What happens if you rub a balloon against your hair? Your hair will stick to the balloon! This happens because electrons from the atoms in your hair move to the balloon, leaving your hair positively charged, and consequently, the balloon becomes negatively charged. The positive and negative charges attract each other, causing your hair to stick to the balloon!
Fig. 1 - Hair sticks to a balloon due to the electric charge between them.
If you observe carefully, you can notice that the hairs try to reach for the ballon even when they are not touching it. This is because the attractive effect caused by the charged balloon extends through space, creating an electric field. Finding a mathematical expression to describe such an electric field can be challenging. However, the key is learning to describe the electric field from multiple charges. If we know how to describe the electric field from a few point charges, we can increase the number of charges and even move on to charge distributed on an object (like the case of the charged balloon!) Let's have a look at how this works!
Properties of the Electric Field From Multiple Point Charges
A point charge is an amount of electric charge existing at a single point. Point charges can be positive or negative, and charges with the same sign repel each other, while charges with opposite signs attract each other. It is useful to use point charges to model objects that are very small compared to the distances involved in a specific problem, or model a large object using a great number of point charges.
A point charge is an amount of electric charge concentrated at a single point.
The electric field is a vector quantity that we use to describe the effect of an electric charge or system of electric charges. We do so by specifying the force that the system of charges would exert on a test charge at each position. Because of its vector nature, an electric field has both magnitude, and direction. The magnitude of the electric field of a point charge is proportional to the amount of electric charge and inversely proportional to the square of the distance from it. For a positive charge, the electric field lines point away from the charge, while for negative charges, they point toward it, as shown in the image below.
An electric field is a vector quantity that represents the force that a (positive) test charge would feel at any position relative to the source.
In general, we can sketch the electric field of a single or a pair of point charges directly. However, when we have multiple point charges close to each other, the electric field becomes hard to visualize because of the contributions of each charge. To find the resulting electric field at a certain location, we must find the vector sum of the electric fields from each point charge. This means that electric field lines will never cross, but rather, every location has a vector that represents the net electric field at that point. The image below shows an example of the electric field from four point charges.
Notice that the electric field lines always point towards the negative charges and away from the positive ones. Close to the charges, the electric field lines are closer together. As the distance from the charges increases, so does the separation between the field lines. The separation between the field lines represents the magnitude of the electric field. Thus, the magnitude of the electric field is greater in locations where the field lines are close together, near the charges, as shown in the image below.
Formula for the Net Electric Field
As mentioned above, the net electric field from multiple charges at a certain location is found by taking the vector sum of the electric fields from the charges. For a single point charge, the magnitude of the electric field is given by:
\[|\vec{E}|=\frac{1}{4\pi\epsilon_0}\frac{|q|}{r^2},\]
where \(q\) is the charge in coulombs, \(\mathrm{C},\) \(r\) is the distance from the point charge to the point of interest in meters, \(\mathrm{m},\) \(\epsilon_0\) is the permittivity of free space that has a value of \(8.854\times10^{-12}\mathrm{\frac{C^2}{N\,m^2}},\) and \(\vec{E}\) is the electric field, which has units of newtons per coulomb, \(\mathrm{\frac{N}{C}}.\)
Consider three charges whose electric fields have magnitudes of \(|\vec{E}_1|,\) \(|\vec{E}_2|,\) and \(|\vec{E}_3|\) at a given point. The net electric field is found by taking the vector sum of each electric field:
\[\vec{E}_\mathrm{net}=\vec{E}_1+\vec{E}_2+\vec{E}_3.\]
If we consider a two-dimensional electric field, the electric field vector from each charge is made up of an \(x\)-component and a \(y\)-component. The magnitude of the net electric field at any point is found by substituting these components into the Pythagorean relation:
\[|\vec{E}_\mathrm{net}|=\sqrt{(E_{x_1}+E_{x_2}+E_{x_3})^2+(E_{y_1}+E_{y_2}+E_{y_3})^2}.\]
In this equation, \(E_{x_1}\), \(E_{y_1}\), and so on represent the electric field of each charge in the \(x\) and \(y\) directions at the point of interest.
We can generalize this for \(n\) number of point charges and rewrite these equations as:
\[\begin{align*}\vec{E}_\mathrm{net}&=\vec{E}_1+\vec{E}_2+...+\vec{E}_n\\[8pt]&=\sum_{i=1}^n\vec{E}_i.\end{align*}\]
\[\begin{align*}|\vec{E}_\mathrm{net}|&=\sqrt{(E_{x_1}+E_{x_2}+...+E_{x_n})^2+(E_{y_1}+E_{y_2}+...+E_{y_n})^2}\\[8pt]&=\sqrt{\left(\sum_{i=1}^nE_{x_i}\right)^2+\left(\sum_{i=1}^nE_{y_i}\right)^2}.\end{align*}\]
Electric Field of Two Opposite Point Charges
A unique example of the vector sum of electric fields is that of two opposite point charges. If two charges are equal in magnitude with opposite charge, they create an electric dipole. The electric field of an electric dipole is called a dipole field. The field lines point from the positive charge to the negative charge, as shown below.
An electric dipole is a pair of equal but opposite charges.
The electric field of an electric dipole is known as an electric dipole field.
What would the magnitude of the electric field be in the center of the dipole? Your initial thought may be that the electric field from the positive and negative charges cancel out, making the magnitude of the electric field zero at this location. However, we must consider the direction of the electric field for each charge. The electric fields for the positive and negative charges both point towards the negative charge, making the magnitude of the electric field twice the magnitude of just one of the charges at this location. The article, "Monopoles and Dipoles" contains an example that finds an expression for the electric field at the center of a dipole.
If two opposite charges are different in magnitude, the resulting electric field will differ from that of an electric dipole. The electric field lines wrap more closely around the weaker charge, as shown in the following image. We can see clearly that field lines are closest together in between the two charges, indicating the greatest field strength there.
A \(4.0\,\mathrm{nC}\) charge is separated by a distance of \(6.0\,\mathrm{cm}\) from a \(-3.0\,\mathrm{nC}\) charge. What is the magnitude of the electric field directly in between the charges (\(3.0\,\mathrm{cm}\) away from each)?
We start by finding the electric field from each charge at that point. Since the electric field points in the horizontal direction at this point, we will just need the \(x\) component in our calculation. As mentioned before, the electric field from both charges points toward the negative charge, so we will solve for the magnitude of the electric field for each charge. The magnitude of the electric field from the positive charge is given by:
\[\begin{align*}|\vec{E}_1|&=E_{x_1}\\[8pt]&=\frac{1}{4\pi\epsilon_0}\frac{|q|}{x_1^2}\\[8pt]&=\frac{1}{4\pi(8.854\times10^{-12}\mathrm{\frac{C^2}{N\,m^2}})}\frac{4.0\,\mathrm{nC}}{(3.0\,\mathrm{cm})^2}\\[8pt]&=\frac{1}{4\pi(8.854\times10^{-12}\mathrm{\frac{C^2}{N\,m^2}})}\frac{4.0\times10^{-9}\,\mathrm{C}}{(3.0\times10^{-2}\,\mathrm{m})^2}\\[8pt]&=4.0\times10^4\,\mathrm{\frac{N}{C}}.\end{align*}\]
Thus the magnitude of the electric field from the positive charge is \(|\vec{E}_1|=4.0\times10^4\,\mathrm{\frac{N}{C}}\) in the positive \(x\)-direction. Similarly, the magnitude of the electric field from the negative charge is given by:
\[\begin{align*}|\vec{E}_2|&=E_{x_2}\\[8pt]&=\frac{1}{4\pi\epsilon_0}\frac{|q|}{x_2^2}\\[8pt]&=\frac{1}{4\pi(8.854\times10^{-12}\mathrm{\frac{C^2}{N\,m^2}})}\frac{3.0\,\mathrm{nC}}{(3.0\,\mathrm{cm})^2}\\[8pt]&=\frac{1}{4\pi(8.854\times10^{-12}\mathrm{\frac{C^2}{N\,m^2}})}\frac{3.0\times10^{-9}\,\mathrm{C}}{(3.0\times10^{-2}\,\mathrm{m})^2}\\[8pt]&=3.0\times10^4\,\mathrm{\frac{N}{C}}.\end{align*}\]
Thus the magnitude of the electric field from the negative charge is \(|\vec{E}_2|=3.0\times10^4\,\mathrm{\frac{N}{C}}\) in the positive \(x\)-direction. Since the electric field components only have the \(x\)-direction, we find the magnitude of the net electric field by taking the sum of the magnitudes:
\[\begin{align*}|\vec{E}_{\mathrm{net}}|&=|\vec{E}_1|+|\vec{E}_2|\\[8pt]&=4.0\times10^4\,\mathrm{\frac{N}{C}}+3.0\times10^4\,\mathrm{\frac{N}{C}}\\[8pt]&=7.0\times10^4\,\mathrm{\frac{N}{C}}.\end{align*}\]
Electric Potential Energy of Multiple Point Charges
When a charge is moving through an electric field, the electric force does work on the charge only if the charge's displacement is in the same direction as the electric field. This means that the work done by the electric force on the charge is path-independent, similar to how the work done by gravity on a falling object is path-independent. Since the work done is path-independent, we say that electric force is a conservative force that gives electric potential energy to the system as it does work on a charge.
The electric potential energy of a charge, \(q_0,\) that is a distance, \(r,\) away from another charge, \(q,\) is given by:
\[U=\frac{1}{4\pi\epsilon_0}\frac{qq_0}{r}.\]
If the charge \(q_0\) moves from one position, \(r_a,\) to a new position, \(r_b,\) the work done would be equivalent to the change in potential energy:
\[\begin{align*}W&=\Delta U\\[8pt]&=\frac{1}{4\pi\epsilon_0}\left(\frac{qq_0}{r_b}-\frac{qq_0}{r_a}\right)\\[8pt]&=\frac{qq_0}{4\pi\epsilon_0}\left(\frac{1}{r_b}-\frac{1}{r_a}\right).\end{align*}\]
The electric potential energy of the charge in the presence of multiple point charges is found by simply taking the sum of the potential energies from each charge:
\[\begin{align*}U&=\frac{q_0}{4\pi\epsilon_0}\left(\frac{q_1}{r_1}+\frac{q_2}{r_2}+...+\frac{q_n}{r_n}\right)\\[8pt]&=\frac{q_0}{4\pi\epsilon_0}\sum_{i=1}^n\frac{q_i}{r_i}.\end{align*}\]
Electric potential energy is usually easier to work with than electric fields because the electric potential energy is a scalar instead of a vector quantity.
Examples of the Electric Field of Multiple Point Charges
Let's do a couple more examples to practice finding the electric field from multiple charges!
Two equal, positive charges of charge \(q\) are separated by a distance, \(d.\) What is the magnitude of the electric field directly in between them?
Since they have the same charge, the magnitude of both charges is equal:
\[|\vec{E}_1|=|\vec{E}_2|\]
The positive charges repel each other, so their electric fields point in opposite directions. Thus, \(\vec{E}_1=-\vec{E}_2.\) The net electric field is then:
\[\begin{align*}\vec{E}_\mathrm{net}&=\vec{E}_1+\vec{E}_2\\[8pt]&=-\vec{E}_2+\vec{E}_2\\[8pt]&=0.\end{align*}\]
Thus, the magnitude of the electric field directly between two equal, positive charges is zero.
Three point charges are positioned as shown in the image below. Two positive charges are on the \(x\)-axis separated by a distance, \(2d,\) and a negative charge is on the \(y\)-axis a distance \(d\) above the \(x\)-axis. The magnitude of each charge is \(q\). Draw a vector to represent the magnitude and direction of the electric field at the origin, and write down its magnitude.
Fig. 7 - Two positive point charges on the \(x\)-axis and one negative charge on the \(y\)-axis.
First, let's draw vectors to represent the electric field from each charge. The vectors will all have the same magnitude. We'll call the electric fields from the positive charges on the left and right, \(\vec{E}_1\) and \(\vec{E}_2,\) respectively. These are shown in blue in the image below. The electric field from the negative charge on top is \(\vec{E}_3,\) which is shown in green.
Fig. 8 - Electric field components of two positive point charges and one negative point charge. All arrows have the same length since every vector has the same magnitude.
When we add these vectors together, \(\vec{E}_1\) and \(\vec{E}_2\) cancel each other out so that there is no horizontal component in the net electric field. That means that \(\vec{E}_3\) is the only component that contributes to the net electric field at the origin. The net electric field vector, \(\vec{E},\) is represented in pink in the image below.
Fig. 9 - Net electric field vector at the origin from three point charges.
The magnitude of the \(\vec{E}\) is then equal to the magnitude of \(\vec{E}_3.\) Thus, substituting in \(d\) for the distance, we can write:
\[\begin{align*}|\vec{E}|&=|\vec{E}_3|\\[8pt]&=\frac{1}{4\pi\epsilon_0}\frac{|q|}{r^2}\\[8pt]&=\frac{1}{4\pi\epsilon_0}\frac{q}{d^2}.\end{align*}\]
Electric Field from Multiple Charges - Key takeaways
References
- Fig. 1 - Hair sticks to balloon (https://commons.wikimedia.org/wiki/File:Attractive-electric-force-between-hair-and-balloon.jpg) by Mike Run (https://commons.wikimedia.org/w/index.php?title=User:MikeRun&action=edit&redlink=1) published by CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/deed.en).
- Fig. 2 - Electric field from positive and negative charges, StudySmarter Originals.
- Fig. 3 - Electric field from four charges, StudySmarter Originals.
- Fig. 4 - Electric field strength, StudySmarter Originals.
- Fig. 5 - Electric field from an electric dipole, StudySmarter Originals.
- Fig. 6 - Electric field from two unequal, opposite charges, StudySmarter Originals.
- Fig. 7 - Three point charges on a coordinate system, StudySmarter Originals.
- Fig. 8 - Electric field components from three point charges, StudySmarter Originals.
- Fig. 9 - Net electric field from three point charges, StudySmarter Originals.
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