Electrostatics Study Guide (page 2)
Analogy is a powerful tool, and we use it often in physics. Remember, we started this journey with the static of mechanical objects. Now we move further from macroscopic to microscopic and analyze interaction between charges and the effect on the surrounding medium.
Electric Charge and Charge Conservation
A basic description of substances leads us to the concepts of electrical charges and the fundamental particle—the electron. Empirically, it has been found that a piece of amber gemstone, rubbed with a piece of animal fur, will create a new type of interaction; it will be able to attract to its surface small pieces of material and dust particles. The material is said to be electrically charged, and the smallest particle of charge is called the electron. The electron was assigned a negative charge of 1.6 · 10–19 Coulombs. These electrons are considered one of the elementary subatomic particles. Electrons (negatively charged) are constituents of atoms, which are electrically neutral (zero net charge). The atom also contains positive charges, or protons, which are positioned in the central part of the atom, the nucleus. The nucleus is also composed of neutral particles, called neutrons.
Contrary to the previous example where the amber was charged up with electrons, other materials tend to charge positively (with protons). This has been summarized in the electrostatic series. Materials such as amber, rubber, and polyethylene charge negatively, whereas paper, cat's fur, and nylon charge positively. The difference between the types of charge can be easily shown by letting two of these materials interact. Materials from the same group repel each other, and materials from the opposite group attract each other. This defines the electrical interaction.
Materials are considered to be conductors, semiconductors, or insulators. The nomenclature is revealing. Conductors will let charge spread in the volume. Semiconductors will vary their electrical properties and sometimes behave as conductors while at other times behaving as insulators. Insulators will localize the charge acquired; their electrons bond strongly to the atoms and make it hard for electrical charge to flow through the material.
A last observation. Charging an object can be achieved by different means: friction, conduction, or induction. Charging by friction involves rubbing two materials against each other and in the process electrons are transferred from one object to the other. Conduction involves a conductive material in which electrons are free to move in the volume. Induction is established between two materials that do not touch, but where the material 'that is charged interacts with particles in the uncharged material, creating a distribution of charge on the surface close to it.
In the neutral state, the atom has an equal number of electrons and protons, and the net charge is zero.
n = p
where n is the number of electrons and p is the number of protons. If an electron is given sufficient energy to be able to break the bond with the atom and leave it, then the atom becomes charged, as it now has one more proton than electrons. The atom becomes positively charged, and we call this new particle a positive ion (n < p). If by friction between two objects, one of the objects is charging positively (losing electrons), we typically retrieve the electrons on the second object. Then the second object, initially neutral, becomes negatively charged. This ion is called a negative ion (n > p). If the two objects are isolated from the exterior (that is, isolated from other objects), then the charges are moving from one to the other, and the total number of electrons and protons will stay the same; the ions simply rearrange between the two objects. We can define this as a new law of conservation, similar to the conservation of energy and conservation of momentum in mechanics and to the conservation of thermal energy in thermodynamics. This new law is the conservation of charge.
After the charge is rearranged between the objects in contact, the state remains unchanged as long as there is no other interaction. This defines a situation of equilibrium similar to previous cases of mechanical and thermal equilibrium where the charge, whether positive or negative, is measured in Coulombs (1 C), and its usual symbol is q. As mentioned previously, an electron has negative charge:
qe = –1.6 · 10–19 C
The proton is equal in charge but has more mass:
qp = +1.6 · 10–19 C
Consider two objects, one made of rubber and the other of nylon that is electrically neutral. You rub the two objects together and the rubber becomes charged with –2.88 · 10–16 Coulombs. What is the nylon's charge, and how many electrons have been shifted to the rubber from the nylon in the process?
Initially, both objects have a zero net charge, and, because the system will be considered to be isolated, the same amount of charge will be retrieved in the final state.
qrubber = –2.88 · 10–16 C
qnylon = ?
qrubber + qnylon = 0 C
–2.88 · 10–16 C + qnylon = 0 C
qnylon = – (–2.88 · 10–16 C)
qnylon = + 2.88 · 10–16 C
The number of electrons that make up the charge of the rubber is:
N = qrubber/qe
N = (–2.88 · 10–16 C)/(–1.6 · 10–19 C) = 1,800 electrons pass from nylon to rubber.
Electric Forces and Coulomb's Law
We have talked about interaction between materials in the electrostatic series. We can summarize that there are two types of electrical interactions: attraction and repulsion. Attraction forces are established between particles of different charge sign (positive and negative charges). Repulsion forces act between charges of the same kind.
The quantitative expression of the electrical force is known as Coulomb's law (from Charles de Coulomb, 1736–1806). If one considers the charges to be pointlike (that is, all charges gather in one point), then the law says: The magnitude of the electrical force exerted by a point-like charge q1 on a charge q2 is proportional to the product of the charges and inversely proportional to the square of the distance between the charges.
Where the proportionality constant is:
k = 9 · 109N · m2/C2
The electric force, as are all other forces, is a vector. The direction is given by the type of particles interacting, as shown in Figures 13.2 and 13.3 and defined previously.
Although the names given to the forces are different to show the action charge and the test-charge (for example, F21 indicates that charge 2 is acting on charge 1), the absolute value is the same.
F21 = F12
Also shown in the figures is the opposite direction of interaction.
In a vector form, the law is expressed as:
One can see that the new formula shows that the direction of the electric force is the same as the direction between the two charges.
The magnitude of the electrical force exerted by a point-like charge q1 on a charge q2 is proportional to the product of the charges and inversely proportional to the square of the distance between the charges.
Two amber beads are brought in proximity after being charged with 10 and 50 electrons. They are placed at a distance of 20 cm from each other. Find the force acting on each bead.
First, convert to the units necessary to work with Coulomb's law and then find the charges. Then, calculate the forces on each object.
R = 20cm = 0.2m
q1 = 10 · (–1.6 · 10–19 C) = –1.6 · 10–18 C
q2 = 50 · (–1.6 · 10–19 C) = –8.0 · 10–18 C
F21 = F12 = ?
F21 = = 9 · 10–9 N · m2/C2 ·
F21 = 9 · 10–9 N · m2/C2 ·
F21 = 28.8 · 10–23N
F21 = F12 = 28.8 ·10–23N
The charges are the same sign, and therefore, the force is of repulsion.
If there are three or more objects charged and interacting, you will have to find the force that one charge is acted upon by all the other neighboring charges, and then find the net force. Remember that the force will be a vector, so the net force is the sum of vectors (value and direction).
Consider three equal charges placed on the x-axis such that the distances between are 100 cm and 300 cm between charge 1 and charge 2 and between charge 1 and charge 3, respectively. Find whether the charge in the middle is at rest or not.
In order to find our answer, we start by converting all values to SI. Then, we will find the net force on the middle charge and see if it is zero or nonzero. In the case the force is zero, the charge is at rest (Newton's second law). If the net force is nonzero, the charge will be accelerated in the direction of the net force.
r12 = 100cm = 1m
r13 = 300cm = 3m
r23 =300 cm – 100 cm = 2 m = 2 · r12
q1 = q2 = q3 = q
Fnet = ?
Charge 2 is acted upon by two forces because of repulsion with the charges 1 and 3. The forces are F23 exerted by charge 3 on charge 2 and F21 exerted by charge 1 on charge 2. As you can see from the figure, they are in opposite directions.
If we consider the positive direction as in the figure, then F21 is positive and F23 is negative.
As mentioned in the problem statement:
r23 = 2 · r12 = 2 · r21
The last term in the equality is possible to write because we are interested in the absolute value and not the direction (which was considered in drawing the figure).
Fnet = F23 + F21
Fnet = –F23 + F21
This result tells us that the charge is acted upon by a net force and subsequently will be accelerated by an acceleration that can be calculated based on Newton's second law if the mass of the charge is known:
Fnet = m · a
The force is in the direction of F21 , meaning the object is accelerated toward the positive x direction.
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