1110 Phys Chapter 1
Electric Force and Electric Fields
The electromagnetic force between charged particles is one of the fundamental forcesof nature.
1- Properties of Electric Charges
A number of simple experiments demonstrate the existence of electric forces andcharges. For example, after running a comb through your hair on a dry day, you willfind that the comb attracts bits of paper. The attractive force is often strong enough tosuspend the paper. The same effect occurs when certain materials are rubbed together,such as glass rubbed with silk or rubber with fur.
Figure (1)
Another simple experiment is to rub an inflated balloon with wool. The balloon then adheres to a wall, often for hours. When materials behave in this way, they are said to be electrified, or to have become electrically charged. Next Figure shows that "Rubbing a balloon against your hair on a dry day causes the balloon and your hair to become charged".
Figure (2)
In a series of simple experiments, it was found that there are two kinds of electriccharges, positive and negative charges. We identify negative charge as that type possessed by electrons and positivecharge as that possessed by protons. To verify that there are two types of charge,suppose a hard rubber rod that has been rubbed with fur is suspended by a sewingthread, as shown in Next Figure (3) . When a glass rod that has been rubbed with silk isbrought near the rubber rod, the two attract each other (Fig. 3-a). On the otherhand, if two charged rubber rods (or two charged glass rods) are brought near eachother, as shown in Figure 3-b, the two repel each other. This observation shows thatthe rubber and glass have two different types of charge on them. On the basis of theseobservations, we conclude that
" charges of the same sign repel one another andcharges with opposite signs attract one another."
Figure (3)
Then we can say that "the electric charge on the glass rod is called positive and that on the rubber rod is called negative. Therefore, any charged object attracted to a charged rubber rod (or repelled by a charged glass rod) must have a positive charge, and any charged object repelled by a charged rubber rod (or attracted to a charged glass rod) must have a negative charge".
Another important aspect of electricity that arises from experimental observationsis that electric charge is always conserved in an isolated system. That is,when one object is rubbed against another, charge is not created in the process. Theelectrified state is due to a transfer of charge from one object to the other. Oneobject gains some amount of negative charge while the other gains an equal amountof positive charge. For example, when a glass rod is rubbed with silk, as in Figure 1, the silk obtains a negative charge that is equal in magnitude to the positivecharge on the glass rod.
Also Another important aspect that electric charge alwaysoccurs as some integral multiple of a fundamental amount of charge e. In modern terms, the electric charge q is said to be quantized, where q is thestandard symbol used for charge as a variable. That is, electric charge exists asdiscrete “packets,” and we can write q = Ne, where N is some integer. Other experimentsin the same period showed that the electron has a charge -e and the protonhas a charge of equal magnitude but opposite sign +e. Some particles, such asthe neutron, have no charge.
From our discussion thus far, we conclude that electric charge has the following importantproperties:
• There are two kinds of charges in nature; charges of opposite sign attract oneanother and charges of the same sign repel one another.
• Total charge in an isolated system is conserved.
• Charge is quantized.
2 - Charging Objects By Induction
It is convenient to classify materials in terms of the ability of electrons to move through the material:
Electrical conductorsare materials in which some of the electrons are freeelectrons that are not bound to atoms and can move relatively freely through the material. In contrast, materials such as copper, aluminum, and silver are good electrical conductors. When such materials are charged in some small region, the charge readily distributesitself over the entire surface of the material.
electrical insulatorsare materials in which all electrons are bound toatoms and cannot move freely through the material. Materials such as glass, rubber, and wood fall into the category of electrical insulators.When such materials are charged by rubbing, only the area rubbed becomes charged,and the charged particles are unable to move to other regions of the material.
Semiconductors are a third class of materials, and their electrical propertiesare somewhere between those of insulators and those of conductors. Silicon and germanium are well-known examples of semiconductors commonly used in the fabricationof a variety of electronic chips used in computers.
3- Coulomb’s Law
Charles Coulomb (1736–1806) measured the magnitudes of the electric forces betweencharged objects and he found that:-
• The electric force is inversely proportional to the square of the separation r between the particles anddirected along the line joining them;
• The electric force is proportional to the product of the charges q1 and q2 on the two particles;
•The electric force is attractive if the charges are of opposite sign and repulsive if the charges have the same sign;
• The electric force is a conservative force.
We will use the term point charge to mean a particle of zero size that carriesan electric charge. We can express Coulomb’s law as an equation giving the magnitude ofthe electric force (sometimes called the Coulomb force) between two point charges:
Where keis a constant called the Coulomb constant. The value of the Coulomb constant depends on the choice of units. The Coulomb constant ke in SI units has the value
where the constant o (lowercase Greek epsilon) is known as the permittivity of free
space and has the value
The smallest unit of charge e known in nature is the charge on an electron (- e)or a proton (+ e) and has a magnitude
Example 1 (The Hydrogen Atom)
The electron and proton of a hydrogen atom are separated(on the average) by a distance of approximately 5.3 x 10-11 m. Find the magnitudes of the electric force and the
gravitational force between the two particles.
Solution From Coulomb’s law, we find that the magnitudeof the electric force is
Using Newton’s law of universal gravitation and above Table for the particle masses, we find that the magnitude of the gravitational force is
Thus, the gravitational force between charged atomic particles is negligible when compared with the electric force.
Figure (4)
When dealing with Coulomb’s law, you must remember that force is a vector quantityand must be treated accordingly. The law expressed in vector form for the electricforce exerted by a charge q1 on a second charge q2, written F12, is
where rˆ is a unit vector directed from q1 toward q2, as shown in Figure 4-a. Because
the electric force obeys Newton’s third law, the electric force exerted by q2 on q1 isequal in magnitude to the force exerted by q1 on q2 and in the opposite direction;that is,
F21= - F12.
Figure (5)
if five charges are present, as shown in figure(5), then the resultant force exerted by particles 2, 3, 4,and 5 on particle 1 is
Example 2Find the Resultant Force
Consider three point charges located at the corners of a right triangle as shown in Figure 6, where q1= q3 = 5.0 µC,q2= - 2.0 µC, and a = 0.10 m. Find the resultant forceexerted on q3.
Figure (6)
Solution First, note the direction of the individual forcesexerted by q1 and q2 on q3. The force F23 exerted by q2 onq3 is attractive because q2 and q3 have opposite signs. Theforce F13 exerted by q1 on q3 is repulsive because bothcharges are positive.
The magnitude of F23 is
In the coordinate system shown in Figure(6), the attractive force F23 is to the left (in the negative x direction).
The magnitude of the force F13 exerted by q1 on q3 is
The repulsive force F13 makes an angle of 45° with the x axis. Therefore, the x and y components of F13 are equal, with magnitude given by F13 cos 45°= 7.9 N.
We can also express the resultant force acting on q3 in unit vector form as
Figure (7)
Example 3 Where Is the Resultant Force Zero?
Three point charges lie along the x axis as shown in Figure (7). The positive charge q1= 15.0 µC is at x = 2.00 m,the positive charge q2= 6.00 µC is at the origin, and theresultant force acting on q3 is zero. What is the x coordinateof q3?
Solution Because q3 is negative and q1 and q2 are positive,the forces F13 and F23 are both attractive, as indicated in Figure (7). From Coulomb’s law, F13 and F23 have magnitudes
For the resultant force on q3 to be zero, F23 must be equal inmagnitude and opposite in direction to F13. Setting themagnitudes of the two forces equal, we have
Noting that keand q3are common to both sides and so can be dropped, we solve for x and find that
Solving this quadratic equation for x, we find that the positiveroot is x = 0.775 m There is also a second root, x = - 3.44 m. This is another location at which the magnitudes of the forces on q3 are equal, but both forces are in thesame direction at this location.
References
This lecture is a part of chapter 23 from the following book
Physics for Scientists and Engineers (with PhysicsNOW and InfoTrac),
Raymond A. Serway - Emeritus, James Madison University , Thomson Brooks/Cole © 2004, 6th Edition, 1296 pages
.
Problems
(1)Two protons in an atomic nucleus are typically separatedby a distance of 2 x 10-15 m. The electric repulsion forcebetween the protons is huge, but the attractive nuclearforce is even stronger and keeps the nucleus from burstingapart. What is the magnitude of theelectric force betweentwo protons separated by 2.00 x 10-15m?
(2)Three point charges are located at the corners of an equilateraltriangle as shown in Figure 9. Calculate the resultantelectric force on the 7.00- µC charge.
Figure (9)
(3)Two identical conducting small spheres are placed with theircenters 0.300m apart. One is given a charge of 12.0 nCand the other a charge of -18.0 nC. (a) Find the electricforce exerted by one sphere on the other. (b) What If?The spheres are connected by a conducting wire. Findthe electric force between the two after they have come toequilibrium.
Problems solutions
(1)
(2)
(3)
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