2.1 Electric Field
We will now consider the concept of electric fields. The electrostatic force or the Coulomb force, like the gravitational force, is a force that acts at a distance even when the objects are not in contact with one another. To justify such the notion we rationalize the action at a distance by saying that one charge creates a field, which in turn, acts on the other charge.
An electric charge cube produces and electric field everywhere in space. To quantify the strength of the field created by that charge, we can measure the force, a positive test charge Q0, experiences at some point.
In other words, if we consider a positive charge Q is sitting over here, and would like to figure out the electric field that it generates at a specific point in space. Let’s say, some R distance away, from this charge. We introduce a positive test charge to that point and we define the electric field that the source charge generates at this location as E is equal to limit as Q0 goes to 0, Coulomb force divided by the test charge Q0. We take Q0 to be infinitesimally small so that the field Q0 generates does not disturb the source charges. Q0 is infinitesimally small so that its E field does not disturb the source charge or charges.
From this point of view, we look at the problem such that a source charge generates its own electric field and in turn, if we place any other charge in this region then the source charge exerts Coulomb force on that other charge, or charges. If we look at the definition of electric field, which is basically Coulomb force on per unit charge, we see that the unit of electric field will be, in SI unit system, as newton per Coulomb.
And, since force is a vector, all though charge is a scalar, whenever we divide or multiply a vector by a scalar we end up with a new vector. Therefore, electric field is a vector, a quantity. Therefore it has both magnitude and directional properties.
We determine the direction of the electric field vector such that by looking at its definition, we see that its direction is in the same direction with the Coulomb force acting on the test charge. So, we conclude that the electric field vector direction is in the same direction with the Coulomb force exerted on a positive test charge which is located at the point of interest.
From this point of view, then if we consider a positive charge as our source charge, and if we’re interested with the electric field that it generates at a specific point in space, let’s say at this point for example, we just take a positive test charge, Q0, place it at that location, since these two charges are positive charges, our source charge will repel Q0, the test charge. Therefore the direction of the associated Coulomb force will be pointing away like this. And, the related electric field, therefore, will be in the same direction with this force and pointing in the same direction.
If we’re interested with the electric field direction at a different direction, let’s say over here, then we place our positive test charge to that location and, again, since both charges are like charges, Q will repel Q0 along the line which joins these two charges. And, the electric field at this location, therefore, will be in the same direction with this force.
And, applying this method for all the points surrounding the source charge Q, we will see that the electric field vectors are going to be pointing in what we call, radially outward direction. So, they originate from the source charge and go to infinity. So, we conclude that E field of a positive point charge directs radially outward.
Applying the same procedure for a negative point charge, as our source charge, if we are interested with the electric field direction at a certain point in space, we place a positive test charge to that point. And, now, since these two charges are unlike charges our source charge will attract Q0 along the line which joins them, therefore, it will be pointing in this direction. And the electric field of the source charge, therefore, will be in the same direction with this Coulomb force.
And, similarly, if we’re interested in another location, we place our positive test charge to that location since, again, Q and Q0 are unlike charges, Q will attract Q0 with a force of F along the line which joins these two charges. And, the electric field will be in the same direction with this force.
Again, applying this to every point which surrounds the source charge Q, we will see that its electric field is going to be pointing in radially inward direction. Therefore, E field of a negative point charge is radially inward direction.
Now, if we consider a system which consists of a positive and a negative charge, if the magnitude of the charges are equal, such a system is known as electric dipole. So, if we have a positive charge over here, and a negative charge located here, such that their magnitudes are equal, the positive charge, by itself, will generate electric field lines in radially outward direction, whereas the negative one will generate, which are radially inward direction. And, these lines will join and generate an electric field line geometry, something like this.
For the negative charge we have radially outward direction, and these lines are eventually joining to one another. And, for the negative charge they are radially inward direction.
Now, these are the sum of the field lines. The electric field vector is tangent to the field line passing through the point of interest. So in such a system, and if I’m interested with the electric field direction at this location, the corresponding electric field vector will be tangent to the electric field line passing through that point.
And, at this point it is going to be, something like this, for example. At this point it is going to be something like this. Tangent to the field line passing through that point, where here it will be like this, and here it will be like this.
If we look at the properties of electric field lines, the first one comes stated as, the electric field vector is tangent to the field line passing through the point of interest.
The second property is that no two field lines can intersect. It means that we can have only one field line passing through a specific point, we can not have two field lines intersecting each other at that location.
And, the third property is associated with the strength of the electric field. And, that is the number of field lines passing through a units surface is proportional to the strength of the electric field generated at that location.
So, according to this property, then, if we consider a units surface, something like this, for example, the number of lines, electric field lines passing through that surface is directly associated with the strength of the electric field. We can easily see that if we go nearby to the source, or nearby to the charges, we will see that the number of field line passing through a units surface will increase, indicating that the strength of the electric field becomes maximum whenever we are just next to the source.
And, if we go away from the source, we will see that the number of lines passing through this units surface will decrease indicating that the strength of the electric field becomes smaller in comparing to the regions nearby to the charges.
If we have more then one charges in our system of interest, using the superposition principle the total electric field due to a group of charges is equal to the vector sum of the electric fields of individual charges. In other words, if we have a Q1 and Q2 and Q3 and, let’s say some Q sub N in our system of interest. And, if we are interested with the electric field generated at a specific point in space, we simply determine the electric field generated by each one of these charges at the location.
For Q3, for example, E3, something like this, for Q2, E2, something like this, and for Q1, E1, something like this. And, let’s assume that the QN is negative and therefore it will be something like this, and so on so forth. And then, we introduce a proper coordinate system.
And, to be able to get the total electric field we add all these electric field vectors by using the superposition principle in order to obtain the total electric field. In other words, vector, sum of all these electric field vectors will eventually give us the total electric field of the system generated at this location