Path is chosen along a radial line so that becomes simply Edr. This process defines the electric potential of a point-like charge. Note that potential function is a scalar quantity as oppose to electric field being a vector quantity.
Now, we can define the electric potential energy of a system of charges or charge distributions. Suppose we compute the work done against electric forces in moving a charge q from infinity to a point a distance r from the charge Q. The work is given by:. Note that if q is negative, its sigh should be used in the equation! Therefore, a system consisting of a negative and a positive point-like charge has a negative potential energy.
A negative potential energy means that work must be done against the electric field in moving the charges apart!
Now consider a more general case, which deals with the potential in the neighborhood of a number of charges as depicted in the picture below:. Let r 1 ,r 2 ,r 3 be the distances of the charges to a field point A, and r 12 , r 13 , r 23 represent the distance between the charges. The electric potential at point A is:. If we bring a charge Q from infinity and place it at point A the work done would be:.
The total Electric Potential Energy of this system of charges namely, the work needed to bring them to their current positions can be calculated as follows: first bring q1 zero work since there is no charge around yet , then in the field of q1 bring q2, then in the fields of q1 and q2 bring q3. Add all of the work needed to compute the total work. The result would be:. The component of E in any direction is the negative of the rate of change of the potential with distance in that direction:.
Both movements would be like going with nature and would occur without the need of work by an external force. This motion would result in the loss of potential energy. Potential energy is the stored energy of position of an object and it is related to the location of the object within a field. In this section of Lesson 1, we will introduce the concept of electric potential and relate this concept to the potential energy of a positive test charge at various locations within an electric field.
A gravitational field exists about the Earth that exerts gravitational influences upon all masses located in the space surrounding it. Moving an object upward against the gravitational field increases its gravitational potential energy. An object moving downward within the gravitational field would lose gravitational potential energy. When gravitational potential energy was introduced in Unit 5 of The Physics Classroom , it was defined as the energy stored in an object due to its vertical position above the Earth.
The amount of gravitational potential energy stored in an object depended upon the amount of mass the object possessed and the amount of height to which it was raised. Gravitational potential energy depended upon object mass and object height. An object with twice the mass would have twice the potential energy and an object with twice the height would have twice the potential energy. It is common to refer to high positions as high potential energy locations.
A glance at the diagram at the right reveals the fallacy of such a statement. Observe that the 1 kg mass held at a height of 2 meters has the same potential energy as a 2 kg mass held at a height of 1 meter. Potential energy depends upon more than just location; it also depends upon mass.
In this sense, gravitational potential energy depends upon at least two types of quantities:. So it is improper to refer to high positions within Earth's gravitational field as high potential energy positions.
But is there a quantity that could be used to rate such heights as having great potential of providing large quantities of potential energy to masses that are located there?
While not discussed during the unit on gravitational potential energy, it would have been possible to introduce a quantity known as gravitational potential - the potential energy per kilogram. Gravitational potential would be a quantity that could be used to rate various locations about the surface of the Earth in terms of how much potential energy each kilogram of mass would possess when placed there.
Gravitational potential is a location-dependent quantity that is independent of the mass of the object experiencing the field. Gravitational potential describes the effects of a gravitational field upon objects that are placed at various locations within it.
If gravitational potential is a means of rating various locations within a gravitational field in terms of the amount of potential energy per unit of mass, then the concept of electric potential must have a similar meaning.
Consider the electric field created by a positively charged Van de Graaff generator. The direction of the electric field is in the direction that a positive test charge would be pushed; in this case, the direction is outward away from the Van de Graaff sphere.
Work would be required to move a positive test charge towards the sphere against the electric field. The amount of force involved in doing the work is dependent upon the amount of charge being moved according to Coulomb's law of electric force. The greater the charge on the test charge, the greater the repulsive force and the more work that would have to be done on it to move it the same distance. If two objects of different charge - with one being twice the charge of the other - are moved the same distance into the electric field, then the object with twice the charge would require twice the force and thus twice the amount of work.
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