ASSIGNMENTS & SOLUTIONS Gauss's Law
Assignment #1: Sections 24.1
# 1,4,6,8
Assignment #2: Section 24.2
# 9,11,13,17,21
Assignment #3: Section 24.3-4
# 36,24,31,37,39
Assignment #4: Section 24.4
# 42 a-d,43,44,47,48
Chapter 24 Answers to Quick Quizzes 1. 2. 3.
(e) (b) and (d) (a)
Solutions to Even Problems #4:
a. -2.34 kNm2/C
b. 2.34 kNm 2/C
c. 0 ·
q
#6:
o
#8: #36: #24: #42: #44: #48:
ERh a. 713 nC b. 5.7 C a. 0 b. 365 kN/C c. 1.46xl 06 N/C d. 649kN/C outward a. zero b. 1.24x104 N/C outward c. 639 N/C outward d. Same a. 0 b. 79.9x10 6 N/C outward c. 0 d. 7.34x106 N/C outward a. 2.56x1 06 N/C inward b. 0
LEARNING TARGETS Gauss's Law I.
I can explain the concept of electric flux
A. B.
I can state that the electric flux is proportional to the number of field lines penetrating a surface
C.
I can write the equation for the electric flux when the electric field is not uniform
I can calculate the electric flux passing through a surface for a uniform electric 'field
II.
I can explain Gauss's Law mathematically and conceptually
A.
1 can state Gauss's Law mathematically and write the equation
B.
I can state that the flux through a closed surface is proportional to the net charge enclosed by the surface C. I can state that l can state that theE-field at a point on a surface depends on the charges inside and outside the surface zero electric flux does not imply no charge enclosed or a zero electJic field on the surface D.
III. A.
I can apply Gauss's Law to highly-symmetric charge distributions I can determine the electric field due to a point charge, a radially-symmetric charged sphere, a cylindrical
B.
I can determine the electric field for spherical and cylindrical charges that have non-uniform charge
charge, an infinite line charge, and infinite planar charges. densities. IV.
' B. C. D.
I can discuss conductors in electro.static equilibrium 1 can state that tr-;.e electric field is zero inside a conduCtor
I can state that all the charge resides on the surface of a conductor I can state that the electric field just above the surface is perpendicular to the surface I can state that the surface charge is greatet at locations that have the smallest radius of curvature
24.1 Electric Flux1 Electric Flux E (phi) is the product of the electric field that is perpendicular to a surface A. (think of air flow) E BA For non perpendicular surfaces. E BA cos When is the Electric Flux a maximum? When is the Electric Flux a minimum? Flux can be thought of as the amount of electric field vector E perpendicular to the area plane or the amount of area plane perpendicular to the electric field vector E.
E ,i Ei A EA cos
E ,i Ei Ai
The definition of Electric Flux is: E ,i E A E A Surface
This is a surface integral. The evaluation is over the surface . The value of E depends on the field pattern and on the surface. A closed surface divides space into an inside and outside region. The Direction of the area vector is so that the vector points outward from the surface.
24.1 Electric Flux2
When E enter a surface 0 E A 0 . When E leaves a surface 0 E A 0 .
When E is perpendicular to a surface 90 E A 0 .
Quick Quiz 24.1 Suppose a point charge is located at the center of a spherical surface. The electric field at the surface of the sphere and the total flux through the sphere are determined. Now the radius of the sphere is halved. What happens to the flux through the sphere and the magnitude of the electric field at the surface of the sphere? (a) The flux and field both increase. (b) The flux and field both decrease. (c) The flux increases, and the field decreases. (d) The flux decreases, and the field increases. (e) The flux remains the same, and the field increases. (f) The flux decreases, and the field remains the same.
24.1 Electric Flux3 Example 24.1 Consider a uniform electric field E oriented in the x direction in empty space. A cube of edge length, is placed in the field, oriented as shown in Figure. Find the net electric flux through the surface of the cube.
E E A E A E A E A E A E A 1
2
3
4
5
6
E E (cos180o )l 2 E (cos0o )l 2 E (cos90o )l 2 E (cos90o )l 2 ... E E (1)l 2 E (1)l 2 0 0 0 0 E 0
24.2 Gauss’s Law1 E
E dA EdA E dA
k q E e2 4 r 2 4 ke q r 1 But ke 4 o E
German mathematician and astronomer (1777–1855) In addition to his work in electromagnetism, he made contributions to mathematics and science in number theory, statistics, non-Euclidean geometry, and cometary orbital mechanics. He was a founder of the German Magnetic Union, which studies the Earth’s magnetic field on a continual basis.
q
o
The net flux through any closed surface surrounding a point charge q is given by q/o and is independent of the shape of that surface.
E
E dA E E dA 1
1
Gauss’s Law E
E dA
qin
o
Pitfall Prevention 24.1 Zero Flux Is Not Zero Field In two situations, there is zero flux through a closed surface: either (1) there are no charged particles enclosed by the surface or (2) there are charged particles enclosed, but the net charge inside the surface is zero. For either situation, it is incorrect to conclude that the electric field on the surface is zero. Gauss’s law states that the electric flux is proportional to the enclosed charge, not the electric field.
24.2 Gauss’s Law2 Quick Quiz 24.2 Let the net flux through a Gaussian surface is zero. Which of the following four statements could be true? Which of the statements must be true? (a) There are no charges inside the surface. (b) The net charge inside the surface is zero. (c) The electric field is zero everywhere on the surface. (d) The number of electric field lines entering the surface equals the number leaving the surface.
Example 24.2 A spherical gaussian surface surrounds a point charge q. Describe what happens to the total flux through the surface if (A) the charge is tripled, (B) the radius of the sphere is doubled, (C) the surface is changed to a cube, and (D) the charge is moved to another location inside the surface.
(A) The flux through the surface is tripled because flux is proportional to the amount of charge inside the surface. (B) The flux does not change because all electric field lines from the charge pass through the sphere, regardless of its radius. (C) The flux does not change when the shape of the gaussian surface changes because all electric field lines from the charge pass through the surface, regardless of its shape. (D) The flux does not change when the charge is moved to another location inside that surface because.
24.3 Application of Gauss’s Law1 Apply Gauss’s law to highly symmetric problems. 1. The value of the electric field can be argued by symmetry to be constant over the portion of the surface. 2. The dot product E dA in can be expressed as a simple algebraic product E dA because E and dA are parallel. 3. The dot product in E dA zero because E and dA are perpendicular. 4. The electric field is zero over the portion of the surface. Example 24.3 A Spherically Symmetric Charge Distribution An insulating solid sphere of radius a has a uniform volume charge density and carries a total positive charge Q. (A) Calculate the magnitude of the electric field at a point outside the sphere. (B) Find the magnitude of the electric field at a point inside the sphere. (Be able to prove)
24.3 Application of Gauss’s Law2 Example 24.4
A Cylindrically Symmetric Charge Distribution
Find the electric field a distance r from a line of positive charge of infinite length and constant charge per unit length .
Example 24.4
A Plane of Charge
Find the electric field due to an infinite plane of positive charge with uniform surface charge density . Suppose two infinite planes of charge are parallel to each other, one positively charged and the other negatively charged. The surface charge densities of both planes are of the same magnitude. What does the electric field look like in this situation?
Example 24.5
Don’t Use Gauss’s Law Here!
Explain why Gauss’s law cannot be used to calculate the electric field near an electric dipole, a charged disk, or a triangle with a point charge at each corner. The charge distributions of all these configurations do not have sufficient symmetry to make the use of Gauss’s law practical. We cannot find a closed surface surrounding any of these distributions for which all portions of the surface satisfy one or more of conditions (1) through (4) listed at the beginning of this section.
24.44Conductors in Electrostatic Equilibrium1 When there is no net motion of charge within a conductor, the conductor is in electrostatic equilibrium. A conductor in electrostatic equilibrium has the following properties: 1. The electric field is zero everywhere inside the conductor, whether the conductor is solid or hollow. If the field were not zero, free electrons in the conductor would experience an electric force (F =qE) and would accelerate due to this force. This motion of electrons, however, would mean that the conductor is not in electrostatic equilibrium. Therefore, the existence of electrostatic equilibrium is consistent only with a zero field in the conductor.
2. If the conductor is isolated and carries a charge, the charge resides on its surface. A gaussian surface is drawn inside the conductor and can be very close to the conductor’s surface. Tthe electric field everywhere inside the conductor is zero when it is in electrostatic equilibrium. Therefore, the electric field must be zero at every point on the gaussian surface, and the net flux through this gaussian surface is zero. From this result and Gauss’s law, we conclude that the net charge inside the gaussian surface is zero.
3. The electric field at a point just outside a charged conductor is perpendicular to the surface of the conductor and has a magnitude o, where is the surface charge density at that point. If the field vector E had a component parallel to the conductor’s surface, free electrons would experience an electric force and move along the surface; in such a case, the conductor would not be in equilibrium. Therefore, the field vector must be perpendicular to the surface.
E E
qin
E dA EA
o
A o
o
4. On an irregularly shaped conductor, the surface charge density is greatest at locations where the radius of curvature of the surface is smallest. Will be prove later
24.44Conductors in Electrostatic Equilibrium1 Quick Quiz 24.3 Your younger brother likes to rub his feet on the carpet and then touch you to give you a shock. While you are trying to escape the shock treatment, you discover a hollow metal cylinder in your basement, large enough to climb inside. In which of the following cases will you not be shocked? A. You climb inside the cylinder, making contact with the inner surface, and your charged brother touches the outer metal surface. B. Your charged brother is inside touching the inner metal surface and you are outside, touching the outer metal surface. C. Both of you are outside the cylinder, touching its outer metal surface but not touching each other directly. Example 24.7 A Sphere Inside a Spherical Shell A solid insulating sphere of radius a carries a net positive charge Q uniformly distributed throughout its volume. A conducting spherical shell of inner radius b and outer radius c is concentric with the solid sphere and carries a net charge -2Q. Using Gauss’s law, find the electric field in the regions labeled , , and in figure and the charge distribution on the shell when the entire system is in electrostatic equilibrium.
