Physics Bootcamp

Subsection 29.3.1 Definition of Electric Field

Following Faraday, we envision the entire space being permeated by invisible lines. At each point in space, they represent a vector, which we will call electric field vector. We will denote it by symbol \(\vec E\) or \(\vec E_P\text{,}\) where subscript \(P\) refers to the space point of interest. When we place a charge \(Q\) at point P, this charge will experience an electric force whose maganitude is

and the direction of the force is same as that of \(\vec E\) if \(Q\) is positive, and opposite to \(\vec E\) if \(Q\) is negative. The SI unit of electric field will be \(\text{N/C}\text{.}\) We can write force/electric field relation as a vector equation as

By measuring force on charge \(Q\text{,}\) we can deduce the electric field by dividing the measured force by charge.

Note that charge \(Q\) is only “feeling” or “discovering” the electric field that already exists at its location due to other charges somewhere else. We call the \(feeler\) charge \(Q\) the test charge and the other charges whose electric field the test charge is responding to are called the source charge or just charges. The space point P, where we have placed charge \(Q\) is called the field point. It is important to note that:

Electric field at the field point exists regardless of whether or not we test its existence by placing the test charge at that point - it has been generated by other charges.

Subsection 29.3.2 How Electric Field Solves the Action-at-a-Distance Problem?

With the viewpoint of electric force based on the electric field, there is no action-at-a distance. Instead, the electric force on the test charge \(Q\) is due to the local value of the electric field regardless of the source of electric field. When the source charge(s) of the electric field moves, it takes some time before the electric field at point P changes and the charge \(Q\) responds to the new electric field.

From the dynamic theory of electric fields one can show that the time taken to respond to a movement of the source charge will be equal to the time it would take light to travel from the source charge to the field point. The reason we see “instantaneous” response in the lab has to do with the fact that light travels at enormous speed of \(3\times 10^{8}\text{ m/s}\) and it takes only nano seconds in the lab setting, which is an imperceptible time.

Subsection 29.3.3 Electric Field of a Point Charge at Origin

Suppose a charge \(q\) is fixed to the origin of a Cartesian coordinate system. What is the electric field at point P a distant \(r\) from the charge? To find the electric field at P, let us place a positive “feeler” (test) charge \(Q\) at P, and deduce electric field by dividing the force on \(Q\) by the charge \(Q\text{.}\)

The magnitude of the force on \(Q\) is:

Now, we can factor the “feeler” charge \(Q\) and immediately obtain the magnitude of the electric field at P.

The direction of the electric field is the direction of the force on a positively-charged test particle. Therefore, if the “source” charge \(q\) is positive, then the electric field at P would be pointed away from \(q\) and towards P, and when \(q\) is negative the electric field at P would have the direction from P towards \(q\text{.}\) See Figure 29.3.4.

Let me emphasize again that the electric field \(\vec E_P\) exists at every space point P whether or not the test charge is placed there. The test charge \(Q\) only “discovers” an electric field by experiencing a force when placed at that point.

Subsection 29.3.4 Electric Field Lines of a Single Charge

The vector arrows of electric field at various points produce a diagram of directed lines. These lines are called electric field lines. The electric field lines of a single positive charge are straight lines emanating from the charge in all directions as shown in Figure 29.3.4 and of a single negative charge are straight lines that terminate on the charge from all directions.

There is a notion of density of electric field lines that gives a quantitative sense in an electric field line diagram. The density of lines at a point is defined by “number of lines” that pass through a unit area pependicular to the lines at that point. You would draw more lines in the region where electric field is stronger and less in the areas with weaker field.

Figure to the right shows an electric field diagram of a single positive charge. At a distance \(r=1\text{ m}\text{,}\) the lines pass through an area of \(4\pi \text{ m}^2\text{,}\) the area of a spherical surface enclosing the charge. But same number of field lines pass through area \(4\pi \times 4\text{ m}^2\) at distance \(r = 2\text{ m}\text{.}\) Therefore, the electric field at at distance \(r = 2\text{ m}\) is \(\dfrac{1}{4} \) of that at at \(r = 1\text{ m}\text{.}\)

\begin{equation*} E_\text{at 2 m} = \dfrac{1}{4} E_\text{at 1 m}. \end{equation*}