Conceptual Physics

Equipotential surface

Electric field perpendicular to surface

24.11 - Electric potential and a uniform electric field

The concepts of electric potential and electric field are linked. In Concept 1, you see

their relationship illustrated in the context of a uniform electric field. Plate A is negatively

charged and plate B is positively charged. Assume that the plates are large enough

compared to their spacing that the electric field between them is of uniform strength

and direction, pointing from plate B toward plate A.

The electric potential between the plates increases in the opposite direction, from

plate A to plate B. Since it takes positive work to push a test charge against the field

from left to right, the potential energy of the charge increases as it moves from plate A

to plate B. Since the strength of the field is everywhere the same, the electric potential

increases at a constant rate as the charge moves from left to right. The potential’s

change per unit displacement parallel to the field is constant.

Another way of expressing the opposite orientations of the field and the direction of

increasing potential is to say that the field is always directed from regions of higher

potential to regions of lower potential.

We state the proportionality between the change of potential and the displacement as

the first equation in Equation 1, and derive it below. The variable ǻs measures a

displacement parallel to the field lines. The negative sign in the equation reflects the

fact that the potential difference is negative for a displacement ǻs in the same direction

as the field. Movement perpendicular to the field lines results in no change in electric

potential. Again, we stress that the field must be uniform.

In many applications later in this book, we will only be interested in the magnitude of the

potential difference between two points in a uniform field that are separated by a

distanced parallel to the field, and we will write ǻV = Ed.

The second equation in Equation 1 is just the first equation, solved for the electric field.

It proves to be a very useful formulation. It shows how the electric field strength can be

determined when the potential difference between two points in the field is known. This

form of the equation is applied in the example problem below.

Derivation. We will show that for a uniform electric field E, and a displacement ǻs

measured in the direction of the field, the potential difference between two points

separated by ǻs is given by the equation ǻV = íEǻs. This is shown in the illustration

of Equation 1.

Variables

We use the subscripts field and ext in the steps below to distinguish the forces exerted and the work done by the field and by an external force

that moves a test charge against the field.

Displacement parallel to uniform

field

Potential V changes at constant rate

Field directed from higher to lower V

Potential difference in uniform

field

ǻV = potential difference

E = electric field strength

ǻs = displacement parallel to field

uniform electric field E

two points in field P 1 , P 2

test charge qtest

force F

work W

distance parallel to field ǻs

potential difference between points ǻV

(^448) Copyright 2000-2007 Kinetic Books Co. Chapter 24