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?
A non conducting solid cylinder of radius 8.4 cm, length 14.2 cm and volume charge density ρ =b/r, where b = -12 μC/m2, and r is the radial distance in m. The total charge on the cylinder is:
A solid conducting sphere of radius R has a uniform charge distribution, with a density = Ps * r / R where Ps is a constant and r the distance from the center of the sphere. Prove
a) the total charge on the sphere is Q = πPsR ^ 3
b) the electric field of the sphere is given by E = (1 / 4πε0) * (Q / R ^ 4) * (r ^ 2)
Calculate the surface charge density on the outer surface of the spherical conducting shell.
Answer Choices: 0.0096, 2.7, 0.21, 0.35, 0.28.
Units are C/m^2
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