| When two charged objects are brought into proximity they either attract or repel each other with an electric force described by Coulomb's Law. Since this force is conservative; that is, path-independent, it can be expressed as the negative derivative of its associated potential energy function.
As in gravitation where an object's gravitational potential energy is proportional to its mass, m,
a charged object's electrical potential energy is proportional to the magnitude of its charge. The greater the charge placed on an object in a given position in an electric field, the larger its electric potential energy.
The ratio of electric potential energy per unit charge is therefore a property of the electric field and is called the field's electric potential, or voltage (volt = joule/coulomb). If you were to connect together a series of positions having the same voltage you would produce an equipotential surface. Electric potential is a scalar quantity.
Electric Field Strength and Potential We are now going to derive two important relationships between the quantities electric field strength and electric potential.
To derive an expression for the local electric field, E, in terms of its electric potential, V, we begin with the definition of a conservative force and the following two facts:
- the force exerted on a charge by an electric field equals the product of the charge times the field strength (F = qE)
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a volt is defined as electric potential energy per unit charge,
This expression gives rise to an alternate unit for measuring electric fields, volt/meter.
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The electric field is the negative gradient of the potential; that is, field lines point from positions of high potential to points of low potential. The more rapidly the voltage changes the stronger the electric field in that region.
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Note that knowing the potential of one position in a field is insufficient to allow you to calculate the electric field strength at that position. You must know the equation for the potential over a region to take its derivative (rate of change with respect to position) and calculate E.
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Also recall that the negative sign in this formula was originally introduced since a conservative electric force reduces a charged object's electric potential energy as it accelerates the object to positions of lower potential: Δ KEgained + Δ Ulost = 0.
Now, instead of solving for the electric field strength, let's solve for the change in the potential between two positions, a and b, in an electric field.
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Note in this expression that we are starting at a and going to b. The integral notation is read as "the negative integral from a to b of E dr."
Point Charges |