Formulas/Definitions
In Gauss's Law, what quantity does ε0 represent?
Permittivity of free space (vacuum permittivity)
What is the surface integral of E dot dA for a square plane with side length s?
E * s^2
A circular plane, with a radius of 2.2 m, is immersed in an E-Field with a magnitude of 800 N/C. The field makes an angle of 20° with the plane. What is the magnitude of the flux through the plane?
4.2x103 Nm^2/C
The net electric flux through an 3D closed surface is positive 2.2x103 Nm2/C. What is the sign of the charge?
Positive
The net electric flux through a closed surface:
A) depends on the size of the surface.
B) depends on the shape of the surface.
C) is zero only if negative charges are enclosed by the surface.
D) is zero only if positive charges are enclosed by the surface.
E) Is zero if the net charge enclosed by the surface is zero.
E
What type of imaginary surface is used to perform Gauss's Law calculations?
Gaussian surface
True or False: In Gauss's Law for Electricity, the surface integral of the electric field over a closed surface is equal to the total electric charge enclosed by that surface divided by the permittivity of free space.
True
A pyramid with a 6.0 m square base has a height of 4.0 m. If it was placed in a vertical E-field (uniform magnitude of 52.0 N/C), determine the flux through one of the four sides.
468 Nm^2/C
The net electric flux through an 3D closed surface is positive 2.2x103 Nm2/C. How much charge must be inside the surface?
1.9x10-8 C
A sphere of radius R has positive charge Q uniformly distributed throughout its volume. Which of the following is the electric field outside the sphere r > R?
A) 0
B) kQr/R^3
C) kQ/r^2
D) kQ/R^3
E) kQ/r
C
Both sides of Gauss's Law equal what quantity?
Electric flux
What is the electric field of a charged conducting cylinder with radius R and length L
q/2πRLε
A Gaussian cube, 0.5 m along each edge, sits on the axis of a coordinate system such that three of its edges are along the postive x, y, and z axes. Determine the net electric flux through the top face of the cube if there is a uniform E-field of -0.5i + 0.3j acting in that region of space.
0.075 Nm^2/C
A hollow metal sphere of radius 3.4cm has a charge of 19.9nC distributed evenly on the entirety of the surface. Find the surface charge density.
.137 nC/cm^2
Which of the following represents the electric field due to an infinite charged sheet with a uniform charge distribution σ?
a) Zero
b) σ
c) σ/2Ɛ
d) 2σ/Ɛ
e) σ/Ɛ
C
What vector is the E-field dotted with in the integral side of Gauss's Law?
What is the surface integral of E dot dA for a sphere of radius r?
E * 4πr2
What is the E-field enclosed in an infinitely long wire? Derive an expression using Gauss's law given that the wire has a linear charge density p and radius r.
p/2πrε0
Create a cylindrical gaussian surface with length l and radius r.
flux = E * 2πrl = q/ε0 = pl/ε0
p/2πrε0
You have a neutral balloon. If you were to add 21,000 electrons to it, what would its net charge be? (e=1.6x10^−19 = charge of one electron)
-3.36⋅10^−9μC
Charged spheres A and B are near each other and isolated from all other charges. Three Gaussian surfaces are shown in the region surrounding the spheres: surface 1 around sphere A, surface 2 around sphere B, and surface 3 around both spheres. The net flux through surface 2 has twice the magnitude as the net flux through surface 1. The net flux through surface 3 has a magnitude equal to the net flux through surface 1. Which of the following are possible values of the charges 𝑄𝐴 and 𝑄𝐵 , the charges on spheres A and B , respectively?
A) 𝑄𝐴=+𝑞 and 𝑄𝐵=+𝑞
B) 𝑄𝐴=+𝑞 and 𝑄𝐵=+2𝑞
C) 𝑄𝐴=+𝑞 and 𝑄𝐵=−𝑞
D) 𝑄𝐴=−𝑞 and 𝑄𝐵=+2𝑞
E) 𝑄𝐴=−𝑞 and 𝑄𝐵=-2𝑞
D
Correct. Per Gauss’s law, the charge enclosed by a Gaussian surface is proportional to the charge enclosed. Since the flux through surface 2 is twice that through surface 1, the magnitude of the charge on sphere B must be twice the magnitude of the charge on sphere A, |𝑄𝐵|=2|𝑄𝐴|. Since the flux through surface 3 is equal to that through surface 1, the magnitude of the total charge on the two spheres must be equal to the magnitude of the charge on sphere A, |𝑄𝐴|=|𝑄𝐴+𝑄𝐵|. This option indictaes that |𝑄𝐵|=2|𝑄𝐴||2𝑞|=2|−𝑞|2𝑞=2𝑞 and |𝑄𝐴|=|𝑄𝐴+𝑄𝐵||−𝑞|=|−𝑞+2𝑞|𝑞=𝑞.
What type of integral is used in Gauss's Law?
Surface integral
A very long insulating cylinder is hollow with an inner radius of a and an outer radius of b. Within the insulating material the volume charge density is given by: ρ(R)=α/R, where α is a positive constant and R is the distance from the axis of the cylinder. Choose appropriate gaussian surfaces and use Gauss’s law to find the electric field (magnitude and direction) everywhere.
α(b-a)/εR
The concept of flux can also be applied to gravitational fields. Gauss's Law for gravity is:(1/4πG)Φg = -m, where m is the mass contained within a Gaussian surface and the gravitational flux is Φg = ∫g·dA (g is the gravitational field through the surface)
Show that this law is equivalent to Newton's Law of Universal Gravitation.
gAcos180 = -4πGm
-g(4πR^2) = -4πGm
g = Gm/R^2
A thin, nonconducting rod of length 𝐿 is positioned on the x-axis with its left end at 𝑥=0. The rod is charged and has a linear charge density of 𝜆=5𝑥^3. What is the total charge on the rod?
5/4 L^4
A solid, nonconducting sphere of radius 𝑅 has a uniform charge distribution throughout its volume. The electric field at a point a distance 𝑟 from the center of the sphere is 𝐸=𝑘𝑄/𝑟^2 , where 𝑟>𝑅 . Which of the following is a correct illustrates a correct use of Gauss's law to determine the electric field at point where 𝑟<𝑅 ?
A) Zero
B) 𝑄𝑟/𝜖𝑅=𝐸(4𝜋𝑟^2)
C) 𝑄𝑟/𝜖𝑅=𝐸(4𝜋𝑅^2)
D) 𝑄𝑟^3/𝜖𝑅^3=𝐸(4𝜋𝑟^2)
E) 𝑄𝑟^3/𝜖𝑅^3=𝐸(4𝜋R^2)
D