Section 21.9 Electric Field of a Point Charge
Subsubsection Key Ideas
Definition 21.9.2. Electric Field Created by a Point Charge.
The electric field at position \(\vec{r}\) due to a point charge located at the origin is
\begin{equation*}
\vec{E}(r) = k \frac {q_{\text{source}}}{r^2} \hat{r}
\end{equation*}
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\(k \approx 9 \times 10^{9} \frac{\mathrm{Nm^2}}{\mathrm{C^2}}\) is the electrostatic constant, a fundamental constant.
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\(q_{\text{source}} \) represents the source charge generating the field.
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\(r \) is the distance from the charge to the location where you want to know the field. Note that \(\vec{E}\) is a function of \(r\text{:}\) its value depends on where you are in space!
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\(\hat{r} \) is a Unit Vector that points radially away from the source charge. Note that this also depends on where you are in space!
Note 21.9.4. Charge subscripts.
Every electric charge generates an electric field. The term source charge is typically used to refer to a charge that is creating an electric field, represented as \(q_{\text{source}}\text{.}\) A source charge never feels a force from its own field. The term test charge, represented as \(q_{\text{test}}\text{,}\) is typically used to refer to the charge experiencing a force due to this field. It is common to drop the subscript notation and describe charges using just \(q\text{,}\) using context to determine which is the charge creating the field and which is the charge experiencing the field.
Note 21.9.5. The Permittivity Constant.
In the equation above, the electric field is written in terms of the electrostatic constant, \(k \approx 9 \times 10^9 \frac{\mathrm{Nm^2}}{\mathrm{C^2}}\text{.}\) In other contexts, this constant is written as \(k = \frac{1}{ 4 \pi \epsilon_0}\text{,}\) where \(\epsilon_{0} = 8.85 \times 10^{-12} \frac{\mathrm{C^2}}{\mathrm{Nm^2}}\) is a fundamental constant known as the permittivity of free spsace.
The constant \(\epsilon_{0} \) (pronounced “epsilon naught”) is a constant of proportionality that scales how the magnitude of the electric field is related to charge and distance in a vacuum. If there is other matter around that charge, for example air or wood or water, that matter can affect how the electric field is permitted to exist, and thus the permittivity constant will be different.
Analogy with Gravity.
Recall the definition of The Gravitational Field. Note the units of the gravitational constant and then recall the units of the electrostatic constant \(k \text{.}\) Do you notice any similarities?
Subsubsection Activities
Activity 21.9.1. Electric Field Vector Maps.
Recall Practice Sketching Field Maps, in which you made a 2D-map of a vector field. The vector field you mapped is proportional to \(\hat{r}\text{!}\) Map this vector field and describe how this vector field changes (or stays the same) at different points in space. What do you think the map would look like if you could make it three-dimensional?
