagainst the direction of the electric field, you will have to do work. If work is required to
move a positive charge from point A to point B, there is said to be an electrical potential
difference between A and B, with point A being at the lower potential. If one joule of
work is required to move one coulomb of charge from A to B, the potential difference
between A and B is one volt (V).The dimensions of potential difference are ML2T−2Q−1.
All we have done so far is to define the potential difference between two points. We
cannot define “the” potential at a point unless we arbitrarily assign some reference point
as having a defined potential. It is not always necessary to do this, since we are often
interested only in the potential differences between point, but in manycircumstances it is
customary to define the potential to be zero at an infinite distance from any charges of
interest. We can then say what “the” potential is at some nearby point. Potential and
potential difference are scalar quantities.
Suppose we have an electric field E in the positive x-direction (towards the right). This
means that potential is decreasing to the right. You would have to dowork to move a
positive test charge Q to the left, so that potential is increasing towards the left. The
force on Q is QE, so the work you would have to do to move it a distance dx to the right
is −QE dx, but by definition this is also equal to Q dV, where dV is the potential
difference between x and x + dx.
Therefore

E=−

dV
.
dx

2.1.1

In a more general three-dimensional situation,this is written
∂V
∂V 
 ∂V
+k
E = − grad V = − ∇ V = −  i
+j
.
∂x
∂x 
 ∂x

2.1.2

We see that, as an alternative to expressing electric field strength in newtons per
coulomb, we can equally well express it is volts per metre (V m−1).
The inverse of equation 2.1.1 is, of course,

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V = − ∫ E dx + constant .

2.1.3

2.2 Potential Near Various Charged Bodies
2.2.1Point Charge
Let us arbitrarily assign the value zero to the potential at an infinite distance from a point
charge Q. “The” potential at a distance r from this charge is then the work required to
move a unit positive charge from infinity to a distance r.
At a distance x from the charge, the field strength is

Q
. The work required to
4πε0 x 2

Q δx
. The work required to move unit
4πε0 x 2∞ dx
Q
Q
charge from r to infinity is −
∫r x 2 = − 4πε0 r . The work required to move unit
4πε0
charge from infinity to r is minus this.

move a unit charge from x to x + δx is −

V=+

Therefore

Q.
4πε0 r

2.2.1

The mutual potential energy of two charges Q1 and Q2 separated by a distance r is the
work required to bring them to this distance apart from an original infiniteseparation.
This is
QQ
P.E. = + 1 2 .
2.2.2
4πε0 r
Before proceeding, a little review is in order.
Field at a distance r from a charge Q:
Q
,
4πε0 r 2

N C−1 or V m−1

Q
Q
ˆ
r=
r.
2
4πε0 r
4πε0 r 3

N C−1 or V m−1

E=

or, in vector form,

E=

3
Force between two charges, Q1 and Q2:
F=

Q1Q2 .
4πε r 2

N

Potential at a distance r from a charge Q:
V=

Q.
4πε0 r

V

Mutual potential energy between two charges:

P.E. =

Q1Q2
.
4πε0 r

J

We couldn’t possibly go wrong with any of these, could we?

2.2.2

Spherical Charge Distributions

Outside any spherically-symmetric charge distribution, the field is the same as if all the
charge were concentrated at a point in the centre, and so, then, is the potential. Thus
V=

Q
....