Electric Fields from Point Charges

1. From Force to Field: A New Perspective

The idea of a field helps us move away from the slightly mysterious notion of "action at a distance." The gravitational field is a concept you are likely familiar with. The Earth doesn't directly "pull" on the Moon; rather, the Earth's mass generates a gravitational field throughout space. The Moon then interacts with the field at its location, and that interaction is what we perceive as the force of gravity.

We can apply the exact same logic to electric charges.

An electric charge $Q$ creates an electric field $\vec{E}$ in the space around it. If we then place a small, positive "test charge" $q_0$ at some point in this field, it will experience an electric force ${\vec{F}}_e$. The electric field $\vec{E}$ at that point is defined as the force experienced by the test charge divided by the magnitude of the test charge itself.

The unit of the electric field is newtons per coulomb (N/C). This definition is the cornerstone of our lesson. It tells us that the field at a point is the "force per unit charge" that would be experienced there. The field exists independently, whether we place a test charge there to measure it or not.

Conversely, and most usefully, if we know the electric field $\vec{E}$ at a point in space, the force on any charge $q$ placed at that point is simply:

Notice that if the charge $q$ is positive, the force ${\vec{F}}_e$ is in the same direction as the field $\vec{E}$. If $q$ is negative, the force is in the opposite direction to the field.

The following reading explains this concept clearly, drawing the parallel with the gravitational field.

2. The Electric Field of a Point Charge

Now that we have defined the electric field, let's find the field created by the simplest object: a single point charge $Q$.

We start with what we know from the last lesson: the force exerted by a source charge $Q$ on a test charge $q_0$ at a distance $r$ is given by Coulomb's Law:

where $\hat{r}$ is the unit vector pointing from $Q$ to $q_0$.

Using our new definition, $\vec{E}={\vec{F}}_e/q_0$, we simply divide by $q_0$:

This is the electric field of a point charge $Q$. Notice that it only depends on the source charge $Q$ and the distance $r$; the test charge $q_0$ has vanished, as it should.

Your strong background in vector calculus makes the alternative form, $\vec{E}=k\frac{Q}{|\vec{r}{|}^3}\vec{r}$, a powerful tool. The following video provides a masterful and thorough breakdown of how to calculate the electric field vector from a point charge, covering both the intuitive trigonometric approach and the formal vector component approach.

3. The Superposition of Electric Fields

What if we have multiple source charges? Just as with electric forces, electric fields obey the principle of superposition.

The total electric field at any point in space is the vector sum of the electric fields produced by each individual source charge.

Each individual field ${\vec{E}}_i$ is calculated using ${\vec{E}}_i=k\frac{Q_i}{r_i^2}{\hat{r}}_i$ as if it were the only charge present.

This means the problem-solving strategy you learned for calculating net force can be directly applied to calculating the net electric field.

4. Calculating the Net Electric Field

Let's put this all together to solve a typical problem. The strategy is:

1. Draw a Diagram: Sketch the charge configuration and the point P where you want to find the field.
2. Individual Fields: For each source charge $Q_i$, draw the electric field vector ${\vec{E}}_i$ at point P. Remember, it points away from positive charges and towards negative charges.
3. Components: Calculate the magnitude of each field, $E_i=k|Q_i|/r_i^2$. Then, using trigonometry, find the x and y components of each field vector.
4. Vector Sum: Add all the x-components to get $E_{\text{net},x}$ and all the y-components to get $E_{\text{net},y}$.
5. Resultant Field: Calculate the magnitude and direction of the net field using $E_{\text{net}}=\sqrt{E_{\text{net},x}^2+E_{\text{net},y}^2}$ and $\theta =\arctan (E_{\text{net},y}/E_{\text{net},x})$.

Let's walk through a complete example.

To see another excellent demonstration of this process, especially one that emphasizes the pure vector notation from the start, the following video is highly recommended. It tackles a similar problem but frames it entirely in terms of position vectors, which can be a very efficient and powerful method.

For more practice, the document Physics 1100: Electric Fields Solutions provides numerous solved problems that follow this same component-based method. The first problem is a great one to review if you want to see the strategy applied one more time.

Conclusion

In this lesson, we have made a crucial conceptual leap from forces to fields. This is a more abstract but ultimately more powerful way of understanding electrostatic interactions.

Here are the key takeaways:

• The electric field $\vec{E}$ is a vector field that permeates the space around charges. It is defined as the force per unit charge: $\vec{E}={\vec{F}}_e/q_0$.
• A point charge $Q$ creates an electric field given by $\vec{E}=k\frac{Q}{r^2}\hat{r}$. The field points away from positive charges and towards negative charges.
• The superposition principle holds for electric fields: the net field at a point is the vector sum of the fields from all individual source charges.
• The process for calculating the net field is a direct application of vector addition: decompose individual field vectors into components, sum the components, and reconstruct the resultant vector.

In all the problems today, we've dealt with a handful of discrete point charges. What happens if you have billions of charges packed together, forming a continuous line, a flat sheet, or a solid sphere? You can't sum them one by one. In our next lesson, we will see how your knowledge of calculus allows us to transition from a discrete sum ($\sum$) to a continuous integral ($\int$), enabling us to calculate the electric field for continuous charge distributions.