Chapter21 "physics"

Electric Charges

he used rock crystals, various precious stones, and semi-precious gemstones. The experiments led to the development of the idea of electric charge, a fundamental property of matter. Moreover, it was understood that the property is of two types, leading to attractive and repulsive forces between different objects, called electric forces. However, objects like metals could not be subject to electric forces. When electric forces did exist between objects, primarily minerals, they could act over a distance without direct contact.

The idea was developed, leading to the description of electric forces between positive and negative charges, which can flow. It is this model that we use today, with minor modifications.

The experiments also quantified that the electric force between electrically charged objects decreases with distance rapidly.

Procedure

If a used comb is brought close to pieces of paper, they are attracted to it. If a balloon is rubbed against a wall, it sticks to it.

These are examples of the phenomenon of static electricity, which has been known to humankind for thousands of years. For example, ancient Greek literature records static electricity experiments on fur and amber.

When a piece of amber is rubbed vigorously with fur, an attractive force develops between the two. If they are then separated, each attracts other objects, like paper.

However, two such pieces of amber repel each other, as do two such pieces of fur.

However, metallic objects do not experience electrical forces.

These observations suggest that the electric property of matter, called the electric charge, comes in two types: positive and negative. If the interacting objects carry the same sign of electric charge, they repel each other. If they carry the opposite sign, they mutually attract.

Electrical forces can act without physical contact between charged objects.

Sources and Properties of Electric Charge

All objects we see around us consist of atoms, which combine to form molecules. The lightest element in the universe is hydrogen, and a hydrogen atom consists of a positively charged proton and a negatively charged electron. The magnitude of charge that a proton and an electron carry are the same, and it is the fundamental unit of charge. In SI units, it is 1.602 times 10^-19 coulomb.

Most atoms additionally constitute another fundamental particle, the neutron. It carries no electrical charge. A neutron and a proton have almost identical masses, which are about 2000 times greater than the mass of an electron.

Heavier atoms, for example, helium, consist of twice the number of protons and neutrons. In all atoms, protons and neutrons are bound together in a nucleus. Hence, most of the mass of an atom is concentrated in the nucleus. The binding force of mutually repelling positively charged protons and electrically neutral neutrons is not electrical. Instead, it is different from gravitational and electrical forces. This force is called the strong nuclear force.

The relative difference in mass of the nucleus and the electrons makes the electrons freer to move. If an electron is removed from an atom, it becomes positively charged, whereas it becomes negatively charged if an electron is added to it. Such atoms are called ions.

It is empirically observed that electric charge can neither be created nor destroyed; it can only be transferred from place to place, from one object to another. Frequently, we speak of two charges "canceling"; this is verbal shorthand. It means that if two objects with equal and opposite charges are physically close to each other, then the equal and opposite forces they apply on any other charged object nullify, resulting in zero net force on the latter. However, the charges on the objects by no means disappear. The net charge of the universe is constant.

The conservation of electric charge is not just a global phenomenon concerning the entire universe but also a local phenomenon. In principle, if a negative charge disappeared from a laboratory and reappeared on the Moon, charge conservation would still hold. However, this never happens. If the total charge on the lab bench is changing, there will be a measurable flow of charge into or out of the system. Again, charges can and do move around, and their effects can and do cancel, but the net charge in the closed, local environment is conserved. This idea is called the law of conservation of charge.

Procedure

All matter is made up of atoms. An atom consists of three subatomic particles: electrons, protons, and neutrons.

The atomic nucleus consists of positively charged protons and electrically neutral neutrons bound by nuclear force.

The SI unit of charge is coulomb. The magnitude of the charge of an electron is the same as that of a proton, which is 1.602 times 10^-19 coulomb.

This magnitude is the fundamental unit of charge, as all other charges in nature are its integral multiples. Thus, charges are quantized.

Since electrons are negatively charged, an atom is electrically neutral when it has an equal number of protons and electrons; most objects around us are electrically neutral.

The total charge in the universe is found to be conserved. It can neither be created nor destroyed.

Moreover, the electric charge is locally conserved. The amount of charge in a region changes only because of the flow of charge into it or away from it.

Conductors and Insulators

Some materials may easily let electrical charges pass through them, while others obstruct their flow. The former are called conductors and the latter insulators. The atomic structures of materials determine whether they are conductors or insulators of electricity.

Most metals are conductors. Their atomic configuration is such that one or more electron(s) are loosely bound to the nucleus in each atom. Thus, a sea of mobile electrons are available in them, known as free electrons. Their easy movement neutralizes any external charge added to the conductor. Hence, metals cannot harbor excess charge and do not experience electrical forces mutually or with other materials.

Human bodies are good conductors of electricity. For example, excess charges accumulate when someone rubs their shoes against a carpet fiber or an insulator. If they then touch a charged conducting material like a doorknob, they receive an electrical shock because of the rapid flow of charges.

Unlike conductors, insulators have atomic structures that do not allow any electrons to move between atoms freely. Thus, any excess charge added to insulators remains in the material. These excess charges can then lead to electrical forces between insulators. Plastic, wood, glass, and fur are typical examples of insulators.

Procedure

Atoms consist of a positively charged nucleus surrounded by electrons, which form a cloud. Attractive electrical forces bind the nucleus and the electrons together.

In some materials, electrons are loosely bound to the nucleus. For example, a copper atom consists of 29 protons and 29 electrons. However, one electron is loosely bound to the nucleus and is free to move.

The total number of free electrons equals the total number of atoms in the wire, which can move around the entire wire. Thus, when an external source of charge comes into contact with copper, electrons quickly move through the wire. Most metals, like iron, gold, and silver, conduct electricity, and are called conductors.

However, materials like wood, glass, paper, and silk do not have free electrons. Their atomic structure is such that all the electrons are firmly bound to the nucleus. Thus, they are insulated from conducting charges and are called insulators.

Hence, whenever they are charged by rubbing against each other, the excess charges remain on them, leading to electrical forces.

Charging Conductors By Induction

The Earth is a good conductor of electricity, and it is so big that it can be considered an infinite source or sink of charges. It can easily exchange charges with any matter.

Generally, conductors like metals do not allow any excess charge to be present on them. Any excess charge added to metals easily flows away, for example, when a metal is placed on the Earth. This process is called earthing.

However, conductors can be charged by a process called induction. For example, consider charging a glass rod, an insulator, by rubbing it against silk. The positively charged glass rod is brought close to the electrically neutral metallic sphere. Free electrons in the metal are attracted towards it and accumulate at the near end. As more accumulate, others are repelled by the accumulated electrons, finally creating a distribution where no electron feels attraction or repulsion.

The overall charge distribution is such that there are more negatively charged free electrons at the end close to the insulating rod and more positively charged atomic nuclei at the other end. Although the sphere has no net charge, the distribution of charges is such that it is now attracted to the insulating rod. It is said to be polarized, and the charge distribution is called an electric dipole.

If the sphere is then grounded, excess free electrons from the Earth flow to the positive end. If the grounding is removed, the negatively charged distribution close to the glass rod leaves the metallic sphere carrying a net negative charge. This process of charge is called induction.

Procedure

Rub a plastic rod against fur to charge it negatively.

Take a conducting metallic sphere supported by an insulating plastic stand.

When the charged rod is brought close to the neutral sphere, free electrons inside the sphere are repelled away from it.

As more free electrons try to move, they are also repelled by those already moved. The electrons distribute such that there is no net force on any free electron.

More free electrons have accumulated on the far end of the conductor, leaving the closer end carrying a net positive charge. The two poles are oppositely charged or polarized. Such a charge distribution is called a dipole.

If the sphere is connected to the Earth via another metallic wire, the free electrons move to the Earth. When the wire is removed, the sphere carries a net positive charge. This process is called induction.

The negatively charged plastic rod does not lose or gain any charge. Induction results in a charge exchange between the conductor and the Earth.

Coulomb's Law

Experiments with electric charges have shown that if two objects each have an electric charge, they exert an electric force on each other. The magnitude of the force is linearly proportional to the net charge on each object and inversely proportional to the square of the distance between them. The direction of the force vector is along the imaginary line joining the two objects and is dictated by the signs of the charges involved.

Newton's third law applies to the Coulomb force — the force on each charge is equal in magnitude and opposite in direction to the force experienced by the other.

Interestingly, the Coulomb force does not depend on the mass of the objects. It is quantitatively similar to the gravitational force, the difference being that the latter is always attractive.

It is important to note that the electric force is not constant; it is a function of the separation distance between the two charges. If either the test charge or the source charge (or both) move, the separating distance changes; hence, the force changes. An immediate consequence is that the direct application of Newton’s laws with this force can be mathematically tricky. It can usually be done, but more straightforward methods of calculating whatever physical quantity we are interested in are preferred.

The new constant in Coulomb's law is called the permittivity of free space or the permittivity of vacuum. It has a significant physical meaning, related to the speed of light in vacuum.

Procedure

Electrical experiments lead to a mathematical law that quantifies observations. Coulomb's law formulates the force of attraction or repulsion between two point charges.

Consider two electrically charged point masses, with charges q-1 and q-2. They experience the same magnitude of force, called the Coulomb force. It is directly proportional to the product q-1-q-2 and inversely proportional to the square of the distance between them. It acts along the imaginary line joining them.

In the SI units, the proportionality constant is approximately 8.988 times 10⁹. For theoretical reasons, it is described via another constant, epsilon-naught, known as the permittivity of vacuum. Its value is 8.854 times 10^-12 in SI.

If both the charges are positive, or both are negative, they experience equal and opposite force away from each other. If one charge is positive and the other is negative, the force is attractive and equal.

The inverse square nature of the force implies that it is effective only at small distances. Friction is an example of a Coulomb force.

Coulomb's Law and The Principle of Superposition

Coulomb's Law describes the force experienced by two point charges under each other's presence. But what if there are more than two charges? For example, if there is a third charge, does it experience a force that is a simple combination of the individual forces due to the first two charges? Can it be described mathematically?

The Principle of Superposition answers the question. Yes, Coulomb's Law applies to each pair of charges, and the net force on each charge is the vector sum of the individual forces on it due to all the other charges. Hence, the principle of superposition formulates a single vector describing the force experienced by a test charge.

Note that the force need not have been such a simple function of the two forces but could be a more complicated function. Thankfully, nature follows this simple principle.

However, note that the principle also applies to all the other charges. Hence, each pair of forces causes the others to accelerate. Thus, unless they are held together in their positions by external forces, each starts moving, thereby changing the individual forces, that changes the net force on each charge, which in turn changes its acceleration. As a result, the mathematical problem is a difficult one.

Procedure

Coulomb's law describes the force between two point charges, q-1 and q-2. What is the force experienced by a third point charge, Q?

q-1 and q-2 are called source charges, and Q is called the test charge. Let Q experience force F-1 due to q-1, and F-2 due to q-2.

Experiments show that the net force experienced by the test charge is the vector sum of F-1 and F-2. This is called the principle of superposition.

If there are more than two source charges, the force on the test charge can be calculated similarly.

Imagine the test charge is an electron at the origin; q-1 equals +8-e, placed on the x-axis at a distance 2-d from the origin, and q-2 equals +18-e, placed on the y-axis at 3-d from the origin. Then, F-1 and F-2 turn out to be equal in magnitude, directed along the x and y axes, respectively.

Hence, the net force on Q is about 1.4 times this magnitude and is directed at 45 degrees to the x-axis.

Electric Field

Consider two point charges, each exerting Coulomb force on the other. It is possible to describe the Coulomb interaction via an intermediate step by defining a new physical quantity called the electric field.

In the new picture, imagine that the first charge sets up an electric field independent of all other charges in the universe. When another charge comes in its vicinity, the second charge experiences an electric force depending on the electric field at that point. The source charge does not exert any force on itself; however, all the other charges exert force on it.

The SI unit of the electric field is newton per coulomb. Since the electric field is a vector quantity, it is called a vector field. Since the Coulomb interaction follows the principle of superposition, by definition, so does the electric field.

It is to be noted that the electric field is defined by considering the test charge as a point charge. If the charge had a significant spatial extent, its different parts would experience different forces. In reality, since charges are quantized, and the fundamental unit is the charge of an electron or a proton, which are practically point charges, this is a good approximation.

The electric field is defined along the direction of the Coulomb force a positive charge would experience. Since a positive source charge would repel this positive test charge, the electric field of the positive point charge is directed away from it. On the other hand, the positive test charge would be attracted by a negative source charge; hence the electric field of the negative point charge is directed into it.

Procedure

Consider a test charge in the vicinity of many source charges: q-i, where i ranges from 1 to N. The principle of superposition of Coulomb forces implies that the force on Q is the vector sum of the forces due to each q. This expression can be rearranged to express the force as Q times a vector quantity, called the electric field.

If the test charge were different, the electric field would not change. Thus, it helps formulate the effect of source charges independent of the test charge.

By definition, it follows the principle of superposition. At any point, the electric fields of the source charges are vectorially added to give the net electric field.

The direction of the electric field is chosen to be the direction of the force that a positive test charge would experience. Hence, a negative test charge would experience a force opposite to the field’s direction.

Thus, the electric field of a positive point charge is directed away from it, and a negative point charge is directed towards it.

Electric Field of Two Equal and Opposite Charges

Atoms generally contain the same number of positively and negatively charged particles, protons, and electrons. Hence, they are electrically neutral. However, the centers of the positive and negative charges do not always coincide. In such a scenario, the electric field of an atom may not be zero.

A separation of the positive and negative charges can lead to a weak, remnant effect of the positive and negative charges. The expectation is that the more the distance between the positive and negative charges, the more the electric field will be. The exact calculation can be facilitated by considering a positive and negative charge of equal magnitude separated by a distance. Such a system is called a dipole.

Symmetry is a powerful tool for calculating electric fields. When a system looks the same under a particular operation on it, it is said to be symmetric. As far as the electric field perpendicular to the separation direction is considered, an equidistant point from the two charges is symmetric under the operation: swapping the two charges. That is, if the two charges were swapped, the electric field in this direction would remain the same.

However, the field is not symmetrical in the direction of joining the two charges. The magnitudes and directions, both being the same, reinforce each other. Thus, at any point along the plane, there is a resulting electric field along the separation direction, directed towards the negative charge. As expected, its magnitude is proportional to the individual charges and the separation distance.

If the point is far removed from both the charges, the magnitude is approximately inversely proportional to the cube of its distance from the center of the charges. Thus, the dependence is stronger than the magnitude of the field due to individual charges. That is, it is a weaker field. That is why electrical forces appear weaker, although comparatively stronger than gravitational forces. They fall off sharply with the distance between objects.

Procedure

Consider a pair of source charges, one negative and the other positive, fixed along the x-axis and separated by a distance d. Let their midpoint be the origin. Consider a point P at a distance r from the origin.

The electric fields at P are directed toward minus-q and away from plus-q. Resolving the electric field along the x and y directions reveals their y-components have the same magnitude but opposite directions because P is equidistant from the two charges. Thus, the system possesses symmetry of the charges.

In the x-direction, the fields add up, resulting in twice the field due to each charge. Hence, the field is directed toward the negative charge.

At large distances compared to the separation, it is proportional to q-d and inversely proportional to the distance cubed, which is weaker than the individual fields.

Some molecules, like water, contain positive and negative charges separated by a distance. This expression helps calculate their fields.

At very large distances, it is zero, implying the effect of the two charges is canceled.

Continuous Charge Distributions

Imagine a bucket of water. It contains many molecules, of the order of 10²⁶ molecules. Thus, although it contains discrete elements (molecules) at the microscopic level, macroscopically, it can be considered continuous. Small volume elements of water, infinitesimal compared to the bulk of the bucket's volume, still contain many molecules. Under this framework, quantized matter is approximated as continuous for practical purposes.

The electric charge can also be subjected to an analogical treatment. Charges are indeed quantized, and electrons and protons carry the fundamental unit of charge. But macroscopic objects contain many molecules, each containing protons and electrons. Hence, the total charge of a system can be considered a continuous charge distribution while keeping in mind that it's a suitable approximation and not the actual reality.

This kind of approximation lends the consideration of line charges, surface charges, and volume charges. For example, a charged rod can be expressed via its line charge density. Although the other two dimensions, breadth and height, are very much present, they can be ignored if there is no reason to believe that there is a significant gradient of charge along these two dimensions. Thankfully, nature follows the principle of superposition for Coulomb's law, and hence, for the electric field. Each line element of charge can then be thought to be creating its unique field, and the electric fields of all the line elements can be vectorially summed to calculate the rod's total electric field. Instead of a summation, the expression is an integral.

Similarly, for a surface charge distribution, for example, a plane or the outer surface of a spherical conductor, the description is via the surface charge density or the charge per unit surface area. The principle of superposition ensures that its total electric field is then given by a surface integral, that is, an integral over the coordinates describing this surface.

Similarly, if a particular charged body contains charge in bulk, for example, a charged insulating sphere, it is described by a volume charge density. The integral is over the coordinates describing its volume.

Procedure

While studying the motion of water through a pipe, infinitesimal volume elements of water are considered. These elements, although small compared to the total volume of the water, contain many molecules. This large number makes it possible to consider their collection a continuous element.

Similarly, although the charge is quantized, a part of a system's total charge can be considered a continuous element. It contains many individual charges but is small enough compared to the total number of charges in the system. Such an approximation is called a continuous charge distribution.

For example, for a charged metallic rod, the charge per unit line element determines the electric field. The principle of superposition gives the rod's electric field as a line integral over its length.

When a plane is charged, the amount of charge per unit surface area determines its field, a surface integral over its entire surface.

When a volume of charge is studied, the charge density per unit volume determines the field, a volume integral over the entire volume.

Electric Fields

Source: Yong P. Chen, PhD, Department of Physics & Astronomy, College of Science, Purdue University, West Lafayette, IN

An electric field is generated by a charged object (referred to as the source charge) in the space around it, and represents the ability to exert electric force on another charged object (referred to as the test charge). Represented by a vector at any given point in the space, the electric field is the electrical force per unit test charge placed at that point (the force on an arbitrary charge would be the charge times the electric field). The electric field is fundamental to electricity and effects of charges, and it is also closely related to other important quantities such as electrical voltage.

This experiment will use electrified powders in an oil that line up with electric fields produced by charged electrodes to visualize the electric field lines. This experiment will also demonstrate how an electric field can induce charges and how charges respond to the electric field by observing the effect of a charged rod on a nearby soda can.

Principles

 A charged object produces an electric field in the surrounding space. For example, according to the Gauss's law, a point charge Q located at the origin produces an electric field:

(Equation 1)

at any point in the space with a distance r from the charge (at origin r = 0), and the direction of the electric field is along the radial direction (away from the charge if Q is positive, and towards the charge if Q is negative). A collection of charges would produce a total electric field according to the superposition principle, namely the total electric field is the vector sum of the electric fields produced by individual charges. For a uniformly charged sphere with total charge Q, the electric field produced outside the sphere is the same as the electric field (given by Equation 1) due to a point-like charge Q located at the center of the sphere, whereas the electric field inside the sphere would be zero.

If one follows the local direction of the electric field to trace out the vector field lines, these lines (whose tangent reflects the local direction of the electric field, and the density of the lines reflects the strength of the local electric field) are known as "electric field lines". They are fictitious lines that help visualize the distribution and direction of electric fields.

An electric field is closely related to electric potential. An electric field would produce a potential drop (or "voltage drop") along the direction of the field. Conversely, a convenient way to generate an electric field is to apply a potential difference. For example, if two different voltages are applied on two separated conductors (or a nonzero voltage applied on a conductor, while keeping another conductor "grounded" at zero voltage), then an electric field in the space between the two conductors pointing in the direction from the higher voltage conductor to the lower voltage conductor is generated.

An electric field (E) will exert a force,

on a charge (q). The direction of the force is the same as the electric field for positive q, and opposite to the electric field for negative q. If a conductor (such as a metal) containing mobile charges is placed in an electric field, the electric field will push positive charges "downstream" in the direction of the electric field and pull negative charges (such as electrons) "upstream" opposite to the direction of the electric field, until the charges accumulate at the boundary (surface) of the conductor and cannot move further. This results in a separation of negative and positive charges in the conductor in an electric field, a phenomenon also known as "polarization" by the electric field. Even for insulators where charges are much less mobile than those in a conductor, a partial "polarization" (where the negative and positive charges are slightly displaced) can occur in an electric field. The electric field will try to make the displacement from the negative to the positive charges aligned with the direction of the field. If the electric field is spatially inhomogeneous such that the forces on the separated positive and negative charges do not cancel, a net force will be exerted on a polarized object.

Procedure

1. Visualize Electric Field Lines

1. Obtain an electrostatic generator (such as a handheld Static Genecon or a van der Graff generator), a pair of electrodes arranged in a concentric circle configuration, and a pair of electrodes arranged parallel to each other.

2. Obtain a Petri dish or an observation tank, fill it with oil (such as Castor oil), and add electrified/polarizable powders (such as semolina seeds) in the oil.

3. Load the electrodes with the parallel electrode configuration onto the observation tank holder. Connect the two electrodes to the "−" (ground) and "+" (charged) terminals of the electrostatic generator, respectively, as in Figure 1. The connection can be made by cables with clamps.

Figure 1: Diagram showing the schematics of two copper wires connected to an electric generator, the other ends (dipped into an oil) of the wires are connected to a pair of parallel electrodes.

4. Turn the crank of the generator which will put positive charges on the electrode connected to the "+" terminal. Make at least 5 full turns. Observe the behavior of the powders.

5. Use a cable to directly short the "−"and "+" terminals to neutralize the charges. Disconnect the electrode from the terminals.

6. Next, load the concentric circle electrode configuration onto the holder and connect the electrodes to the terminals of the generator again, as shown in Figure 2. Stir the oil in the dish to randomize the powders.

Figure 2: Diagram showing the schematics of two copper wires connected to an electric generator, the other ends (dipped into an oil) of the wires are connected to a pair of electrodes shaped as an inner ring and an outer ring respectively.

7. Crank up the generator (at least 5 turns) and charge the electrodes, and observe the behavior of the powders in the dish.

2. Effect of Electric Field

1. Obtain an empty soda can and rest it on its side (so it can roll freely) on a table

2. Obtain an acrylic rod; rub it with fur to charge it.

3. Bring the rod close to the empty soda can, and observe the response of the soda can.

4. Tear a small strip of paper and bring it to the charged rod, observe its behavior.

Results

For step 1.4, the powder will start to form line patterns between the electrodes as shown in Figure 3. This is because the powders are polarized and will line up with the electric field. They are also attracted toward where the field is stronger, namely closer to the positive electrode. The powders do not move appreciably because the oil is very viscous. The pattern of the powders visualizes the "electric field lines".

Figure 3: Diagram showing representative line patterns that may be formed by the powder, in the oil, aligning to the electric field produced by the charged electrodes corresponding to Figure 1. The line patterns reflect the electric field lines and visualize the electric field.

For step 1.7, the powder outside the center ring (made by the "+" electrode) forms a radial line pattern, as shown in Figure 4. This indicates that an electric field exists outside the inner ring. However, the powder inside the inner ring appears random and does not form aligned patterns. This reflects the fact that the electric field inside the ring is approximately zero.

Figure 4: Diagram showing representative line patterns that form by the powder in the oil in response to the electric field produced by the charged electrodes corresponding to Figure 2. The line patterns reflect the electric field lines and visualize the electric field. Random distribution (lack of line patterns) of the powder inside the inner ring reflects the lack of alignment or lack of sufficient strength of electric fields there.

For steps 2.3 and 2.4, both the soda can and paper strip will be attracted by and move toward the charged rod. This is because both the soda can and paper strip will be polarized by the electric field, and the electric field is stronger closer to the rod and weaker farther away from the rod. Therefore, the charges pulled by the electric field to be closer to the rod, are pulled by a stronger force compared to those opposite charges pushed away from the rod. This produces a net attractive force toward the rod.

Applications and Summary

In this experiment, we have visualized electric fields using electrified powders in an oil that align with the electric field lines. We also demonstrated the effect of an electric field produced by a charge rod to attract polarizable objects toward the rod, i.e., the source of the electric field where the electric field is stronger.

Electric fields are ubiquitous. There are electric fields whenever there are charges or voltage (electric potential) differences. Electric fields provide the force to push charges (usually electrons) to form electrical current in any circuits. Electric fields are also responsible for the sparks we see and experience in dry climate (typically in winter time). When a certain action (for example, rubbing a sweater when removing it) produces a sufficient amount of charges and thus a sufficiently strong electric field, the field can cause transient electrical conduction in air (also known as "electric breakdown", where the electric field is strong enough to not only polarize the air molecules, but to even rip off electrons from air molecules), and cause sparks.

The author of the experiment acknowledges the assistance of Gary Hudson for material preparation and Chuanhsun Li for demonstrating the steps in the video.

Electric Field Line

The three-dimensional representation of the electric field of a positive point charge requires tracing the electric field vectors, whose lengths decrease as the square of their distance from the charge and which point away from the charge at each point. This vector field is no doubt challenging to visualize. The visualization of electric fields becomes quickly intractable as the number of charges increases.

The solution to this problem is to use electric field lines, which are not vectors but describe a vector field.

They are defined such that the magnitude of the electric field at any point is given by the density of the electric field lines around that point. Since the density varies with distance from the charges and is defined uniquely at each point in space, it uniquely describes the field vector's magnitude. That is, there is a one-to-one correspondence between the magnitude of the electric field and the density of the electric field lines, thus justifying the definition.

If the electric field lines are close together, the field's magnitude is large at that point. The magnitude is small if the field lines are far apart at the cross-section.

The direction of the electric field is also uniquely defined by the electric field lines. At any point, the tangent to the electric field line determines the electric field's direction. Thus, the definition necessitates that electric field lines do not crisscross each other.

It is important to note that a single electric field line does not trace the magnitude of the electric field, nor does it represent a specific value of the field. Instead, the significant physical quantity is the density of the field lines.

Although the direction and relative intensity of the electric field can be deduced from a set of field lines, the lines can also be misleading. For example, the field lines drawn to represent the electric field in a region must, by necessity, be discrete. However, the actual electric field in that region exists at every point in space.

Procedure

Consider the two-dimensional projection of a positive point charge's electric field. The vectors point away, and their magnitudes decrease with distance.

For a dipole, this field distribution becomes challenging to visualize. For a large number of charges, it becomes intractable.

An alternative visualization via electric field lines solves this problem.

They are defined as lines with direction, such that at any point, the tangent to the electric field line gives the direction of the electric field.

The magnitude of the electric field is given by the density of the electric field lines, the number of field lines per unit cross-sectional area perpendicular to the field.

This representation does not require tracing the lengths of the vector arrows but still traces the electric field at any point uniquely.

For example, the electric field lines of a dipole indicate that the field points away from the positive charge and into the negative charge.

If the magnitude of the positive charge is larger than the negative charge, the field lines become denser near the former.

Properties of Electric Field Lines

The definition of electric field lines greatly eases the visualization of electric fields, a vector field, especially in the presence of many charges. The one-to-one correspondence between the electric field and the electric field lines necessitates that the field lines follow some rules.

For one, the electric field of a positive charge must originate from it. That is because its electric field points away from it. Moreover, since the magnitude of the field asymptotes to zero at infinity, the field lines in the presence of a single positive charge must also extend to infinity.

For a negative charge, the field lines are precisely the opposite. Hence, they come in from infinity and culminate on it.

Since the electric field of a point charge is proportional to its magnitude, so is the number of electric field lines in its vicinity.

By definition, the field line density at any point in space is proportional to the electric field at that point. Also, the electric field vector is tangent to the field line at that point. Now, this implies that electric field lines can never cross each other.

Imagine a point where electric field lines cross. That implies there are two directions of the field at that point. This further implies that a test charge placed at that point would experience a net force that has two directions. Since that is impossible, the hypothesis is ruled out.

Procedure

Electric field lines have specific properties.

In the presence of a positive charge, the field lines originate on it and extend to infinity. For a negative charge, they come in from infinity and culminate on it, indicating the force a positive test charge would experience in its vicinity.

Since the field of a charge is directly proportional to its magnitude, the number of field lines is also proportional to it.

The electric field is always tangential to the electric field line.

Field lines can never cross. If they did, it would imply two different directions of the field, which is impossible.

For a pair of positive charges of the same magnitude, the field lines originate from each and extend to infinity. In between, they point opposite to each other and effectively cancel, implying the electric field is small or zero.

In a dipole, the field lines of the negative charge are reversed, thus reinforcing the field lines in that region.

A constant field is represented by straight, parallel, and uniformly spaced field lines.

Electric Dipoles and Dipole Moment

Consider two charges of equal magnitude but opposite signs. If they cannot be separated by an external electric field, the system is called a permanent dipole. For example, the water molecule is a dipole, making it a good solvent.

Theoretically, studying electric dipoles leads to understanding why the resultant electric forces around us are weak. Since electric forces are strong, remnant net charges are rare. Hence, the interaction between dipoles helps us understand electrical interactions in ordinary objects around us.

When a permanent dipole is placed in a uniform electric field, it experiences no net force. However, it orients itself along the field so that the positive charge end points toward it. This interaction between the dipole and the electric field can be scrutinized by calculating the net torque it experiences.

Simple vector algebra leads to an interesting observation. The net torque depends on a cross-product of a new vector, called the electric dipole moment, and the electric field. The dipole moment is proportional to the magnitude of each charge and the separation between them.

If a molecule has a large separation between its positive and negative charge centers, it has a higher dipole moment. If the charge separated is itself high, its dipole moment is larger.

The dipole moment points from the negative charge to the positive charge. In the absence of any other torque, the dipole rotates and aligns with the field. The observation implies that the potential energy is associated with the orientation of the dipole with respect to the external electric field.

Procedure

When two equal but opposite point charges are held together, the pair is called a dipole. If it does not separate due to external forces, it is called a permanent dipole.

In a water molecule, the centers of negative and positive charges are close but do not coincide, making it a permanent dipole.

In a uniform electric field, the charges experience forces that are equal but opposite, resulting in zero net force. However, the dipole experiences a net torque.

Choose any origin, and consider the position vectors of the charges. The torque on each component is obtained.

Upon vector addition, the net torque is given by the cross product of a new vector with the electric field. It is the product of the charges' magnitude and the displacement vector of the positive charge with respect to the negative charge. This is called the electric dipole moment.

The torque rotates the dipole along the field.

If it is anti-parallel, it experiences zero net torque but is in an unstable equilibrium, as slight deviations lead to a torque that rotates it parallel.

Induced Electric Dipoles

A permanent electric dipole orients itself along an external electric field. This rotation can be quantified by defining the potential energy because the external torque does work in rotating it. Then, the potential energy is minimum at the parallel configuration and maximum at the antiparallel configuration. While the former is a stable equilibrium, the latter is an unstable equilibrium.

Since the absolute value of potential energy holds no physical meaning, its zero value can be chosen as per convenience. It is chosen to be in the configuration when the electric dipole is perpendicular to the field.

The polarity of molecules determines whether they are a good solvent. For example, water is a good solvent for common salt because it attracts the positive and negative ions toward the opposite charge centers inside it, thereby breaking apart the sodium chloride crystals.

Not all molecules, however, possess a permanent electric dipole. If the positive and negative charge centers are located at the same point, there is no separation. For example, the molecular structure of carbon dioxide is symmetric in the charge distribution. Other organic compounds, such as methane, are also non-polar.

However, in the presence of an external electric field, the charge centers are separated because the negative charge center shifts toward the electric field, while the positive charge center shifts away from the field. Since this happens for all molecules, the external field induces a polarity across the substance.

It is interesting to note that the induced dipole moment somewhat nullifies the electric field. This scenario arises near the dipole, where its electric field is opposite to the external electric field. This mechanism reduces the electric field inside dielectrics.

Procedure

When a permanent dipole, like a water molecule, is placed in an external electric field, the torque orients it along the field.

Its potential energy change is given by the negative work done to rotate it, which is an integral of the torque and differential angle's dot product. By choosing the potential energy to be zero at the perpendicular orientation, the potential energy at any angle is obtained. It is the negative dot product of its dipole moment and the field, which can be plotted.

Not all molecules are permanent dipoles. In a carbon dioxide molecule, the negative charge center coincides with the positive charge center, resulting in zero dipole moment.

In the presence of an external electric field, the positive charge is repelled away from the field while the negative charge is pulled toward it, resulting in a net dipole moment.

The charge separation induces an electric field. In its vicinity, it is opposite to the external field, while at large distances, it reinforces it.

The net electric field is the vector sum of the two fields.

Key Takeaways on Electric Charges

1. Types of Electric Charges: Electric charge comes in two types: positive and negative. Like charges repel each other, while opposite charges attract.

2. Coulomb's Law: The electric force between two charges is proportional to the product of their charges and inversely proportional to the square of the distance between them. This law helps calculate the force acting between point charges.

3. Electric Field: An electric field surrounds a charged object and represents the force felt by other charges placed in that field. It is defined as the force per unit charge.

4. Conductors and Insulators: Conductors (like metals) allow free movement of charge, while insulators (like wood and plastic) restrict charge movement.

5. Charging by Induction: A conductor can be charged without direct contact by bringing a charged object close, causing a redistribution of charges due to the electric field.

6. Dipoles: A dipole consists of two equal and opposite charges separated by a distance and experiences torque in an electric field. The dipole moment quantifies this separation.

7. Conservation of Charge: Electric charge is conserved; it cannot be created or destroyed, only transferred between objects.

8. Electric Field Lines: The direction of electric field lines represents the direction of the electric field, while their density indicates the strength of that field.