Understanding Electric Fields, Potential Energy, and Electric Potential
Electric Fields: Concepts and Characteristics
A field is generated by a physical magnitude if, at any moment, a value can be designated to that magnitude in all the region’s points where the field acts. The value of a scalar field depends on the point in space being considered, for example, the distribution of temperature in space. Meanwhile, a vector field associates a vector to each point in space with a magnitude, direction, and sense. An example of this is the gravitational force.
“An electric field is said to exist in a region of space in which an electric charge experiences an electric force.” (Tippens, 2007)
This means that when a space surrounds a charge, and that space is affected, an electric field exists.
Electric field 
at a point is defined in terms of the force 
that a small positive charge (+q) experiences when placed exactly at the point of interest. Because electric force is a vector quantity (it has magnitude, direction, and sense), the electric field is also a vector quantity.
The magnitude of an electric field E can be determined by introducing the previous formula into Coulomb’s Law, resulting in the following equation:
With this equation, you can calculate an electric field at a point without placing a +q test charge. The direction would be towards Q if Q is negative and away from Q if it is positive.
Example 6.1
What would be the magnitude of an electric field at a point located 12 cm away from a charge of -10µC?
Because the charge is negative, the electric field’s direction would be moving towards Q.
Electric Field Lines
These are imaginary lines drawn parallel to the electric field at any point of interest.
The electric field lines will always go from the positive charge to the negative charge.
| Electric Field Lines | Some Examples of Field Lines for Several Electric Fields | |
A very useful form to represent graphically any field is to draw lines that go in the same direction of the field in several points. This is made through field lines. | One Particle Field | |
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| The field produced by one positive charge has field lines that will go outward from the charge and will fade to infinity. | The field produced by a negative charge has field lines that will point inward the negative charge. |
| Two Particle Field | |
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| Field lines from two charges with different signs (positive and negative) originate from the positive charge and end at the negative charge. | In this case, field lines from two positive charges are represented. |
The electric field goes in a tangential direction (i.e., that they touch in a point) from the field.
The places in which the field’s value is higher are those in which the lines are found closer, and vice versa.
By convention, the lines should go from the positive charges towards the negative charges. In the absence of another charge, the lines should finish on infinity.
The number of lines in a field is proportional to the charge’s magnitude.Source: Icarito. Consulted on January 1st, 2010 at: http://www.icarito.cl/
Resulting Electric Field on a Discrete Distribution of Punctual Charges
When more than one charge is in an electric field, the resulting field is the vector sum of all the fields of each charge.
Potential Energy
In a constant electric field, electric work is the work needed to move one charge from point A to point B, defined by:
Where q is the charge [C], E is the electric field’s magnitude [N/C], and d is the distance it travels from point A to point B [m]. The unit of work is the Joule [J].
“Electric potential energy EP equals the work performed against the electric forces to carry the charge +q from infinity up to that point.” (Tippens, 2007)
As for work, its unit is the Joule.
When a positive charge moves against the electric field, the potential energy increases, whereas when a negative charge moves, the potential energy decreases.
Example 7.1
There are two charges separated by 13 cm in a uniform electric field. q1 has a 5µC charge and q2 a -8 µC charge. Find the electric potential.
Electric Potential
“The electric potential V at a point located at a distance d from a charge Q equals the work per unit performed against the electric forces to transport a positive charge +q from infinity up to that point.” (Tippens, 2007)
Electric potential at point A is given by:
Its unit is J/C or Volt.
The potential has a negative sign when it occurs due to a negative charge and a positive sign when due to a positive charge.
Example 7.2
Obtain the potential at point A that is located 9 cm from a charge of 4µC.
Electric Potential Calculation
The resulting potential for a punctual charge system at a given point equals the algebraic sum of the corresponding potentials of each charge.
“The potential difference between two points is the work per positive charge unit that the electric forces make to move a small test charge from the point with the highest potential to the point with the lowest potential.” (Tippens, 2007)
