PHYSICS

a
E E +
a :mmr
b
Fig. 15.1. Schematic diagram
of the electric field: a - for two Fig. 15.2. Uniform (a) and
equal and opposite charges; nonuniform (b) electric field
b - for point charges
are related to the electric field in any region of space in the
following manner:
1. The electric field vector (E) is tangent to the electric field
line at each point;
2. The number of lines per unit area through a surface
perpendicular to the lines is proportional to the strength of the
electric field (or the magnitude of the charge) in that region.
3. The lines must begin on positive charges and terminate on
negative charges, or at infinity in the case of an excess of charge;
4. No two field lines can cross.
An important field for consideration is the uniform electric
field (fig. 15.2, a); a field that has the same magnitude and direction
at all points. In the opposite situation, the field is non-unifornm
(fig. 15.2, b).
15.2.4. Electric Potential
Instead of dealing directly with the potential energy of a
charged particle, it is useful to introduce the more general concept
of energy per unit charge.
Suppose a test charge (qo) moves from A to B under the influence
of a field (E ). The work (dA) done by the electric force (qE) on
the test charge for an infinitesimal displacement (di) is given by:
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dA = F .dx = qoE .di: (15.4)
By definition, the work done by a conservative force equals the
negative of the change in potential energy (dU); therefore:
dA = - dU = - qaE ·di (15.5)
A force is conservative if the work done by the force acting on
a particle moving between two points is independent of the path
the particle takes between the points.
For a finite displacement of the test charge between points
A and B, the change in the potential energy is given by:
-%I B U =U B
-U A = j;·dX
(15.6)
A
where U and U
B A
are the initial and final potential energies,
respectively.
The potential difference (,1cp) between points A and B is defined
as the change in potential energy divided by the test charge (qo):
U B U A
ticp = cp E- dx
B - CPA = - - - = -I . x (15.7)
qo qo A
The quantity which equals the energy per unit charge [rp = ~)
is called the electric potential (U).
qo
The unit of potential is the Joule per Coulomb, which is equal
to a volt (1 V = 1 J/c)·
15.2.5. Obtaining E from the Electric Potential
The electric field (E) and the potential (U) are related by
Equation (15.7). To calculate the electric field if the electric potential
is known in a certain region, apply Equation (15.7) when
there is a uniform electric field:
B B B
L1 qJ = - IE. dx = - I E cos OOds = - IEds (15.8)
A A • A
Since E is constant, it can be removed from the integral sign,
giving:
B
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