This document gives a clear introduction to the concept of electric potential and potential differences. The potential created by various charge distributions are discussed. This document is suitable for any graduate course in basic electrostatics. Adequate number of problems with solutions are pro...
Introduction to Electrostatics. The Electric
potential
1 The Electric potential
Figure 1
The work done on moving a charge Q over a distance dl in the electric field is
dW = −F.dl = −Q E.dl (1)
The negative sign is used to indicate that the work is being done by an external agent.
The total work done in moving the charge from A to B in the electric field is
ZB
W = −Q E.dl (2)
A
1
,By dividing this quantity by Q, one obtains the potential energy per unit charge. This value is
referred to as the potential difference VAB between A and B. i.e.,
ZB
W
VAB = =− E.dl (3)
Q
A
The unit of potential difference is Joules per Coulomb, commonly referred to as Volts (V).
A negative value of VAB indicates that the field performs work on the charge. Conversely, a
positive value of VAB signifies that the work is done by an external agent that moves the charge
within the field. VAB is independent of the path taken by the charge.
From the figure, the electric field produced by a point charge Q placed at the origin is
Q
E= r̂ (4)
4πε0 r2
where r̂ is the unit vector along r.
Now the potential difference between A and B is
ZrB
Q
VAB = − r̂.drr̂ (5)
4πε0 r2
rA
Q 1 1
= − (6)
4πε0 r2 rB rA
Note that in the figure,
dl = r + ∆r − r = ∆r = drr̂ + dtt̂ (7)
where drr̂ represents the component along the radial direction, while dt t̂ denotes the com-
ponent in the transverse direction. The electric field vector E is oriented along the radial
direction, which results in the transverse component’s contribution to the dot product of E and
dl being nullified, as it is orthogonal to the radial direction. This could be understood in the
following way also. Since the point charge has a spherically symmetric field, we could use
the spherical co-ordinate system. The electric field vector always points in the radial direction.
So E = Er r̂. Any displacement vector in spherical co-ordinate can be expressed in terms of
the radial, azimuthal and polar unit vectors. Therefore, dl = drr̂ + r dθ θ̂ + r sinθ dφ φ̂ . So we
have, E.dl = Er r̂.drr̂.
In Eq.6, if we let rA = ∞ and rB = r, we have,
Q 1 1
V∞B = − (8)
4πε0 r2 r ∞
Q
= (9)
4πε0 r
This quantity is called the absolute potential or simply the potential of the point B, if it were
at a distance r from the origin.
2
, Therefore, we can characterize the potential as the difference in potential between a given
point and a reference point where the potential is considered to be zero (specifically, the point
at an infinite distance from the charge is assigned a potential of zero). For practical applica-
tions, any location where the influence of the charge is minimal can be regarded as the point
of zero potential.
An alternative definition of potential at a point located at a distance r from the origin is the
work performed per unit charge by an external force in moving a test charge from infinity to
that specific point.
i.e.,
Zr
V=− E.dl (10)
∞
Therefore, we can express the potential difference as the variation in absolute potential
between the two points.
Thus,
VAB = VB − VA (11)
If the point charge is situated at a location defined by the position vector r0 rather than at the
origin, the potential at a point indicated by the position vector r is given by
Q
V= (12)
4πε0 |r − r0 |
Consider a scenario where multiple point charges, denoted as Q1 , Q2 , ..., Qn , are located at
positions r1 , r2 , ..., rn . The electric potential at a specific point represented by r, resulting
from the influence of all these charges, can be expressed as follows:
Q Q Q
V= + + ................ + (13)
4πε0 |r − r1 | 4πε0 |r − r2 | 4πε0 |r − rn |
n Qj
1
= ∑ (14)
4πε0 j=1 |r − rj |
For continuous charge distributions,
1 ρL (r0 ) dl0
Z
V(r) = for line charge (15)
4πε0 |r − r0 |
L
1 ρS (r0 ) dS0
Z
V(r) = for surface charge (16)
4πε0 |r − r0 |
S
1 ρV (r0 ) dV0
Z
V(r) = for volume charge (17)
4πε0 |r − r0 |
V
(18)
The quantities that are marked with primes indicate the source points, while those without
primes pertain to the field points.
3
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