Chapters 22 and 23 Level I:
Note: When studying the following material, make sure to completely redraw the figures on your notebook and write the formulas as you proceed. This will help you learn more. Make sure to draw horizontal fraction bars when you use one.
Electric Charge and Electric Field:
In brief, electrons are negative charges and protons are positive charges. An electron is considered the smallest quantity of negative charge and a proton the smallest quantity of positive charge.
Two negative charges repel. Also, two positive charges repel. A positive charge and a negative charge attract each other (all experimentally verified.)
Point Charge: An accumulation of electric charges at a point (a tiny volume in space) is called a point charge.
Note: When an atom loses an electron, the separated electron forms a negative charge, but the remaining that contains one less electron or consequently one more proton becomes a positive charge. A positive charge is not necessarily a single proton. In most cases, a positive charge is an atom that has lost one or more electron(s).
SI Unit of Electric Charge: Coulomb
If two like point charges are separated by 1m and the repulsion force between them is 9.0x109N, each charge is called 1 Coulomb, shown as 1C. It has been shown that it takes 6.25x1018 electrons to form 1C of negative electricity.
Problems: Make sure to solve these problems before going any further. For answers click here.
Q1. It takes 6.25x1018 protons to form 1C of positive charge. Find the charge of each proton.
Q2. How many electrons are there in -1mC, -1µC, -1nC, and -1pC?
Q3. Calculate the surface area of a sphere whose radius is (a) 1.00ft, (b) 2.00ft, (c) 3.00ft, and (d) 4.00ft. Note that the surface area of a sphere is given by ( A = 4πr2)
Gold Leaf Electroscope:
An electroscope (See the figure shown below) is a device that detects the existence of electric charges on objects. It is a made of a small compartment mostly made of glass (a glass jar, for example) with a metal rod inserted into it through an insulator cap. The end of the rod that is inside the compartment has two small metal foils (aluminum , gold, or another metal) hinged to it that are free to open up like the wings of a butterfly. The outer end is connected to a metal sphere or a pan. When a charged object (no matter positive or negative) is brought into contact with the outer sphere or pan, some of the charges get transferred to the foils via the metal rod. The foils become charged up with like charges that repel each other causing the foils to separate and open up. That is how the foils indicate that some electric charges are transferred to them. Even if a charged object is held near the sphere or the pan with no physical contact, the foils still open up, but if the object is taken away from the pan, the foils drop down again. Why? Click on the following link and watch the process: http://www.youtube.com/watch?v=2PmWlPjV6n0&feature=related .
The Force Between two Point Charges:
The force between two point charges q1 and q2 separated a distance r has a magnitude given by the
Coulomb's Law: F = ( kq1q2) / r2 , where k = 8.99x109 Nm2/C2. For simplicity in calculations, we may often show 9x109 instead of 8.99x109.
The direction of the force is along the line that connects the two point charges as shown below:
Let red denote positive and blue negative.
Note that the first two figures show like charges that repel. The third figure shows unlike charges that attract.
Example 1: Find the magnitude and direction of the force between a 25.0-μC charge and a 40.0-μC charge when they are separated by a distance of 30.0 cm. Both are point charges.
Solution: F = (kq1q2 )/ r2 ; F = (9x109)(25x10-6)(40x10-6) / 0.32 ; F = 100. N, directed away from each other. This applies to the first of the above 3 figures.
Q4. Find the magnitude and direction of the force between a -50.0-μC charge and a -20.0-μC charge when they are separated by a distance of 3.00 cm. Let both be point charges. First Solve. For answers click here.
Q5. Two balloons (basketball size) are connected by an East-West light thread and are positively charged. What do you expect to observe if you cut the thread? If the balloons were both negatively charged what would happen?
Q6. If two golf balls have equal and opposite charges on them and they are brought in touch with each other, what will be the charge on each after contact? For answers click here.
Q7. A golf ball has +12µC of charge on it and another has -17µC. If they are brought into contact, what will be the charge on each afterwards. The golf balls are identical.
Example 2: In the figure shown, find the force on charge q3. Assume three significant figures.
Solution: q1 attracts q3 in the direction shown. q2 repels q3 in the direction shown. The magnitudes of F13 and F23 are (all in SI units):
F13 = kq1q3 / r13 2 = (9.0E9)(20E-6)(50E-6) /(52 +52) = 9 /50 = 0.180N at 135˚ w.r.t. the pos. x-axis.
F23 = kq2q3 / r23 2 = (9.0E9)(40E-6)(50E-6) /(52 +52) = 18 /50 = 0.360N at 45˚ w.r.t. the pos. x-axis.
Note: "w.r.t." means with respect to.
From here on, we use what we learned in Physics I : Vector Addition.
Rx = F13x + F23x = 0.180cos(135) + 0.360cos(45) = 0.127N
Ry = F13y + F23y = 0.180sin(180-45) + 0.360sin(45) = 0.382N
R = 0.403N,
θ = 71.6°
|Do R and θ appear proportional to the above figure?|
Q8. Redraw the above example assuming all charges are positive. Also, use the same y-distances of 5.0m, but change the x-distance to 8.0m. Calculate (a) the angle that each of F13 and F23 makes with the positive x-axis. (b) Find the magnitudes F13 and F23. (c) Determine the magnitude and direction of the resultant of F13 and F23. Note: If you feel you are not ready for this problem now, do it after you go through "Test Yourself 1", completely. For answers click here.
Test Yourself 1:
1) Like charges (a) repel. (b) attract. (c) neither a nor b. click here
2) Unlike charges (a) repel. (b) attract. (c) neither a nor b. click here
3) A charge is considered a point charge if (a) its dimension with respect to the distances over which its effect is to be studied is relatively small. (b) it has a zero diameter. (c) both a & b.
4) The force of a point charge on other charges around it that are at the same distance has (a) the same magnitude and direction. (b) the same magnitude only. (c) different magnitude and different directions. click here
5) Charge +q1 is at (0,0) and +q2 at (5, 0). The force of +q1 on +q2 points (a) West. ( b) East. (c) North.
6) Charge +q1 is at (0,0) and +q2 at (5, 0). The force of +q2 on +q1 points (a) West. ( b) East. (c) North.
7) Charge +q1 is at (0,0) and +q2 at (0,- 4). The force of +q1 on +q2 points (a) South. ( b) East. (c) North.
8) Charge +q1 is at (0,0) and +q2 at (0,- 4). The force of +q2 on +q1 points (a) South. ( b) East. (c) North.
9) Charge -q1 is at (0,0) and +q2 at (- 4, 0). The force of -q1 on +q2 points (a) South. ( b) East. (c) West. click here
10 ) Charge -q1 is at (0,0) and +q2 at (- 4, 0). The force of +q2 on -q1 points (a) South. ( b) East. (c) West.
11) Charge +q1 is at (-3,0) and -q2 at (0, 3). The force of +q1 on -q2 points (a) Southwest. ( b) Northeast. (c) North.
12) Charge +q1 is at (-3,0) and -q2 at (0, 3). The force of -q2 on +q1 points (a) Southwest. ( b) Northeast. (c) South.
13) the correct angle for the force in Question 11 is (a) 45º. (b) 135º. (c) 225º. click here
14) the correct angle for the force in Question 12 is (a) 45º. (b) 135º. (c) -45º. click here
15) Charge +q1 is at (-7,0) and +q2 at (0, 3). The angle for the force of q1 on q2 is (a) 23.2º. (b) 203.2º (c) -46.4º.
16) The distance between q1 and q2 in Question 15 is (a) 6.32 units. (b) 7.62 units. (c) 5.62 units. click here
17) The distance between points (0,5) and (5,0) is (a) 52 + 52 = 50 units. (b) (52 + 52)1/2 = 7.07 units. (c) 5 + 5 = 10 units.
18) The force of 25.0μC at (0,7.00m) on -12.0μC at (11.0m,0) is (a) 15.9mN, 32.5º . (b) 15.9mN, -32.5º . (c) 15.9mN, 147.5º.
19) The force of - 45μC at (0,- 4.0m) on 32μC at (9.0m,0) is (a) 0.13N, 204º. (b) 0.13mN, 24º. (c) 0.13N, 14º. click here
20) The force of -50.0μC at (-10.0m, 0) and 80.0μC at (10.0m, 0) on 20.0μC at (0, 0) is (a) 0.234N, 180º. (b) 0.234N,-180º. (c) both a & b. click here
21) The force of +50.0μC at (-10.0m, 0) and +50.0μC at (10.0m, 0) on 20.0μC at (0, 0) is (a) 0.180N, 180º. (b) 0.180N,-180º. (c) zero.
22) The force of +50.0μC at (-10.0m, 0) and +50.0μC at (0, 10.0m) on 20.0μC at (0, 0) is (a) 0.127N, 45º. (b) 0.127N,-45º. (c) 0.180N, 180º. click here
23) The force of +40.0μC at (0, -3.00m) and +20.0μC at (0, 3.00m) on 50.0μC at (4.00m, 0) is (a) 0.217N, 42º. (b) 0.217N,-42º. (c) 0.891N, 14.0º. click here If you learned how to do this problem, go back to Q8.
Electric Conductivity of Materials:
Classification of Electrons:
There are 3 types of electrons: bound electrons, valence electrons, and free electrons.
Bound electrons are the inner shells electrons that are under strong Coulomb forces from nucleus and difficult to detach.
Valence electrons are the outer shells electrons and participate in chemical reactions. They are easier to remove from the atoms.
Free electrons do not belong to any particular atom. They flow in between atoms under the repulsive forces from the electron clouds of different atoms and the smaller attraction forces from the nuclei of the closest atoms. Conductivity of a substance depends on the number of free electrons of that substance.
Classification of Materials:
From the point view of conduction, materials are classified as conductors, semiconductors, and insulators. The electric conductivity of a substance depends on its number or abundance of free electrons.
Metals are conductors. A metal contains a large number of free electrons.
Nonmetals are insulators. A nonmetal contains few free electron.
Semiconductors are alloys of metals and nonmetals. The have controlled conduction properties depending on their metal percentages.
If electricity (accumulation of negative or positive charges) can not flow easily, it causes localized charges and forms static electricity. This happens when a bunch of electrons, for example, is given to an insulator. Because of lack of free electrons in the insulator, the transferred electrons stay locally and do not distribute in the insulator quickly. They form static electricity.
If a conductor (mounted on an insulator), is given a number of excess electrons, the electrons distribute themselves in that conductor; however, the insulator mounting stops the electrons from flowing into the mounting and it becomes a boarder for the free electrons. In the conductor part, since the excess free electrons repel each other, they locate themselves as far from each other as possible. For a sphere, the farthest possible distance is a uniform distribution of charges over its external surface. For other shape objects, it depends on the geometry.
The following figure shows a metal sphere as well as an oval-shaped metal object, both on insulator mountings. 12 electrons are removed from the sphere and given to the oval. The sphere becomes positive and the oval negative. Note the higher concentration of electrons at the farthest possible distance, the sharper edges.
Charging of an Object:
An object may be given electric charges in two ways: 1) by direct contact, and 2) by induction.
1) Charging by contact:
When a charged object is brought into contact with an uncharged (electrically neutral) object, part of its charges flow onto the uncharged object and make it partially charged. The transfer proportion depends on the shapes of the two objects. For example, if the two objects are two identical metal spheres with insulator mountings, they share the charge equally. For asymmetric and unequal objects, the reasoning is more complicated and involved. The following figure shows the simple case of two identical metal spheres on insulator mountings a) before contact, b) during contact, and c) after separation.
2) Charging by Induction:
Charging by induction means charging without contact. The Earth may be considered as being electrically neutral. Adding a certain number of positive or negative charges to the Earth does not affect its neutrality. Earth is so huge that the charges on the objects do not count at all compared to the charges that the Earth contains. That is why Earth is electrically neutral for our experiments. We can easily transfer some charges to it or take from it and it will not be affected. If an electrically charged sphere (on an insulator mounting) is connected with a conductor (a metal wire) to the ground, it gets discharged either by transferring some electrons to the Earth or pulling some from it. The following figure shows how a positively charged sphere and a negatively charged one become discharged by being connected to the Earth.
Charging an Object Positively by Induction: If a plastic rod is rubbed against wool, it becomes negatively charged. If the rod is brought near a neutral metal sphere that is on an insulator mounting, it repels the free electrons of the sphere to the far end of it. At the same time, the rod attracts the positive charges of the sphere to the very near end of it. If the far end is connected to the Earth by a wire, the electrons flow to the ground while the positive charges are held captive by the rod. When the connection with the ground is cut off, the rod may be taken away leaving the sphere with positive charges. The process is shown below:
Charging an Object Negatively by Induction: To be explained by students with appropriate figures.
Electric Field (E):
Anywhere there is an electric charge, q1 , there exists the property of attraction or repulsion on other charges placed around it. This effect of attraction or repulsion is called the electric field of q1. The electric field of charge q1 at Point P, depends on the amount of q1 and 1/r2 where r is the distance from the point charge. We may come up with a formula for electric field (E) as
E1 = kq1/r2 (1)
E1 is the magnitude of the electric field of charge q1 at Point P.
Again, k is called the Coulomb's constant. Its value is k = 8.99x109 Nm2/C2. The unit for electric field is N/C.
The way the electric field strength (E) of a point charge q weakens with (r) is like the way light intensity weakens as we move away from a light bulb. Suppose you have built an empty sphere out of glass that has a surface area of 1 ft2 and has a tiny light bulb at its center. Also suppose that you have made another glass sphere which radius is twice the first one and is around the first sphere. It is easy to show that when you double the radius of a sphere, its area quadruples (4 ft2 ). If the two glass spheres are concentric, you can see why the light intensity at the outer sphere is 1/4 of the light intensity at the inner sphere. You have already figured it out that the same amount of light energy that passes through the inner sphere must reach the outer sphere and pass through it as well. Since the same energy is given to an area 4 times greater; therefore, the intensity becomes 4 times weaker. This is an example of weakening as (1/r2). What would happen to the light intensity (brightness) if you made glass spheres with radii 3x, 4x, 5x, 6x, and so forth? Again as you have correctly visualized, the light intensity (brightness) would become 1/9, 1/16, 1/25, 1/36 and so forth ( Fig. 1).
Problem: Calculate the electric field strength around a 25μC point charge at the following different distances: (a) 10.0cm, (b) 20.0cm, (c) 30.0cm, and (d) 50.0cm.
Test Charge : One unit of positive charge is called a test charge. Force per unit charge is called Electric Field Strength (Metric unit being N/C, of course). When a test charge is placed at different points around a positive charge +q, it will be repelled by a force. The farther the test charge, the weaker the force of repulsion. Note that the direction of the repulsive force is always along the line connecting +q and the test charge and acts outward as shown below:
Fig.1: field strength of a ( +q ) charge. Visualize a test charge ( +1 unit ) at the tail of each vector.
For a negative charge (-q ), a similar situation is shown below. Visualize a test charge ( +1 unit ) at the tip of each vector. When a test charge is placed at different points around a negative charge -q, it will be attracted by a force. The farther the test charge, the weaker the force of attraction. Note that the direction of the attraction force is always along the line connecting -q and the test charge and acts inward as shown below:
Fig.2: Field Strength of a ( -q ) charge.
Electric Field Orientation: The electric field orientation of a ( +q ) and a ( -q ) charge separated by a distance L is shown below. Such two equal and opposite charges form the so-called an "electric dipole."
Note: The meaning of each field line is as follows: if a test charge ( +1 unit of charge ), is placed on a field line, it will move on that curved line being constantly repelled from the (+q) and attracted toward the (-q) until it is absorbed by (-q ).
The space around these two charges contains infinite number of points. Each point experiences the effects of two fields, one from the ( +q ) charge and one from the ( -q ) charge. Since each field acts along the line that connects the charge to a given point, vector addition must be employed in order to find the resultant field. The following example clarifies the need for vector addition:
Homework: In the figure shown, find the resultant field at each point where there is a charge. For example, when finding the field at where q1 is, suppose q1 is nonexistent and find the resultant field by q2 and q3 at that point.
Uniform Electric Field:
An electric field is called uniform if its strength does not change with distance. The electric field of a point charge is not uniform, because it strongly weakens when distance from the charge increases. It weakens proportional to (1/r2) and its field lines diverge or open up very quickly in space. There are other physical quantities that vary as (1/r2). Gravitational effect of a mass, M, also weakens as (1/r2). Such quantities are said to follow the (1/r2) law. Is it possible to create an electric field that does not change with distance? The answer is "yes". If two parallel metallic plates are separated by a distance and connected to a battery, one plate accumulates some negative charges while the other plate accumulates equal amount of positive charges. The electric field in between the plates and specially away from the edges will essentially be uniform and the electric field lines become parallel. Such a device forms the so-called parallel-plate capacitor. The following figure shows the difference between the non-uniform field of a point charge and the uniform field of a parallel-plate capacitor.
Field lines diverging, field weakens with distance Field lines parallel, strength the same everywhere
If charge q is placed at a point where the electric field is E, it will experience a force (F) such that
F = qE
This is because of the fact that, for example, for a point charge we may write the Coulomb's law as
F = q2 (kq1/r2) or, F = q2E1
E1 is the electric field of charge q1. q2 is in the field of q1.
F = q E
Example 3: In the figure shown, find (a) the force on the oil drop of mass 2.0 micrograms carrying a charge of 450nC at 1.0cm from the positive plate, (b) the work it does as it moves to reach the negative plate, (c) its K.E. just before hitting the negative plate, and (d) its speed before striking the negative plate.
(a) F = qE = (450nC)(16000 N/C) = 7.2 mN (milli Newtons)
(b) W = Fd = (7.2 mN)( 0.05m) = 360 µJ (micro Joules)
(c) K.E. = W = 360 µJ Note: K.E. = (1/2)Mv2
(d) Solving for v, we get: v = SQRT (2K.E./M) = 0.60 km/s
Test Yourself 2:
1) Electrons are classified as (a) low Negative, medium negative, and high negative. (b) bound, valence, and free. (c) both a & b. click here
2) Bound electrons are (a) the outer shell ones. (b) the ones at the nucleus. (c) the inner shell ones. click here
3) Free electrons are those that (a) do not belong to any particular atom and move in between atoms under the influence of repulsion from the electronic clouds of atoms or the weaker attraction from the nuclei of the surrounding atoms. (b) are constantly being freed from the atoms. (c) neither a nor b. click here
4) Valence electrons are those that (a) may be freed from the outermost shells of atoms. (b) participate in chemical reactions. (c) both a & b. click here
5) Electrically, materials are classified as (a) conductors and insulators. (b) conductors, semiconductors, and insulators . (c) semiconductors and insulators . click here
6) Conductors contain a large # of free electrons. (a) True (b) False click here
7) Semiconductors (a) are alloys of metals and nonmetals. (b) contain a controlled # of free electrons . (c) both a & b.
8) Insulators (a) contain a very low number of free electrons. (b) are nonmetals. (c) both a & b. click here
9) The type of materials that can best hold electric charges locally on them are (a) metals. (b) semiconductors. (c) nonmetals.
10 ) As soon as electric charges are given to a metal, (a) they stay locally where they are placed at. (b) they distribute throughout the metal object or might flow to the ground. (c) they always stay on the metal . click here
11) In order for electric charges given to a metal piece stay on it (a) the metal piece must be on an insulator mounting. (b) the metal piece must be on an insulator mounting but wired to the ground. (c) the metal piece must be initially neutral. click here
12) When electric charges are given to a metal sphere that is located on an insulator, the charges (a) flow to the lowest point of the sphere where the mounting is. (b) slowly distribute, but unevenly. (c) quickly distribute evenly over the sphere's surface.
13) A positively charged object is the one that (a) has protons distributed over its surface. (b) has more protons than electrons. (c) has lost a number of electrons. (d) b & c. click here
14) A metal sphere has 6.0μC and another identical one has 14.0μC on it. The charge on each after being brought into contact is (a) 4.0μC. (b) 0. (c) 10.0μC. click here
15) A metal sphere has 26.0μC and another identical one has -18.0μC on it. The charge on each after being brought into contact is (a) 4.0μC. (b) 8.0μC. (c) 14.0μC. click here
16) A metal sphere has -20.0μC and another identical one has -10.0μC on it. The charge on each after being brought into contact is (a) -30.0μC. (b) +30.0μC. (c) -15.0μC. click here
Problem: Two identical metal spheres A and B are on their insulator mountings, both initially neutral. A is on the left and B on the right. They are first brought into contact. A negatively charged rod is then held close to the left of A. To answer the following questions, drawing a figure is absolutely necessary.
17) The rod makes the left of sphere A (a) negative. (b) positive. (c) neutral. click here
18) The rod makes the right of sphere B (a) negative. (b) positive. (c) neutral. click here
19) The reason why the right of sphere B becomes negative is that (a) the negative rod repels the free electrons in the connected spheres to the farthermost distance possible. (b) electrons cannot flow through both spheres . (c) positive charges flow through both spheres to the left of A. (d) both a & c. click here
20) If you connect the right of sphere B to the neutral Earth, (a) the electrons flow to the Earth. (b) the Earth does not accept those electrons. (c) that end becomes positive. click here
21) If while the negative rod is still held near the left of A, the connection to the Earth of B is removed, (a) the left sphere is positive and the right one neutral. (b) both spheres are positive. (c) the left one is negative while the right one is positive.
22) If the negative rod is moved far away, (a) both spheres become negative. (b) both spheres become positive. (c) one sphere becomes negative and one positive. click here
23) The positive charges of the two sphere in contact stay (a) at the point where they are in contact. (b) at the left of each sphere. (c) equally at the left of A and the right of B. click here
24) If the spheres are separated, the charges on each (a) distribute themselves evenly on each sphere. (b) stay where they are. (c) fall toward the mounting on each sphere.
25) The electric field of a point charge (a) is uniform. (b) is non-uniform. (c) varies with 1/r. click here
26) The electric field of a point charge varies (a) with 1/r2. (b) with 1/r3. (a) does not vary with distance and is constant.
27) The electric field of charge q1 at a distance r is mathematically (a) E = kq1/r2. (b) E = kq1/r. E = constant.
28) The way the light of a light bulb weakens as we move away from it follows the (a) 1/r2 law (b) 1/r law. (c) 1/r3 law.
29) According to 1/r2 law, doubling our distance from a light bulb, the light energy we receive at each eye becomes (a) 1/4. (b) 1/2. (c) neither a nor b. click here
30) According to 1/r2 law, quadrupling our distance from a light bulb, the light energy we receive at each eye becomes (a) 1/4. (b) 1/16. (c) 1/32.
31) The electric field between two parallel plates oppositely charged (a) follows 1/r law. (b) follows 1/r2 law. (c) is constant and does not depend on the distance from either plate. click here
32) The coulomb's formula F = kq1q2/r2 may be written as (a) F = (kq1/r2) q2 (b) F = (E1) q2 where is the electric field of q1 at r. (c) both a & b.
33) According to the previous question, (a) Force = Field X charge. (b) F = Eq. (c) both a & b. click here
34) Two charges of q1 = 35μC and q2 = 45μC are placed at different distances from the negative plate of a parallel-plate capacitor (that has a uniform electric field in between its plates). Draw a figure for it. The field strength is 2000. N/Coul. The force on the charges are: (a) .070N and .070N. (b) .070N and .090N. (c) .090N and .090N.
35) If the distance of q1 from the negative plate in the previous question is 4.0cm, the work done on q1 as it is pushed toward the negative plate by the field is (a) .0056J. (b) .0063J. (c) .0028J. click here
36) Since .0028J of work is done by the field, we may say that the potential energy of q1 at 4.0cm from the negative plate is (a) .0028J (b) .0028N (c) .0028watts.
37) The energy of q1 as it speeds up toward the negative plate becomes more of (a) elastic type. (b) potential type. (c) K.E. type. click here
38) Due to conservation of energy, the K.E. of q1 just before striking the negative plate is (a) .0028N (b) .0028J (c) .0028w.
39) If q1 is on a mass of 12μgrams, its mass in kg is (a) 12E-3 kg. (b) 12E-6 kg. (c) 12E-9 kg. click here
40) Knowing the K.E. of q1 at the negative plate, its speed at that point is (a) 320m/s. (b) 863m/s. (c) 680m/s.
Problem: Two identical small Styrofoam balls that are 2.2grams each are hung by two strings (each 1.0 long) from the same point. When each is given charge Q, the angle between each string and vertical line becomes16°. Find the amount of charge Q.
Continuous Charge Distributions:
The formula E = kq/r2 in its vector form applies to point charges. To find the electric field of a charge distribution at point P in space, it must be divided into infinitesimal charges dq (that can act as point charges) and dE, the field of each dq, be calculated at that point. The integral of such dE's will then give the desired field.
|The Electric Field Along a Slender
In the figure shown, find the electric field of the uniformly charged slender rod of length L at P that is at a distance a from End B of it.
Solution: Because of the uniform charge distribution on the slender rod, if charge Q is divided by the rod's length L, we get the linear charge density λ = Q/L in units of C/m. An extremely tiny segment of length dx meters has therefore a charge equal to dq = λdx on it in Coulombs.
|The Electric Field Around an Infinite
Line of Charge
Calculate the electric field intensity at a distance R from an infinite line of charge with a linear charge density of λ C/m.
Solution: An extremely tiny segment of length dl meters carries a charge equal to dq = λdl Coulombs.
The field of dl at P is dE = kdq/r2 that is
dE = kλdl /r2. (dE is a vector). (1)
For every dl at a point like A, there is a dl at a point like B where A and B are symmetric with respect to the line labeled R. Such symmetric dl's generate dEs that have have equal and opposite y-components at P and they cancel each other's effect. We end up adding the x-components of dEs only to come up with the total field at P. Each x-component is
dEx = (dE) cosθ. (dEx is a scalar).
The magnitude of differential field at P becomes [from (1)]:
dEx = kλdl cosθ/r2. (2) Variables dl and r2 must now be expressed in terms of θ.
The way dEx is completely expressed as a function of θ is shown on the right.
|6) Find the electric
field intensity, E, at point P
that has a distance y from the center of a
non-conducting disk of radius a that has a
surface charge density of s
For every point on the ring, there is an opposite point on the
other side of it that forms an equal and opposite x-component of
dE at P. Such x-comps.
cancel each other and make ΣdEx
= 0 at P. Each y-comp. is
See if Equations (1), (2), and (3) on the right make any sense to you. Note that dq is the charge on each ring (differential area).
If you substitute (2) in (1) and the result in (4), you will get (5).
Electric Field of a Dipole:
| As was mentioned an electric dipole is made of two equal and opposite charges
that are a distance d away. Any molecule that has
its center of positive charges away from its center negative charges forms
H2O and NaCl are examples. Such molecules experience a torque on them when placed inside an external electric field causing them to rotate and align themselves along the direction of the external field. To find the electric field of a dipole itself, let's place +Q and -Q at +a and -a on the y-axis, and find the field they generate on the perpendicular bisector of the line that connects them, as shown.
The net field at A, a distance r from the origin is the resultant of E+ and E- as shown. Since these two vectors have equal magnitudes, the resultant E becomes straight downward. The x-components of E+ and E- cancel. Their equal y-components are:
Torque on a Dipole:
When an electric dipole is inside a uniform electric field as shown, the external field exerts two equal but opposite forces on the dipole causing it to rotate and align itself along the field lines of the external field. If the angle between the dipole moment P and the external field lines is θ, the magnitude of torque τ is
τ = 2F(d/2)sinθ = qE(d)sinθ = PEsinθ
or, τ = P x E (all vectors, cross product)
where P = qd is the dipole moment.
1) Referring to Example 4, show that for a limited rod length, L, but large values of a, the derived formula reduces to E=kQ/a2.
2) Rework the integral of Example 5 to obtain an expression for E for a line of charge that is not infinitely long. Let the angle vary from -θ to +θ instead of from -π/2 to +π/2.
3) In the figure shown, (a) obtain an expression for the electric field at C if the rod segment is circular and uniformly charged with a linear charge density of l C/m, and (b) show that for a semicircular rod the field is E = 2kλ/R.
|4) In the figure shown, determine the minimum horizontal speed to the right at which an electron must be ejected near the negative plate such that it will not be absorbed by the positive plate before leaving the uniform electric field of 3600N/c that exists in between the plates. Me=9.108x10-31kg and e=1.602x10-19C.||
5) In Example 6, Equation (7), what result do you obtain if you let the disk's radius, a, approach infinity? In doing this, you are verifying that when the disk becomes an infinite plane of charge, the electric field becomes E = s/2ε0.
Find the electric field intensity in between two parallel and infinite sheets of charge with uniform surface charge densities of s and -s by adding the electric field intensity of the individual fields.
Draw Part (c) on paper and draw the net electric field in Regions 1, 2, and 3.
7) In Example 6, Equation (7), what result do you obtain if you let the the distance of point P from the disk, y, becomes very large compared to a?
8) Find the electric field at a distance r from the center of an electric dipole if the point is along the line that connects the opposite charges.