# Electric Charge Field and Potential

Electricity. The word probably brings to your mind images of lights turning on, the process of charging your cell phone, or even a lightning bolt crossing the skies. For example, did you know that a lightning bolt can carry as much as $$1$$ billion volts of electricity? Obviously, we all know the dangers of lightning but just what does it mean for lightning to carry $$1$$ billion volts of electricity, and why is it so dangerous?

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Table of contents

In this article, we'll explain these very questions by exploring topics like electric charge, field, and potential. First, we'll emphasize the definitions for electric charge, field, and potential. Then, we'll look at their relationships, how to calculate them, and what properties they have through some examples.

Fig. 1 - Lightning bolts can carry $$1$$ billion volts of electricity.

## Electric Charge, Field, and Potential Definitions

Let's start things off with some key definitions. These concepts are going to be used constantly throughout the article and in physics in general, so its important we have a good grasp on them.

### Electric Charge

The fundamental quantity involved in electricity is electric charge.

Electric charge is a fundamental physical quantity and a property of matter, which can take positive and negative values, determining how a particle is affected by electric or magnetic fields.

We represent electric charge as $$q$$ and it is measured in units of Coulombs $$\mathrm{C}$$. Alternatively, we can describe the charge of particles using the fundamental charge unit, equal to the charge magnitude of an electron or proton $$1.6\times10^19\,\mathrm{C}$$. So a proton has a charge of $$+1$$ and an electron $$-1$$. A neutron is neutral with $$0$$ charge, this is because it is composed of smaller particles, called quarks, whose charge sums to zero. Similarly atoms are neutral because they have an equal number of protons and electrons whose charges balance. However, other fundamental particles like photons or neutrinos have no intrinsic charge, meaning they do not interact with the electromagnetic force at all.

### Electric Field

As we touched on previously, electric charges interact with electric fields. An electric field can be a tricky concept to wrap your head around so lets take a closer look.

An Electric Field is a physical vector quantity that takes values at every point in space, defining the force experienced by a test charge sitting at that point in space. Electric charges are sources of electric fields.

Electric fields, alongside magnetic fields, are understood to be manifestations of a more general electromagnetic field. Electric fields exert conservative forces on charges, meaning that the work done by an electric field on a charge is independent of the path taken by the charge, only on its initial and final position.

### Electric Potential

As the electric field produces conservative forces, they can also be explained in terms of something called electric potentials.

Electric Potential is a scalar quantity assigned to each point in space that defines how much energy or work would be done by the electric field in moving a unit test charge from some reference point to that point in space.

Potential results from what is called a conservative force — the work done in moving a particle from one point to another in the field is independent of the path it takes. Gravity is another example of a conservative force. It will do the same amount of work on an object no matter how that object gets to a certain height.

It is important to note that potential is always defined in relation to some reference point; we cannot measure an absolute potential. Hence, we are usually interested in the potential difference between two points. We represent electric potential with a capital $$V$$ and measure it with the unit of Volts $$\mathrm{V}$$, equivalent to Joules per Coulomb $$\frac{\mathrm{J}}{\mathrm{C}}$$.

We know that like charges repel and opposites attract. Therefore, let's illustrate the concept of electric potential with two point charges: one positive and the other negative.

Fig. 2 - Electrons move from areas of low potential to areas of high potential.

Fig. 2 may seem a bit counterintuitive. An electron moving closer to a proton will decrease in potential energy as it moves toward the proton, converting its energy to kinetic energy. So it would appear that the low potential should be near the proton. However, by convention, potentials are defined as the energy required to move a positive charge to that position. Hence, the high potential is near the proton as it would take a lot of energy to overcome the repulsive force between the two positive charges.

We can define the potential energy $$U$$ of the electron, with charge $$q_e$$, in terms of the potential $$V$$ using the equation$U=Vq_e.$

As $$q_e$$ is negative, the electron's potential energy will also be negative, showing that as the electron moves towards regions of high potential, it loses potential energy converting it into kinetic energy.

Now, let's switch out our electron for a proton starting near our first proton. We'll call the proton we replaced for the electron $$\text{Proton}_1$$.

Fig. 3 - Protons move from areas of high potential to areas of low potential.

Since we measure potential in terms of the amount of work done on a positive charge, placing two positive charges near each other increases their potential. Moving two like charges closer together takes a lot of energy because like charges naturally want to repel. Therefore, to force two like charges together, we have to do work on the system or put energy into it; this means that areas of high potential exist near positively charged particles or objects, and areas of low potential exist around negatively charged particles or objects. Likewise, it takes energy to separate two unlike charges; work is needed to pull the mutually attracting bodies away from one another.

## Electric Charge, Field, and Potential Relationships

Before we dive into the calculations of electric charge, field, and potential, we will explore an analogy to help us better conceptually understand these concepts. For this section of the article, we'll first think of potential, force, and field in terms of gravity to gain an intuition and then apply this to electricity. Take a look at Fig. 1 below, showing a cyclist preparing to ride up a steep hill.

Fig. 4 - We can relate electric and gravitational potential to aid us in our understanding.

With gravitational potential energy, a mass increases its potential energy as its height increases; because the Earth's gravity acts as a downward force on it. Therefore, it takes work to go against this force and increase an object's height. This is why the cyclist looks so worried, to maintain their kinetic energy as they cycle up the hill, they will have to put more work in (via their legs) to counteract the increasing gravitational potential as they move from point $$\text{A}$$ to point $$\text{B}$$. However, once they have reached the top of the hill at$$\text{B}$$, the hard part is done and now that gravitational potential is going to make life much easier for them on the way to $$\text{C}$$. At the top of the hill, they have some amount of gravitational potential energy, as they push off down the hill this gravitational potential energy is converted into kinetic energy. Converting potential energy into kinetic energy allows the cyclist to ride down the hill without putting any work in themselves, as they move with the gravitational field rather than against it.

Fig. 5 - Electricity concepts are very similar to gravitational concepts.

Electric potential $$V$$ works in a very similar way. Take a look at Fig. 5; we have two electrically charged plates, the red plate is positively charged, and the blue is negatively charged. We can think of these plates a bit like the ground or the top of the hill in Fig. 4. The big difference here is that which plate is the ground and which is the top of the hill depends on the sign of the charge moving between the plates. For positive charges, the 'ground' is the negative plate as they have zero potential energy when they're located at this plate. For negative charges, the ground is the positive plate. Let's consider the path $$\text{A}\rightarrow\text{B}\rightarrow\text{C}$$ taken by a positive particle. At point $$\text{A}$$, the particle has the lowest potential energy (it's on the ground), to move the particle to $$\text{B}$$ external work must be done to move the particle against the electric field increasing its potential energy. At $$\text{B}$$ the particle has its highest potential energy, like the biker does when at the top of the hill, and so when moving to $$\text{C}$$ the electric field does work on the particle converting potential energy into kinetic energy as the particle is pulled towards $$\text{C}$$ by the negative plate. For a negative particle, the path is essentially the same but in reverse, with $$\text{A}$$ the 'top of the hill' and $$\text{B}$$ the ground.

Fig. 6 - $$\text{A}$$ moves against the field, so it drops in potential and has to have work done on it to move.

We can illustrate this with field lines representing a field's force vectors, demonstrating the direction and magnitude of the force experienced by a particle in that field. For electric fields, we draw field lines for positive particles by convention, as can be seen in Fig. 6. Note how the arrows are of equal length; this is because the electric field between two parallel plates is a Uniform Field, having equal strength at every point. Drawing in the field lines highlights how the electric and gravitational examples are similar; in Fig. 7 we see the gravitational field lines pointing down toward the earth's surface. Again, the arrows are of equal length because we can approximate the gravitational field as uniform near the Earth's surface.

Fig. 7 - Going from $$\text{A}$$ to $$\text{B}$$ is moving against the field; therefore, following this path increases the cyclist's potential and requires work to be done.

## Charge, Field, and Potential Calculations

Above, we compared electric charge, field, and potential qualitatively. Now, we will compare them mathematically, looking at some of the defining equations.

### Electric Force and Point Charges

For point charges, the equation for electric force is given by Coulomb's Law:

$$|\vec F_E | = \frac{1}{4\pi \epsilon_0} \frac{|q_1 q_2 |}{r^2},$$

where $$\vec F_E$$ is the net electric force acting on the charges, $$\epsilon_0=8.85\times 10^{-12}\,\mathrm{C^2/(N\,m^2)}$$ is a constant known as the permittivity of free space, $$q_1$$ and $$q_2$$ are the charges of the particles, and $$r$$ is the distance between them.

Note that the magnitude of the force is directly proportional to the product of the charges but inversely proportional to the square of the distance.

### Electric Force and Electric Field

If we are considering any general electric field, the electric field strength $$\vec{E}$$ is related to the force $$\vec{F}_E$$ experienced by a charge $$q$$ in that field using the equation$\vec{E}=\frac{\vec{F}_E}{q}.$

For example, if we consider the electric field produced by a charge $$q_1$$, then we can use Coulomb's Law and the previous equation to find the field strength\begin{align}\vec{E}&=\frac{\vec{F}_E}{q_2}\\&=\frac{1}{q_2}\frac{1}{4\pi \epsilon_0} \frac{|q_1 q_2 |}{r^2}\\&=\frac{1}{4\pi \epsilon_0} \frac{|q_1|}{r^2}\end{align}

### Electric Potential

As discussed previously, we can also describe electric fields with the quantity of electric potential. If we are considering the electric potential associated with the field surrounding a point charge $$q$$, then the potential is given by

$$V = \frac{1}{4\pi \epsilon_0} \frac{q}{r},$$

where $$V$$ is the electric potential, $$q$$ is the charge producing the field, and $$r$$ is the distance from the place we want to measure the electric potential to our charge.

## Electric Charge, Field, and Potential Properties

The easiest way to see electrical properties is to compare them to their gravitational counterparts. So we can continue making connections to gravity as shown in Table 1.

Table 1 - Comparing electricity to gravity.

 Physical Concept Gravity Electricity Unit Mass $$m$$ in $$\mathrm{kg}$$ Charge $$q$$ in $$\mathrm{C}$$ Distance Height $$h$$ in $$\mathrm{m}$$ Separation distance $$r$$ in $$\mathrm{m}$$ Field Gravitational Field $$\vec g$$ in $$\mathrm{N/kg}$$ Electric Field $$\vec E$$ in $$\mathrm{V/m}$$ or $$\mathrm{N/C}$$ Potential $$V$$ for both types $$V = \frac{U}{m}$$ $$V = \frac{U}{q}$$ Force from a Uniform Field $$F_g = m\vec g$$ $$F_E = q\vec E$$ Force on two particles $$F_g = \frac{Gm_1 m_2}{r^2}$$ $$|\vec F_E | = \frac{1}{4\pi \epsilon_0} \frac{|q_1 q_2 |}{r^2}$$ Potential Energy for a Uniform Field $$U$$ $$U=mg\Delta h = F_g \Delta h$$ $$U=Fr$$

Here we highlight a few key points from the table above:

• Both gravity and electric forces need a unit on which to act: for gravity, its mass, and for electric forces, its charge.
• The force of a gravitational or electric field equals the unit we work with times the field.
• The force acting on two particles (whether charge or mass) is equivalent to multiplying our units by some constant and dividing by the distance between them squared.
• If the field is Potential energy equals the conservative force times the separation distance (whether the height from the ground or the distance between two particles).

## Electric Charge, Field, and Potential Examples

Let's recap what we've learned with some examples.

An electron is directly placed inside a constant electric field of magnitude $$150\,\mathrm{N/C}$$. What is the electric force of the electric field on the electron? The charge of an electron is $$-1.602\times 10^{-19}\,\mathrm{C}.$$

Fig. 8 - A constant electric field of $$150\,\mathrm{N/C},$$ the gray arrows show the direction of the field.

Remember that our formula for the force of an electric field is

$$\vec{F}_E = q\vec E.$$

So, to find the force, we can plug in our values. Let's represent the left direction as negative and the right as positive.

\begin{align*} \vec{F}_E &= q\vec E \\ \vec{F}_E &= (-1.602 \times 10^{-19} \,\mathrm{C})(150\,\mathrm{N/C}) \\ \vec{F}_E &= -2.4\times 10^{-17}\,\mathrm{N}. \\ \end{align*}

Therefore, the electron feels a force of $$2.4\times 10^{-17}\,\mathrm{N}$$ to the left.

Now let's look at an example involving two particles.

Find the magnitude of the electric force that the proton exerts over the electron shown in Fig. 9. What is the magnitude of the electric potential that the proton generates at the electron's position? Use $$-1.602\times 10^{-19}\,\mathrm{C}$$ for the charge of an electron and $$1.602\times 10^{-19}\,\mathrm{C}$$ for the charge of a proton. Remember that a nanometer equals $$10^{-9}$$ meters.

Fig. 9 - Note that the graph measures distance in nanometers.

Before we begin solving, we need to find the distance between our two charges. Note that the grid above has units of nanometers. To find the distance between the proton and the electron, we can use the Pythagorean theorem:

$$r = \sqrt{(4\,\mathrm{nm})^2 + (4\,\mathrm{nm})^2} = 4\sqrt{2} \,\mathrm{nm}.$$

Now, we'll solve for the electric force using our equation for two point charges:

\begin{align*} |\vec F_E | &= \frac{1}{4\pi \epsilon_0 } \frac{|q_1 q_2 |}{r^2} \\ &= \frac{1}{4\pi (8.85\times 10^{-12}\,\mathrm{C^2/(N\,m^2)})} \frac{(1.602\times 10^{-19}\,\mathrm{C})^2}{(4\sqrt{2}\,\mathrm{nm})^2} \\ &= \frac{1\,\mathrm{N\,m^2}}{4\pi (8.85\times 10^{-12}\,\mathrm{C^2})} \frac{2.566\times 10^{-38}\,\mathrm{C^2}}{(4\sqrt{2} \times 10^{-9}\,\mathrm{m})^2} \\ |\vec F_E| &= 7.21\times 10^{-12}. \\ \end{align*}

Finally, let's find the electric potential $$V$$ that the proton generates at the electron's position:

\begin{align*} V&= \frac{1}{4\pi \epsilon_0 } \frac{q}{r} \\ &= \frac{1}{4\pi (8.85\times 10^{-12}\,\mathrm{C^2/(N\,m^2))}} \frac{1.602\times 10^{-19}\,\mathrm{C}}{4\sqrt{2}\times 10^{-9}\,\mathrm{m}} \\ V &= 0.25\,\mathrm{V}. \\ \end{align*}

The example above gives us the amount of electric potential that exists between a proton and an electron that are $$4\sqrt{2}$$ nanometers apart, coming out to about $$0.25\,\mathrm{V}$$. To put that in perspective, your average lightbulb holds about $$120\,\mathrm{V}$$, so about $$473$$ times the electric potential between our proton and electron. But wait, didn't we say that a lightning bolt generates $$1$$ billion volts of electricity? That's a lot of light bulbs! Nearly $$8\,333\,333$$ of them!

## Electric Charge Field and Potential - Key takeaways

• Electric charge is a physical quantity carried by objects or particles that can either be positive or negative.
• Neutral objects have an equal amount of negative and electric charge, netting their total electric charge to zero. There are, however, some exceptions to this rule because some fundamental particles carry no charge and are neutral in that respect.
• Electric fields propagate from a source object and exert an electric force on charged particles placed inside it.
• An electric force is a push or a pull on an object or particle due to that particle or object's electric charge.
• Potential energy is energy inherent in an object due to its physical characteristics or position relative to other objects.
• Electric potential is the energy needed to move one unit of positive electric charge to a certain reference point.
• You can relate gravitational concepts to electrical ones.
• Electric field lines branch out of positive charges and converge on negative charges. Therefore, an electron would flow against electric field lines.
• An electric force is exerted along the direction of electric field lines.
• For point charges, our electric force equation is:

$$|\vec F_E | = \frac{1}{4\pi \epsilon_0} \frac{|q_1 q_2 |}{r^2}$$

• $$\vec F=q\vec E$$ is the equation for the force on a charged particle or object in an electric field.

• We can find the electric potential due to a point charge (relative to a reference point at infinity) by using the equation below:

$$V = \frac{1}{4\pi \epsilon_0} \frac{q}{r}$$

## References

1. Fig. 1 - Photo of City Buildings Under Lightning Strike (https://www.pexels.com/photo/photo-of-city-buildings-under-lightning-strike-2693284/) by Nick Kwan (https://www.pexels.com/@nickkwanhk/) is licensed by Pexels License (https://www.pexels.com/license/)
2. Fig. 2 - Electrons, Protons, and Potential, StudySmarter Originals
3. Fig. 3 - Multiple Protons and Potential, StudySmarter Originals
4. Fig. 4 - Gravitational Potential and the Biker, StudySmarter Originals
5. Fig. 5 - Electric Potential Between Two Plates, StudySmarter Originals
6. Fig. 6 - Electric Field Between Two Plates, StudySmarter Originals
7. Fig. 7 - Gravitational Field and the Biker, StudySmarter Originals
8. Fig. 8 - Electron Moving through a constant electric field, StudySmarter Originals.
9. Fig. 9 - Proton and Electron Separated by Nanometers, StudySmarter Originals

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##### Frequently Asked Questions about Electric Charge Field and Potential

How are electric potential and electric field related?

Electric potential is the energy needed to move one unit of positive electric charge to a certain reference point against an electric field.

What is meant by potential of a charge in an electric field?

The potential of a charge in an electric field is the amount of work needed to move one unit of that particle's positive electric charge.

Is electric potential equal to the electric field?

Electric potential is not equal to the electric field. Electric potential is related to electric potential energy and electric field relates to the electric force. Electric potential is a scalar quantity but electric field is a vector quantity.

What is the difference between electric field and electric potential?

An electric field is a vector quantity that represents the force that a (positive) unit test charge would feel at any position relative to the source. On the other hand, electric potential is the energy needed to move one unit of positive electric charge to a certain reference point against an electric field.

How is Electric Charge, Field and Potential calculated?

Electric charge is not calculated; it is measured. You can calculate the electric field by dividing the electric force by the particle's charge inside the field. You can find the potential by dividing the electric potential energy by the particle's charge.

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