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# Electric Force

Did you know that laser printers use electrostatics to print an image or text onto a sheet of paper? Laser printers contain a rotating drum, or cylinder, that gets positively charged using a wire. A laser then shines on the drum and creates an electrostatic image by discharging part of the drum in the shape of the image. The background around the image remains positively charged. Positively charged toner, which is a fine powder, is then coated onto the drum. Since the toner is positively charged, it only sticks to the discharged area of the drum, not the background area that is positively charged. The sheet of paper you send through the printer is given a negative charge, which is strong enough to pull the toner from the drum and onto the sheet of paper. Right after receiving the toner, the paper is discharged with another wire to keep it from sticking to the drum. The paper then passes through heated rollers, which melt the toner and fuse it with the paper. You then have your printed image! This is just one example of how we use electric forces in our daily lives. Let's discuss the electric force on a much smaller scale, using point charges and Coulomb's law, to understand it more fully!

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Fig. 1 - A laser printer uses electrostatics to print an image on a sheet of paper.

## Definition of Electric Force

All material is made up of atoms, which contain protons, neutrons, and electrons. Protons are positively charged, electrons are negatively charged, and neutrons have no charge. Electrons can be transferred from one object to another, causing an imbalance of protons and electrons in an object. We call such an object with an imbalance of protons and electrons a charged object. A negatively charged object has a greater number of electrons, and a positively charged object has a greater number of protons.

There is an electric force in a system when charged objects interact with other objects. Positive charges attract negative charges, so the electric force between them is attractive. The electric force is repulsive for two positive charges, or two negative charges. A common example of this is how two balloons interact after rubbing both of them against a blanket. Electrons from the blanket transfer to the balloons when you rub the balloons against it, leaving the blanket positively charged and the balloons negatively charged. When you put the balloons next to each other, they repel and move away from each other, since they both have a total negative charge. If you instead put the balloons on the wall, which has a neutral charge, they will stick to it because the negative charges in the balloon attract the positive charges in the wall. This is an example of static electricity.

Electric force is the attractive or repulsive force between charged objects or point charges.

We can treat a charged object as a point charge when the object is much smaller than the distances involved in a problem. We consider all the mass and charge of the object to be located at a singular point. Numerous point charges can be used for modelling a large object.

Electric forces from objects that contain large numbers of particles are treated as non-fundamental forces known as contact forces, such as normal force, friction, and tension. These forces are fundamentally electric forces, but we treat them as contact forces for convenience. As an example, the normal force of a book on a table results from the electrons and protons in the book and the table pushing against each other, so that the book cannot move through the table.

## Direction of the Electric Force

Consider the electric force between two point charges. Both point charges exert an equal, but opposite electric force on the other, signifying that the forces obey Newton's third law of motion. The direction of the electric force between them always lies along the line between the two charges. For two charges of the same sign, the electric force from one charge on the other is repulsive and points away from the other charge. For two charges of different signs, the image below shows the direction of the electric force between two positive charges (top) and a positive and negative charge (bottom).

Fig. 2 - The electric force from charges of the same sign is repulsive and from different signs is attractive.

## Equation for the Electric Force

The equation for the magnitude of the electric force, $$\vec{F}_e,$$ from one stationary charge on another is given by Coulomb's law:

$|\vec{F}_e|=\frac{1}{4\pi\epsilon_0}\frac{|q_1q_2|}{r^2},$

where $$\epsilon_0$$ is the permittivity constant that has a value of $$\epsilon_0=8.854\times10^{-12}\,\mathrm{\frac{F}{m}},$$ $$q_1$$ and $$q_2$$ are the values of the point charges in coulombs, $$\mathrm{C},$$ and $$r$$ is the distance between the charges in meters, $$\mathrm{m}.$$The electric force, $$\vec{F}_e,$$ has units of newtons, $$\mathrm{N}.$$

Coulomb's law states the magnitude of the electric force from one charge on another charge is proportional to the product of their charges and inversely proportional to the square of the distance between them.

To find the electric force from one charge on another charge, we first calculate the magnitude of the force using Coulomb's law. Next, we add the direction of the force based on whether the force is attractive or repulsive so that the electric force is expressed as a vector:

$\vec{F}_e=\frac{1}{4\pi\epsilon_0}\frac{|q_1q_2|}{r^2}\hat{r},$

where $$\hat{r}$$ is a unit vector in the radial direction. This is especially important when we find the total electric force acting on a point charge from multiple other point charges. The net electric force acting on a point charge is simply found by taking the vector sum of the electric force from multiple other point charges:

$\vec{F}_{e_{net}}=\vec{F}_{e_1}+\vec{F}_{e_2}+\vec{F}_{e_3}+...$

Notice how Coulomb's law for charges is similar to Newton's law of gravitation between masses, $$\vec{F}_g=G\frac{m_1m_2}{r^2},$$ where $$G$$ is the gravitational constant $$G=6.674\times10^{-11}\,\mathrm{\frac{N\cdot m^2}{kg^2}},$$ $$m_1$$ and $$m_2$$ are the masses in $$\mathrm{kg},$$ and $$r$$ is the distance between them in meters, $$\mathrm{m}.$$ They both follow the inverse square law and are proportional to the product of the two charges or masses.

## Force of an Electric Field

Electric and gravitational forces are different than many other forces that we are accustomed to working with because they are non-contact forces. For example, while pushing a box down a hill requires you to be in direct contact with the box, the force between charges or spherical masses acts from a distance. Because of this, we use the idea of an electric field to describe the force from a point charge on a test charge, which is a charge that is so tiny that the force it exerts on the other charge does not affect the electric field.

Consider the force by a test charge, $$q_0,$$ from a point charge, $$q.$$ From Coulomb's law, the magnitude of the electric force between the charges is:

$|\vec{F}_e|=\frac{1}{4\pi\epsilon_0}\frac{|qq_0|}{r^2}.$

The magnitude of the electric field is found by taking the electric force divided by the test charge, $$q_0,$$ in the limit that $$q_0\rightarrow0$$ so that $$q_0$$ does not affect the electric field:

\begin{align*}|\vec{E}|&=\frac{F}{q_0}\\[8pt]&=\frac{1}{4\pi\epsilon_0}\frac{|qq_0|}{q_0r^2}\\[8pt]&=\frac{1}{4\pi\epsilon_0}\frac{|q|}{r^2}.\end{align*}

This is the equation for the magnitude of the electric field of a point charge. The direction of the electric field depends on the sign of the charge. The electric field always points away from positive charges and towards negative charges.

When a charge, $$q,$$ is placed in an electric field, we can find the electric force on the charge using the same relation as before:

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

If the charge is positive, the force on it points in the same direction as the electric field. If the charge is negative, they point in opposite directions, as shown in the image below.

Fig. 3 - Electric force on a positive charge and a negative charge in the presence of an electric field.

## Examples of the Electric Force

Let's do a couple of examples to practice finding the electric force between charges!

Compare the magnitudes of the electric and gravitational forces from an electron and a proton in a hydrogen atom that are separated by a distance of $$5.29\times10^{-11}\,\mathrm{m}.$$ The charges of an electron and proton are equal, but opposite, with a magnitude of $$e=1.60\times10^{-19}\,\mathrm{C}.$$ The mass of an electron is $$m_e=9.11\times10^{-31}\,\mathrm{kg}$$ and the mass of a proton is $$m_p=1.67\times10^{-27}\,\mathrm{kg}.$$

We'll first calculate the magnitude of the electric force between them using Coulomb's law:

\begin{align*}|\vec{F}_e|&=\frac{1}{4\pi\epsilon_0}\frac{|q_pq_e|}{r^2}\\[8pt]&=\frac{1}{4\pi\epsilon_0}\frac{|e(-e)|}{r^2}\\[8pt]&=\frac{1}{4\pi\epsilon_0}\frac{e^2}{r^2}\\[8pt]&=\frac{1}{4\pi(8.854\times10^{-12}\,\mathrm{\frac{F}{m}})}\frac{(1.60\times10^{-19}\,\mathrm{C})^2}{(5.29 \times10 ^{-11}\,\mathrm{m})^2}\\[8pt]&=8.22\times10^{-8}\,\mathrm{N}.\end{align*}

Since an electron and a proton have opposite signs, we know the force is attractive so that the forces point towards each other.

Now, the magnitude of the gravitational force is:

\begin{align*}|\vec{F}_g|&=G\frac{m_pm_e}{r^2}\\[8pt]&=\left(6.674\times10^{-11}\,\mathrm{\frac{N\cdot m^2}{kg^2}}\right)\frac{(1.67\times10^{-27}\,\mathrm{kg})(9.11 \times 10^{-31}\,\mathrm{kg})}{(5.29\times10^{-11}\,\mathrm{m})^2}\\[8pt]&=3.63*10^{-47}\,\mathrm{N}.\end{align*}

We conclude that the electric force between the electron and the proton is much stronger than the gravitational force since $$8.22\times10^{-8}\,\mathrm{N}\gg3.63\times 10^{-47}\,\mathrm{N}.$$ We can generally ignore the gravitational force between an electron and a proton since it's so small.

Consider the three point charges that have equal magnitude, $$q$$, as shown in the image below. They all lie in a line, with the negative charge directly between the two positive charges. The distance between the negative charge and each positive charge is $$d.$$ Find the magnitude of the net electric force on the negative charge.

Fig. 4 - The net electric force from two positive charges on a negative charge in the middle of them.

To find the net electric force, we take the sum of the force from each of the positive charges on the negative charge. From Coulomb's law, the magnitude of the electric force from the positive charge on the left on the negative charge is:

\begin{align*}|\vec{F}_1|&=\frac{1}{4\pi\epsilon_0}\frac{|q(-q)|}{d^2}\\[8pt]&=\frac{1}{4\pi\epsilon_0}\frac{q^2}{d^2}.\end{align*}.

The force between them is attractive, so it points towards the positive charge in the negative $$x$$-direction and has a minus sign:

$\vec{F}_1=-\frac{1}{4\pi\epsilon_0}\frac{q^2}{d^2}\hat{x}.$

The magnitude of the electric force from the positive charge on the right on the negative charge is equal that of $$\vec{F}_1$$:

\begin{align*}|\vec{F}_2|&=|\vec{F}_1|\\[8pt]&=\frac{1}{4\pi\epsilon_0}\frac{q^2}{d^2}.\end{align*}.

The force between them is also attractive, so it points towards the positive charge in the positive $$x$$-direction:

$\vec{F}_2=\frac{1}{4\pi\epsilon_0}\frac{q^2}{d^2}\hat{x}.$

Thus the vectors are equal in magnitude, but opposite in direction:

$\vec{F}_1=-\vec{F}_2.$

Taking the sum of these, we find the net electric force on the negative charge to be:

\begin{align*}\vec{F}_\mathrm{net}&=\vec{F}_1+\vec{F}_2\\[8pt]&=-\vec{F}_2+\vec{F}_2\\[8pt]&=0\,\mathrm{N}.\end{align*}

## Electric Force - Key takeaways

• Electric force is the attractive or repulsive force between charged objects or point charges.
• Forces such as normal force and friction are fundamentally electric forces, but we treat them as contact forces for convenience.
• Two point charges exert equal, but opposite electric forces on each other, signifying that the forces obey Newton's third law of motion.
• The direction of the electric force between two charges lies along the line between them. For charges of the same sign, the force is repulsive, and for charges of the opposite sign, it is attractive.
• Coulomb's law states the magnitude of the electric force from one charge on another charge is proportional to the product of their charges and inversely proportional to the square of the distance between them: $$|\vec{F}_e|=\frac{1}{4\pi\epsilon_0}\frac{|q_1q_2|}{r^2}.$$
• We use an electric field to describe the force felt on a test charge by a point charge.

## References

2. Fig. 2 - Repulsive and attractive electric force, StudySmarter Originals.
3. Fig. 3 - Electric force on charges in electric field, StudySmarter Originals.
4. Fig. 4 - Net electric field on three charges, StudySmarter Originals.

#### Flashcards in Electric Force 12

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What is electric force?

Electric force is the attractive or repulsive force between charged objects or point charges.

How do I find the electric force?

We find the magnitude of the electric force using Coulomb’s law, and we find the direction of the electric force based on whether the force is attractive between opposite charges or repulsive between like charges.

What are the units of electric force?

Electric force has units of newtons (N).

How are electric force and charge related?

Coulomb's law states the magnitude of the electric force from one charge on another charge is proportional to the product of their charges.

Which factors affect the electrical force between two objects?

The electric force between two objects is proportional to the product of their charges and inversely proportional to the square of the distance between them.

## Test your knowledge with multiple choice flashcards

Forces such as the normal force and friction are fundamentally ____, but we treat them as contact forces for convenience.

The electric force between two charges of opposite signs is ____.

The direction of the electric force between two charges lies ____ the line between them.

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