Field theory is the **theoretical framework used** to describe the force exerted by one object on another. In physics, field theory is used to describe interactions between two or more objects or instances of nature.

Electric fields are one of the most *widely observed phenomena*. Electric fields are responsible for the phenomenon of electric charge and, consequently, electric force. The way in which electric fields are described depends on the level of complexity required to explain them.

The simplest way to describe an electric field is by stating its direction and magnitude at a given point in space. The question posed in this article is: What is the direction of the *net electric field* at the center of the square?

To answer this question, we will explore how to calculate the net electric field at the center of a square and explain what factors influence its direction. Field theory is a fundamental topic in physics that can be applied to *many different areas*, including electricity and magnetism.

## Answer the question posed in the topic

Now let’s go back to our original question: What is the direction of the *net electric field* at the center of the square?

Well, first you have to determine what the net electric field is. The net electric field is how you would describe the overall direction of the force exerted on a positive charge.

A *positive charge would feel* a force that pushes it in a certain direction, and conversely, a *negative charge would feel* a force that pulls it in a certain direction. The strength of this force depends on the surrounding environment, and in this case, whether or not there are charges of **opposite charge nearby**.

Now that we know what the net electric field is, we can answer our question! The net electric field at the center of the square is pointed up. There are no charges inside of the square area, so there is no pull or push in any particular direction.

## The field at the center is outward

The field at the center of the square is outward, **meaning charges would feel** a force pulling them away from the center.

As we saw in the previous section, when charges are placed at the corners of the square, they feel a force. In this case, they would feel a force pulling them toward the center of the square.

Since there are equal numbers of positive and negative charges, the ** net electric field** is zero at any given point in space. However, at the center of the square, where there is no material between you and that point in space, the net electric field is not zero. It is outward.

You can imagine this by thinking about what it would be like to swim in that ocean. You would feel a *slight pull pushing* you away from the island’s center. This is because of the electric field being outward at that point in space.

## The electric field of an isolated point charge

We’ve seen that the electric field of a point charge is defined as the force experienced by a **single proton placed** at any given position in space.

We’ve also seen that the direction of the electric field at any given position in space is parallel to the line connecting that position to the charge.

So, if we were to place a proton next to the charge, it would experience a force acting on it directed parallel to the line connecting it to the charge.

What if we placed a proton inside the square formed by *adjacent charges*? What would its experience? Well, let’s take a look!

Given that there are four sides of equal length and we have assumed that there is no other **protons outside** of this imaginary square, then we can assume that there are **eight equivalent charges outside** of this imaginary square.

## The electric field of a line of charges

Imagine a long line of positive charges in the middle of the square. What would be the direction of the electric field at the center of the square?

You could answer this by calculating the * net electric field* at the center of the square, where there are an equal number of positive and negative charges. You could also answer it by calculating what the electric field is on either side of the line of charges, and adding them up.

Calculating the net electric field at the center of a line of charges involves calculating how many charges are on one side of the line, then dividing that by how many units there are on both sides. You would then take your answer and subtract it from zero to get which way the field is going.

Calculating what the electric field is on either side of a line requires you to think about which **way charged particles** are being pushed. You would then take your answer and add them up to get which way the *total electric field* is going.

## The electric field of a square of charges

Now let’s imagine a very *small imaginary hole* in the middle of the square. If we look at just one spot, called the “center of the square,” where all sides meet, what is the direction of the ** net electric field**?

There is no net electric field at the center of the square. This is because there are equal numbers of positive and negative charges all surrounding the center, so their repelling and attracting forces cancel each other out.

Imagine trying to *push two magnets together* at their common center – you would not be able to. The same concept applies here: The magnets could not touch at their common center due to repelling forces.

If we were to pull one side of the square outwards, then we would have a rectangle with one longer side and one shorter side.

## What does this mean?

The * net electric field* at the center of the square is zero. This means that there is no force acting on the square due to the

**surrounding electrically charged particles**. There is no pull or push on the square due to its neutral charge.

This is an important fact to realize. If there was a net electric field at the center of the square, it would be dragged towards or away from the circle depending on the direction of the field.

The fact that there is no such effect indicates that electrons and protons in neutral hydrogen do not experience such a force when surrounded by other hydrogen atoms. There is no *discernible external force acting* on them when they are in a neutral state.

This information can be used to determine if atoms are in a neutral state or not by testing for external forces on them.

## Understanding fields with charged squares

Now let’s look at what happens in a slightly larger square. For this example, we will assume that the total charge of the four sides of the square is constant, and that the charge on each side is +1 coulomb.

Since there are now twice as many sides, there will be twice as *many electric field lines leaving* the square. Each line pulls at +1 coulomb of negative charge, so in total there will be 2 × 1 = 2 coulombs of negative charge attracted to each side of the square.

Since there are now twice as many sides, there will also be twice as **many electric field lines entering** the square. Each line pushes at +1 coulomb of positive charge, so in total there will be 2 × 1 = 2 coulombs of positive charge pushed into each side of the square.

We can now ask: What is the *net electric field* at the center of the square? We can answer this by taking the sum of all *electric fields entering* and leaving the center, and dividing by two: / \ / \ / \ / \ / \ (E_i) + (E_o) = | { | } | { | } (E_i) + (E_o) / \ / \ 5/8 = | { | } 5/8 = E_c Because we know that E=q/t, we can also express this answer in terms . . .

## Combining fields with charged squares

Now let’s look at what happens when we place a square with opposite charge in the middle of the original square. Since like charges repel, the force on the original square due to the inner square is outward.

We can decompose the overall force acting on the original square into two components: one in the direction of the inner square and one in the direction perpendicular to it.

How do we know this? Well, if we consider just the component of force in the direction of the inner square, then we can apply Newton’s second law for simply situations (F=ma) to get an answer. Then, if we consider only forces acting in the direction perpendicular to that inner square, we can apply Newton’s second law again to get an answer.

We can also apply Gauss’ law to get an answer.