CP CALC A region in space contains a total positive charge Q that is distributed spherically such that the volume charge density ρ ( r ) is given by ρ ( r ) = 3 αr / 2 R f o r r ≤ R / 2 ρ ( r ) = α [ 1 − ( r / R ) 2 ] f o r R / 2 ≤ r ≤ R ρ ( r ) = 0 f o r r ≥ R Here α is a positive constant having units of C/m 3 , (a) Determine α in terms of Q and R . (b) Using Gauss’s law, derive an expression for the magnitude of the electric field as a function of r . Do this separately for all three regions. Express your answers in terms of Q . (c) What fraction of the total charge is contained within the region R /2 ≤ r ≤ R ? (d) What is the magnitude of E → at r = R /2? (e) If an electron with charge q ' = − e is released from rest at any point in any of the three regions, the resulting motion will be oscillatory but not simple harmonic. Why?
CP CALC A region in space contains a total positive charge Q that is distributed spherically such that the volume charge density ρ ( r ) is given by ρ ( r ) = 3 αr / 2 R f o r r ≤ R / 2 ρ ( r ) = α [ 1 − ( r / R ) 2 ] f o r R / 2 ≤ r ≤ R ρ ( r ) = 0 f o r r ≥ R Here α is a positive constant having units of C/m 3 , (a) Determine α in terms of Q and R . (b) Using Gauss’s law, derive an expression for the magnitude of the electric field as a function of r . Do this separately for all three regions. Express your answers in terms of Q . (c) What fraction of the total charge is contained within the region R /2 ≤ r ≤ R ? (d) What is the magnitude of E → at r = R /2? (e) If an electron with charge q ' = − e is released from rest at any point in any of the three regions, the resulting motion will be oscillatory but not simple harmonic. Why?
CP CALC A region in space contains a total positive charge Q that is distributed spherically such that the volume charge density ρ(r) is given by
ρ
(
r
)
=
3
αr
/
2
R
f
o
r
r
≤
R
/
2
ρ
(
r
)
=
α
[
1
−
(
r
/
R
)
2
]
f
o
r
R
/
2
≤
r
≤
R
ρ
(
r
)
=
0
f
o
r
r
≥
R
Here α is a positive constant having units of C/m3, (a) Determine α in terms of Q and R. (b) Using Gauss’s law, derive an expression for the magnitude of the electric field as a function of r. Do this separately for all three regions. Express your answers in terms of Q. (c) What fraction of the total charge is contained within the region R/2 ≤ r ≤ R? (d) What is the magnitude of
E
→
at r = R/2? (e) If an electron with charge q' = −e is released from rest at any point in any of the three regions, the resulting motion will be oscillatory but not simple harmonic. Why?
An infinitely long cylindrical conducting shell of outer radius r1 = 0.10 m and inner radius r2 = 0.08 m initially carries a surface charge density σ = -0.15 μC/m2. A thin wire, with linear charge density λ = 1.1 μC/m, is inserted along the shells' axis. The shell and the wire do not touch and there is no charge exchanged between them.
A) What is the new surface charge density, in microcoulombs per square meter, on the inner surface of the cylindrical shell?
B) What is the new surface charge density, in microcoulombs per square meter, on the outer surface of the cylindrical shell?
C) Enter an expression for the magnitude of the electric field outside the cylinder (r > 0.1 m), in terms of λ, σ, r1, r, and ε0.
QUESTION 1
Problem:
An infinitely long cylindrical conductor has radius R and uniform surface charge density o. In terms of R and o,
what is the charge per unit length A for the cylinder?
Answer:
A = 2
Density, density, density. (a) A charge -328e is uniformly distributed along a circular arc of radius 6.00 cm, which subtends an angle of 72°. What is the linear charge density along the arc? (b) A charge -328e is
uniformly distributed over one face of a circular disk of radius 3.50 cm. What is the surface charge density over that face? (c) A charge -328e is uniformly distributed over the surface of a sphere of radius 2.00 cm. What
is the surface charge density over that surface? (d) A charge -328e is uniformly spread through the volume of a sphere of radius 3.30 cm. What is the volume charge density in that sphere?
(a) Number
Units
(b) Number
Units
(c) Number
Units
(d) Number
Units
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Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning