Chapter 25: Electric Potential
Learning Goals
- Understand the definition of Electric Potential Energy and Electric
Potential. Be aware of the difference between these terms.
- Be able to calculate potential differences for uniform fields, fields due
to point charges and continuous charge distributions.
- Know how to obtain electric fields from electric potentials, and
vice-versa.
- Know how to treat electric potentials due to charged conductors.
Electric Potential
Recall that energy methods utilizing quantities such as
kinetic and potential energy can provide solutions to
problems that would be very difficult to solve using
Newton's second law. Electric potential is defined
in terms of the work required to move a small positive
test charge q0 in the presence of an
electrostatic force:
Definition: the electric potential difference
VB - VA between two points A and B
in a region of space that contains an electric field is
the work per unit charge done against the field
required to move a small positive test charge q0
from point A to point B. In symbols:
The units of electric potential are Joules/Coulomb = Volt (V).
Notice that electric potential, like potential energy, is
a scalar (not a vector!) quantity.
Special Case Formulas for Specific Electric Fields
Be careful when applying these formulas - they only apply to
specific situations!
where d is a displacement in the direction of the
field E.
where V is now interpreted as the absolute
(relative to zero at infinity) potential.
Potential Energy
The potential energy stored in a distribution consisting
of two point charges is equal to
where r12 is the distance between the two charges. The sum
of all such terms for any particular assembly of discrete
charges equals the total stored energy of the charge
distribution.
Conductors
Since E=0 inside conductors in electrostatic
equilibrium, their volumes are equipotential
volumes.
Relationship of Electric Fields to Electric Potentials
From the definition of electric potential, it is evident that
the electric field may be obtained from the gradient of the
electric potential. In particular, for Cartesian coordinates
and so on.
Continuous Charge Distributions
A continuous distribution of charge may be divided into
infinitesimal dq elements to give
