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Electric Potential(Voltage)
Mathematics tells us that we can express a conservative vector field by means of a scalar
field. Specifically, we know that the vector field can be expressed as the gradient of a
scalar field. Since a static electric field is a conservatice field, it is of conceptually
(and mathematically) simpler to express an electric field in terms of an electric
potential that we call voltage. This allows us to work with the field as s scalar field
instead of a vector field. We define the relationship between the two as follows :
E = (grad) V
E
is the electric field
V
is the scalar field
Since we also know that the gradient of a scalar field is in the direction of maximum
increase, the E lines must be perpendicular to equipotential (equal voltage) surfaces.
This V is the same potential (voltage) that you have dealt with in other classes, only now
you can see how it is derived from an electric field. In order to calculate the voltage
between two points (a and b), we can use our definition of V and get:
/b

Vb  Va =   E * dL Volts

/a
Calculations of Electric Potential(V) fields(as well as the eletric intensity(E)fields)
will usually involve integrating over the entire distribution. As there are many different
ways of doing this for different geometries, please consult your textbook for more
information.
One of the most useful things about the electric potential is how it can be used with
boundary conditions to solve for different types of fields even when only know the voltage
at the boundaries. This is done by applying Poisson's equations and Laplace's equation.
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