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?
Problems 11-13 refer to the following situation. A nonuniform, but spherically symmetric,
distribution of charge has a charge density p(r) given as follows:
Problem 11:
What is the constant po?
Q
πR³
a.
Po and R are positive constants. The total charge of the distribution is Q.
Problem 12:
What is the E-field for r R?
1 Q
b.
4περ 12
a.
p(r)
= {Po (7) .
20
πR³
1 Qr²
4περ R4
1 QR
4περ 13
C.
C.
r R
C.
30
πR³
1 Qr³
4περ R5
QR²
1
4περ 14
d.
d.
d.
4Q
πR³
1 Qr4
4πεο R6
QR³
1
Απο 15
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.
Problem:
An infinitely long cylindrical conductor has radius R and uniform surface charge density Ơ. In terms of R and o, what is the
charge per unit length A for the cylinder?
Answer:
A = 2
Chapter 22 Solutions
University Physics with Modern Physics (14th Edition)
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