Electric charge and force

Chapter 1 of Electricity, Magnetism and Circuits

= 1. Electric charge and force =

The picture shows a Wimshurst machine, which was invented around 1880. Earlier, in the 18th century, various other electrostatic generators had already been invented, all using friction to generate static charges; the big innovation introduced by Wimshurst was the separation of charges by electrostatic induction instead of friction, which allows it to attain higher charges using purely mechanical means. There are several physical principles involved in a Wimshurst generator, which will be treated in later chapters: Leiden jars, dielectric strength, etc.

The transport of electric charges is present in many phenomena from our daily experience. That phenomenon can be easily studied using a roll of adhesive tape. You will also need two pencils or pens. Cut a strip of tape of approximately 20 cm and stick it to the surface of a table, leaving a small part off the table to allow you to stick one of the pencils and pull off the tape from the table holding it with the pencil and without touching the tape. As you pull off the tape from the table, electric charges are transferred to it and will stay there as long as the tape does not touch other objects.

Do the same thing with a second strip of the same tape, on the same table. Using the same materials, we expect the second strip of tape to have the same type of charge as the first one. Bring the two strips of tape close together, approaching them by the sides of the tapes that have no glue, to prevent them from getting stuck. You should observe the repulsive force that exists among charges of the same type. Bring one of the strips of tape close to other nearby objects (approaching them from the side of the tape without glue); the tape should be attracted to any other objects that have no charge.

Discard the strips of tape already used and prepare two new strips. This time stick one of them to the table and then stick the second one on top of the first one. Pull both strips off the table and at the same time separate them from each other. One of the strips of tape will extract charges from the other, so the two will have opposite charges. You should observe a strong attractive force between the two strips, because they now have charges of different types. Observe also that any of the two strips, independently of its type of charge, should be attracted to objects without charge.

1.1. Atomic structure
All objects are made of atoms. Each atom has a nucleus with two types of particles, protons and neutrons, tightly packed into the very small nucleus. Among protons exist a force which is repulsive and called electric force. Neutrons do not feel any electric forces either among themselves or from the protons.



Around the nucleus there are other particles, the electrons, which are much smaller than protons and neutrons and with a mass approximately 2000 times smaller than them. The distance between the electrons and the nucleus is approximately 100 000 times larger that the size of the nucleus. An electric force, with the same properties and strength of the electric force among protons, is also observed among electrons; it is also a repulsive force. Between an electron and a proton there is also an electric force, which has the same strength as the force between two electrons or two protons, but this force is now attractive. That attractive electric force between the electrons and the nucleus keeps the electrons bound to the atom. The protons are kept bound to the nucleus, in spite of their repulsive electric force, due to another type of nuclear force called strong due to its strength higher than that of the electric force.

Thus, there are two types of electric charges; the charge of a proton and the charge of an electron. Neutrons have no charge at all. The electric force among charges of the same type is repulsive, while the force among different types of charge is attractive. The magnitude of the charge of a proton or an electron is the same, because the forces between two electrons, two protons or an electron and a proton have all the same strength.

If an atom has the same number of protons and electrons it will not produce any resultant force on a distant particle with any type of charge, since the forces produced by the protons and electrons are of the same magnitude but with opposite directions. We thus say that the atom is neutral, as if it did not have any particles with charge. Therefore, it has been found convenient to treat the two types of charges as positive and negative and to interpret the neutrality of an atom with the same number of electrons and protons as a consequence of the sum of the charges of its electrons and protons being equal to zero. Historically, the proton has been assumed to have positive charge and the electron negative.

The SI unit used to measure electric charge has been dubbed coulomb, denoted by a capital C. The charge of any proton, called elementary charge, has been measured to be:


 * $$e = 1.602\times 10^{-19} \;\mathrm{C}$$

electrons have a charge $-e$ with the same absolute value but opposite sign.

1.2. Charging by rubbing
A high amount of energy would be required to extract a proton or neutron out of a nucleus. That would only happen inside stars or in particle accelerators where the high-energy conditions inside a star can be reproduced. However, removing one electron from a neutral atom takes a much lower amount of energy; the neutral atom would become a positive ion, with a net charge equal to $e$. About the same energy would be needed to insert an extra electron into that neutral atom, leaving a negative ion with a net charge $-e$.

The outermost electron in atoms or molecules of different materials have different binding energies, which means that it is easier to extract or insert an electron in some materials than in others. When two different materials are in close contact, electrons will pass from the material where they have a lower binding energy to the other; the electric potential energy they gain in the provides the mechanical energy it takes for their transfer.

In the experiments with adhesive tape described in the previous section, the tape's glue and the table's surface are very close to each other so electrons will pass from one to the other. If they were both neutral initially, after their separation one of them will have a negative charge (additional electrons) and the other a positive charge with the same absolute value (less electrons).

Different materials can be ordered into a triboelectric sequence (see table), in which the materials at the top of the sequence loose their electrons more easily.



When two objects are rubbed, the atoms or molecules in their surface will come into close contact and electrons will pass from the one which is higher in the triboelectric sequence into the other that is lower. For instance, if we rubbed a glass rod with a silk rag electrons would pass from the rod to the rag; the rod would become positively charged and the rag negatively charged. However, if we rubbed a rabbit with the same glass rod, electrons would pass from the rabbit's fur into the rod; the rod would now become negatively charged while the rabbit fur would be positively charged.

1.3. Properties of the electric charge
Electric charge, as well as mass, is an intrinsic property of matter. One difference between mass and charge is that there are two different types of charge, while there is only one kind of mass. There are also particles that have no mass and particles that have no charge. Two very important properties of the electric charge are its quantization and conservation.

Charge quantization. In high-energy collisions among particles, such as the ones among cosmic rays with atoms in the upper atmosphere or among particles in accelerators, several new particles are produced different from the proton, neutron and electron. All elementary particles ever observed have charges that are integer multiples of the elementary charge $e$ (1.602×10-19 C). Thus, the charge of any object in nature is believed to be an integer multiple (either positive, negative or zero) of the elementary charge.

In electrostatic experiments, such as the one described with adhesive tape, the charges we deal with are usually equivalent to a very large number of elementary charges. Therefore, in those cases we would not introduce measurable errors by assuming that charge can change continuously, rather than by discrete amounts.

Charge conservation. In any observed physical process, the total initial and final charges are the same. In the cases where we are only dealing with a transfer of electrons from one atom to another, that is trivially true. However, in the high-energy collisions where some particles are destroyed and new ones are created, it is remarkable that the total charge is always conserved. If new particles with positive charge are produced, other particles with negative charge must also be produced.

1.4. Force among point charges
In the 18th century, Benjamin Franklin found out that a charged metallic cup, insulated on its base so it would not loose its charge, would not produce any force in small objects placed inside the cup, while the same objects would feel a force when placed outside and near the cup.

In the previous century, Isaac Newton had provided a mathematical proof that the net gravitational force produced by an spherical, uniform, hollow sphere would vanish at any point inside it. That result is a consequence of the dependence of the gravitational force on the inverse of the square of the distance.

Franklin then interpreted the result of his experiment as an experimental proof that the electric force is also proportional to the inverse of the square of the distance between the objects. The magnitude of the gravitational and electric forces are given by similar mathematical expressions, but the electric force can be either attractive or repulsive:


 * The force between two charges of the same sign is repulsive.
 * The force between two charges of opposite signs is attractive.

Several years after Franklin's experiments, Charles Coulomb conducted more precise experiments to measure the electrostatic forces between two point charges (a point charge is a distribution of charges in a small region).




 * $$ \boxed{F = \dfrac{k|q_1||q_2|}{K\;r^2}}$$

where $r$ is the distance between the charges, $q_1$ and $q_2$ are the values of the two charges, $k$ is a constant called Coulomb's constant and $K$ is the dielectric constant of the medium between the two charges. In vacuum the dielectric constant is 1 and the dielectric constant of dry air is very close to that value; thus, if the medium between the charges is dry air, we can safely ignore the dielectric constant $K$. In SI units, the measured value of Coulomb's constant is.


 * $$k = 9.0\times 10^9\ \mathrm{\frac{N\cdot m^2}{C^2}}$$

Various gases and liquids have different dielectric constants bigger than 1; therefore, the electric force between two charges inside a gas or a liquid will be smaller than in vacuum.

Example 1.1


Three point charges are held together by strings forming a rectangular triangle as shown in the figure. (a) Find the tension in the string that links the 7.4 nC and 9.3 nC charges. (b) If the 5.6 nC charge were removed, would the tension found in the previous line increase or decrease?

Solution. (a) The diagram with the forces acting on the 7.4 nC charge particle (dubbed as particle number 3) is as follows:


 * [[Image:cargas_3fios_forcas.png|none]]

where $\vec{F}_{13}$ and $\vec{F}_{23}$ are the electrostatic forces produced by particles 1 and 2, with charges of 9.3 nC and 5.6 nC, and $\vec{T}_{13}$ and $\vec{T}_{23}$ are the tensions in the two strings that link particle 3 to those two charges. For particle 3 to remain in equilibrium, the necessary conditions are:


 * $$F_{13} = T_{13} \qquad\qquad F_{23} = T_{23}$$

Before making any calculations, it is convenient to convert the value of constant $k$ into the same system of units used in the problem data (nC and cm):


 * $$k = 9\times 10^9\ \mathrm{\frac{N\cdot m^2}{C^2}}

= 9\times 10^9\; \frac{10^6\;\mu\mathrm{N}\times 10^4\;\mathrm{cm}^2}{10^{18}\;\mathrm{nC}^2} = 90\ \mathrm{\frac{\mu N\cdot cm^2}{nC^2}}$$

Thus, assuming that the particles are surrounded by air, the tension in the string linking particles 1 and 3 will be:


 * $$T_{13} = F_{13} = \frac{k \;|q_1|\,| q_3|}{r^2} = \frac{90\times 7.4\times 9.3}{1^2 + 1.5^2}\ \mu\mathrm{N} = 1.9\ \mathrm{mN}$$

(b) The tension would not change since, as we showed in the previous line, $T_{13}$ does not depend on the force $F_{23}$ produce by the particle with charge of 5.6 nC.

1.5. Electric field
A different way of explaining the electrostatic force among two charged particles consists of assuming that each electric charge creates a field of forces around it, which will act upon other particles with charge. If we place a particle with charge $q_0$ at a point where there exists an electric field, the result will be an electric force $\vec{F}$; the electric field $\vec{E}$ is defined as the force by unit charge:


 * $$\boxed{\vec{E} = \dfrac{\vec{F}}{q_0}}$$

Therefore, the electric field at a point is a vector pointing in the direction and orientation of the force that a positive unit charge would feel, when place at that point.

In the other way around, if we determine that at a given point there is an electric field $\vec{E}$, we can compute teh electric force acting on a particle with charge $q$ placed at that point: the force will be $\vec{F} = q\,\vec{E}$. We just need to know the field, in order to calculate the force; we do not need to know which charges gave rise to that field. In the SI unit system, the electric field has units of newton over coulomb (N/C).

As we saw, the electric force produced by a positive point charge $Q$ over a second positive, test-charge $q_0$ is always repulsive, and with a modulus that decreases as the square of the distance. Hence, the electric field produced by a positive point-charge $Q$ are vectors pointing away from the charge, as shown in the left-hand side of the figure.



A more convenient way to represent that vector field consists of drawing some field lines, as it was done on the right-hand side of the figure. At each point, the field line that goes through that point will point in the direction of the field. The value of the field will be higher in the regions where the field lines come closer together.

To compute the value of the electric field created by a point charge $Q$ at a given point, a test charge $q_0$ is placed at that point and the force exerted on that charge is divided by the value of the charge, $q_0$. Thus, using Coulomb's law we obtain the expression for the radial component of the electric field due to the point charge $Q$:


 * $$E = \dfrac{k\,|Q|}{r^2}$$

where $r$ is the distance from the charge $Q$ which creates the field, until the point at which the field is being calculated. The sign of the charge $Q$ will indicate whether the field is repulsive ($Q>0$) or attractive ($Q<0$).



The electric field created by a single point charge is usually too weak to be measured. The electric fields we observe in nature are the sum of the fields created by many point charges; the individual vector fields must be added to get the total field. In chapter 6 we will come back to the problem of computing the field due to a system of many point charges; for the time being, will be more concerned with the effects of the resulting field.

The field lines for the field due to a system of many point charges will not necessarily be straight lines, as the the figures above; we can have curved field lines as in the figure on the side.

Example 1.2
The electric force on a test point charge of 5 nC, at a given point is 2×10-4 N ''in the positive direction along the x axis. Find the electric field at that point. What would be the force on an electron placed at the same point?''

Solution. From the force we obtain the electric field:


 * $$\vec{E} = \frac{\vec{F}}{q_0} = \frac{2\times 10^{-4}}{5}\,\vec{e}_x\ \left(\mathrm{\frac{N}{nC}}\right) =4\times 10^4 \;\vec{e}_x\ \left(\mathrm{\frac{N}{C}}\right)$$

The electric force on an electron at the same point would be:


 * $$\vec{F} = -e\,\vec{E} = -1.60\times 10^{-19} \times

4\times 10^4\;\vec{e}_x = -6.4 \times 10^{-15}\;\vec{e}_x \;(\mathrm{N})$$

1.6. Conductors and insulators
In some materials, such as metals, the outermost electron on each atom is free to move throughout the material; thus, there is a very dense "cloud" of electrons (conducting electrons), which can move inside the object. That kind of materials are called conductors.

If a conductor is placed in a region where there is an electric field, since the electronic cloud has negative charge, it will move in the opposite direction of the electric field. That motion will give rise to an excess of negative charge in one end of the object and an excess of positive charge (lack of electrons) in the opposite end. If the net charge of the conductor is zero, the absolute value of the charges at the two ends will be the same. Those charges of opposite signs in the two ends of the conductor will give rise to an internal electric field in the opposite direction of the external field; when enough charges have been displaced, the internal field will be strong enough to cancel the external field and the motion of the electronic cloud will stop.

A material which is not a conductor is called an insulator; inside an insulator the electric charges cannot move from one end to the other. The figure shows a positively-charged bar placed near a conducting sphere mounted on an insulating support. The electronic cloud moves in the direction of the bar, leaving positive charges in the region farther away from the bar and the same amount of negative charge in the region closer to the bar. If the support was not an insulator, the positive charges would not remain in the sphere but they would move as far away from the bar as they could go through the support.

If the bar had negative charge, instead of positive, the position of the positive and negative charges in the sphere would be the opposite. Once those charges of opposite signs are accumulated on the surface of the sphere, the total electric field inside the sphere will become zero; hence, the atoms inside the sphere will not feel any effect from the charged bar. In both cases (positively or negatively charged bar), the charges on the surface of the sphere closer to the bar will be attracted to the bar and that attraction will be stronger than the repulsion felt by the other type of charges on the surface farther away from the bar; therefore, an external charged object exerts an attractive force on an conductor, without net charge, placed on an insulating support.

If we did the same experiment with an insulating sphere, there would not be any accumulation of charges on the surface, the field inside the sphere would not become zero and thus, each atom inside the sphere would feel the effect of the electric field due to the charged-bar. As a result, each individual atom will be polarized, namely, its own electron cloud will be displaced in the opposite direction of the field. In this case (positively-charged bar), the electron cloud of the atoms will no longer be centered on the nucleus but in a point closer to the bar; each atom will become a small electric dipole, namely, a system with negative and positive charges of the same value but at different points.

The figure shows some of those dipoles inside the insulating sphere. If we used bars with charges of different signs, the side of the dipoles closer to the bar will always be the ones with charge of opposite sign to that of the bar charge. As a result, the attractive force acting on one end of the dipole will be stronger than the repulsive force acting on the other end and the net force will be attractive. The attractive forces acting in all atoms add up together producing an attractive force on the sphere. Hence an insulator is always attracted towards external charged objects, independently of the sign of the charge in that object.

1.7. Charging by induction
A method which can be used to charge two conductors, leaving them with equals amounts of opposite-sign charges, is the induction method shown in the figure.



The two conductors, mounted on insulated supports, are placed in contact with each other. An external charged object is then brought close to one of the conductors, as shown in the figure above. The electric filed produced by that charged object will polarize the charge in the two conductors, leaving charge of the opposite sign to the external charge closer to the external object and charge of the same sign as the external object in the conductor that is farther apart from the external object. The two conductors are then separated, keeping the external object near one of them. Finally, the external charged object can be removed and since the charges polarized in the two conductors can no longer recombine, both conductors will keep their charges with opposite signs.



The Wimshurst electrostatic generator uses induction to generate charges. The two conductors which are put into contact are two small metallic sheets glued to a insulator disk, on the same diameter of the disk, where there is a metal bar with metal brushes that touch the two sheets. As the disk is rotating, each pair of opposite sheets will remain in contact only for a small time.

It will be enough to have a small charge in any of the metal sheets, to induced charges of opposite signs in two other sheets on the other disk (the generator has two disks rotating in opposite directions). Those two sheets will then induce charges in four sheets on the first disk and so on.

There are two collectors placed near the disk, in such a way that after a charge is induced on a metal sheet, that sheet will rotate to the region where the brushes of the opposite disk are placed, those inducing charges on the sheets on that other disk, and then passes near the collector where it will loose its charge that will go into a Leyden jar. When the charges stored in the two Leyden jars are big enough, they will produce a spark in the generator and the jars will discharge.

1.8. Questions
To check your answer, click on the letter.


 * 1) A bar with positive charge is placed near a sheet of paper without any charge. Thus, the force on the paper will be:
 * A . Attractive.
 * B . Repulsive.
 * C . Zero.
 * D . Dependent on whether the bar is made of a conductor or an insulator.
 * E . Attractive if the paper is dry or zero if the paper is wet.

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