2.2 Electrical Field of a Point Charge
Now let us try to determine the electric field for point charge. Let’s assume that our source charge is a positive point charge q and we are interested to determine the electric field some r distance away at point p. In order to do this, we will choose a positive test charge q zero and place it at the point of interest.
Since both of these charges are like charges, our source will repel the test charge q zero in radially outward direction with a Coulomb force of f sub c. Now we know that the electric field that it generates at the point of interest should be the same direction with the Coulomb force.
In other words it has to be radially outward direction. The magnitude of this force is from Coulomb’s law, 1 over 4 Pi Epsilon zero times the product of the magnitude of the charges, q q zero over r squared.
If we’d like to express this in vector form, then we can introduce a unit vector in radial direction denoted as r unit and multiply the magnitude of this force by that unit vector r. From the definition of electric field, we have electric field is equal to Coulomb force per unit charge. In this case, per unit test charge.
Substituting the explicit value of Coulomb force, we will have 1 over 4 Pi Epsilon zero q q zero over r squared times r unit divided by q zero. Q zeroes at the numerator and the denominator will cancel, leaving us electric field of a point charge is equal to 1 over 4 Pi Epsilon zero charge divided by the square of the distance to the point of interest.
Therefore if we are interested with the electric field generated by a point charge q some r distance away from the charge at a point p, that electric field is going to be in radially outward direction. Its magnitude is going to be equal to Coulomb constant times magnitude of the charge q divided by square of the distance to the point of interest. R is the distance between the source charge and point of interest.
The expression over here in the box is also sometimes known as Coulomb’s law in terms of electric field. Now, if we have more than one point charges in our system and if we are interested in the net electric field at a specific point for more than one point charges, then simply we calculate the electric field due to each one of these charges at the point of interest and then add these electric field vectors vectorially to be able to get the total electric field.
Therefore e total, or the net electric field at the point of interest will be the vector sum of the electric fields generated by each individual charge at the point of interest. Following example is going to be related to the superposition of the electric field vectors.