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Tuesday, January 27, 2009
Mastering Physics Solutions Use Wisely
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The Electric Field of a Ball of Uniform Charge Density
A solid ball of radius
r_b
has a uniform charge density
rho
.
Part A
What is the magnitude of the electric field
E(r)
at a distance r_b" title="" v:shapes="_x0000_i1028" align="middle"
border="0" height="16" width="38">from the center of the ball?
Express your answer in terms of
rho
,
r_b
,
r
, and
epsilon_0
.
The Electric Field of a Ball of Uniform Charge Density
Part A
What is the magnitude of the electric field
E(r)
at a distance r_b" title="" v:shapes="_x0000_i1484" align="middle"
border="0" height="16" width="38">from the center of the ball?
Hint 1. Gauss's law
Gauss's law can be written as
\oint \vec{E} \cdot d \vec{A} = \frac{q_{\rm encl}}{\epsilon_0 }
,
where
d\vec A
refers to an infinitesimal element of an imaginary Gaussian surface,
q_encl
is the net charge enclosed in the
Gaussian surface, and
epsilon_0
is the permittivity of free space. Always choose a Gaussian surface that matches the symmetry
of the problem. Implicit in the question "what is
E(r)
?" is the assumption that the electric field depends only on distance from
the origin, which is also the center of the charged ball. Also, the ball has uniform charge density. Therefore, the electric field
must point either radially outward or radially inward, since by symmetry there is no possibility for the electric field to point in
any other direction. Given the symmetry of this problem, the best Gaussian surface to use is a sphere centered at the origin.
Since the electric field is the same at all points on this surface, the constant
E(r)
can be "pulled out" of the integrand. The left
side of Gauss's law reduces to
E(r)A(r)
, where
A(r)
is the surface area of a sphere with radius
r
.
Hint 2. Find
q_encl
The
q_encl
in Gauss's law refers to the net charge enclosed inside the Gaussian surface. What is
q_encl
here?
Hint 1. What is the volume of the sphere?
If a body has uniform charge density
rho
, the charge in a volume
V
is
\rho V
(this formula is the same as that for the mass of a
sphere of uniform mass density). What is the volume of a sphere with radius
r
?
Express your answer in terms of
pi
and
r
.
V
=
(4/3)*pi*r^3
Express your answer in terms of
rho
,
pi
, and
r_b
.
q_encl
=
(4/3)*pi*r_b^3*rho
Express your answer in terms of
rho
,
r_b
,
r
, and
epsilon_0
.
E(r)
=
{\rho}{\cdot}\frac{\left(r_{b}{^{3}}\right)}{3{\cdot}epsilon_{0}{\cdot}r^{2}}
Correct
my answers
The Heavenly Messenger: Mastering Physics Solutions Use Wisely
http://rocketscient1st.blogspot.com/2009/01/masteringphysicssolutions...
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2/6/2011 8:28 PM
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ANSWER:
E(r)
=
{\rho}{\cdot}\frac{\left(r_{b}{^{3}}\right)}{3{\cdot}epsilon_{0}{\cdot}r^{2}}
Notice that this result is identical to that reached by applying Coulomb's law to a point charge centered at the origin with
q=\rho V
. The field outside of a uniformly charged sphere does not depend on the size of the sphere, only on its charge. A
uniformly charged sphere generates an electric field as if all the charge were concentrated at its center.
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 Winter '11
 MEZL,CARRIER
 Electrostatics, Electric charge, charge density, Heavenly Messenger

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