The Concept of the Electric Field
In the 19th century, Michael Faraday, one of the greatest experimenters in the field of electricity, had a peculiar way of working: he filled his laboratory with wires, charged spheres, and small containers with conductive liquids. A famous anecdote tells that, in his obsession to visualize the “lines of force” surrounding an electric charge, he spilled iron filings all over the lab floor, leaving it covered in patterns that resembled modern art. His colleagues, confused, thought he had lost his mind, but Faraday was outlining one of the most revolutionary concepts: the electric field. In this article, we will explore how these ideas, born from genius and experimentation, allow us to map and understand the invisible interactions governing electricity. If you’ve ever wondered how to see the intangible, this journey is for you.
Learning Objectives:
By the end of this lesson, the student will be able to
- Understand the concept of the electric field and its relationship with electric force through Coulomb’s Law.
- Apply the definition of the electric field to solve problems related to point charges.
- Analyze the principle of superposition in discrete and continuous charge distributions to calculate electric fields.
- Evaluate the integration of linear, surface, and volumetric distributions to determine electric fields in complex configurations.
- Solve practical exercises including configurations such as charged rods, charged rings, and infinite charged planes.
CONTENT INDEX:
What is the electric field?
The electric field and charge distributions
Exercises
What is the Electric Field?
When we place a source charge somewhere in space, we can sense its presence using a test charge due to the electric force it experiences. This force is studied through Coulomb’s Law. Based on this, we say that the source charge “floods the space” with a property, an Electric Field, which is responsible for producing the electric force.
To measure the electric field of a charge q at a certain point \vec{r} in space, we need to place a test charge q_0 at that location. The electric field is described as the amount of electric force felt by the test charge q_0 per unit charge.
\vec{E}_q(\vec{r}) = \displaystyle \frac{\vec{F}_{q\to q_0}(\vec{r})}{q_0}
But when we proceed this way, we overlook the fact that the test charge should also have its own field, and this will overlap with the field of the source charges. To address this issue, we define the electric field through the limit:
\displaystyle\vec{E}_q(\vec{r}) = \lim_{q_0 \to 0} \frac{\vec{F}_{q\to q_0}(\vec{r})}{q_0}
Using the definition through the limit, we ensure that the field of the test charge q_0 does not interfere with the measurements of the field of charge q. Now, recalling Coulomb’s Law, the electric field of a charged particle q is given by:
\displaystyle\vec{E}_q(\vec{r}) = \lim_{q_0 \to 0} \frac{1}{q_0} \frac{1}{4\pi \epsilon_0} \frac{qq_0}{\|\vec{r} -\vec{r}^\prime\|^2} \frac{\vec{r} -\vec{r}^\prime}{\|\vec{r} -\vec{r}^\prime\|} = \frac{q}{4\pi\epsilon_0} \frac{\vec{r} -\vec{r}^\prime}{\|\vec{r} -\vec{r}^\prime\|^3}
And from this, it follows that
\displaystyle \vec{F}_{q\to q_0}(\vec{r}) = q_0 \vec{E}_q(\vec{r})
The Electric Field and Charge Distributions
Since the electric field is studied in terms of force and satisfies the principle of superposition, we can analyze the fields of different charge distributions.
Discrete Distributions
Let us consider a distribution of n discrete charges q_1, q_2, \cdots, q_n with positions \vec{r}^\prime_1, \vec{r}^\prime_2, \cdots, \vec{r}^\prime_n. If we want to calculate their field at a point \vec{r} in space, we have:
\displaystyle\vec{E}_q(\vec{r}) = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^n q_i \frac{\vec{r} -\vec{r}_i^\prime}{\|\vec{r} -\vec{r}_i^\prime\|^3}
This is the sum of all individual fields.
Continuous Distributions
There are three types of continuous charge distributions, each associated with the number of parameters needed to describe their spatial arrangement. These are linear, surface, and volumetric distributions.
Linear Distribution
In a linear charge distribution, each line element of the charged body has a linear charge density \lambda(\vec{r}^\prime)=dq(\vec{r}^\prime)/dl, such that the electric field element is given by:
d\vec{E}(\vec{r}) =\displaystyle \frac{dq(\vec{r}^\prime)}{4\pi\epsilon_0} \frac{\vec{r} - \vec{r}^\prime}{\|\vec{r} - \vec{r}^\prime\|^3} = \frac{\lambda(\vec{r}^\prime)}{4\pi\epsilon_0} \frac{\vec{r} - \vec{r}^\prime}{\|\vec{r} - \vec{r}^\prime\|^3}dl
Integrating this expression gives:
\displaystyle \vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int_{\mathcal{C}} \lambda(\vec{r}^\prime) \frac{\vec{r} - \vec{r}^\prime}{\|\vec{r} - \vec{r}^\prime\|^3}dl
Where \mathcal{C} is the parametric representation of the curve describing the shape of the charged body.
Surface Distribution
In a surface charge distribution, each surface element of the charged body has a surface charge density \sigma(\vec{r}^\prime)=dq(\vec{r}^\prime)/dS, such that the electric field element is given by:
\displaystyle d\vec{E}(\vec{r}) = \frac{dq(\vec{r}^\prime)}{4\pi\epsilon_0} \frac{\vec{r} - \vec{r}^\prime}{\|\vec{r} - \vec{r}^\prime\|^3} = \frac{\sigma(\vec{r}^\prime)}{4\pi\epsilon_0} \frac{\vec{r} - \vec{r}^\prime}{\|\vec{r} - \vec{r}^\prime\|^3}dS
Integrating this expression gives:
\displaystyle \vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \iint_{\mathcal{A}} \sigma(\vec{r}^\prime) \frac{\vec{r} - \vec{r}^\prime}{\|\vec{r} - \vec{r}^\prime\|^3}dS
Where \mathcal{A} is the parametric representation of the surface describing the shape of the charged body.
Volumetric Distribution
In a volumetric charge distribution, each volume element of the charged body has a volumetric charge density \rho(\vec{r}^\prime)=dq(\vec{r}^\prime)/dV, such that the electric field element is given by:
\displaystyle d\vec{E}(\vec{r}^\prime) = \frac{dq(\vec{r}^\prime)}{4\pi\epsilon_0} \frac{\vec{r} - \vec{r}^\prime}{\|\vec{r} - \vec{r}^\prime\|^3} = \frac{\rho(\vec{r}^\prime)}{4\pi\epsilon_0} \frac{\vec{r} - \vec{r}^\prime}{\|\vec{r} - \vec{r}^\prime\|^3}dV
Integrating this expression gives:
\displaystyle \vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \iiint_{\mathcal{V}} \rho(\vec{r}^\prime) \frac{\vec{r} - \vec{r}^\prime}{\|\vec{r} - \vec{r}^\prime\|^3}dV
Where \mathcal{V} is the parametric representation of the volume describing the shape of the charged body.
Exercises:
Charged Rod
Consider a rod of length L uniformly charged with a charge Q and positioned vertically. Determine the electric field of the rod at a horizontal distance x from the center of the rod.
Charged Ring
Consider a ring of radius R uniformly charged with a charge Q placed on the xy-plane. Determine the electric field at a height z from the center of the ring.
Infinite Charged Plane
Consider an infinite plane uniformly charged with a surface charge density \sigma. Determine the electric field at a distance L from the plane.