22.53 CALC A nonuniform, but spherically symmetric, dis-tribution of charge has a charge density p(r) given as follows:Ho-(1-2rp(r)= Pofor r RRfor r RP(r) 0where po 30/TR is a positive constant. (a) Show that the totalcharge contained in the charge distribution is Q. (b) Show that theelectric field in the region r 2 R is identical to that produced by apoint charge Q at r = 0. (c) Obtain an expression for the electricfield in the region r = R. (d) Graph the electric-field magnitude E(e) Find the value of r at which the electric fieldis maximum, and find the value of that maximum field.as a function of r.slab of insulatingnar-22.5 22.56 CALC A nonuniform, but spherically symmetric, distri-bution of charge has a charge density p(r) given as follows:4rP(r) = Afor r R3R,P(r)= 0for r Rwhere po is a positive constant. (a) Find the total charge containedin the charge distribution. Obtain an expression for the electricfield in the region (b) r z R; (c) rs R. (d) Graph the electric-field magnitude E as a function of r. (e) Find the value of r atwhich the electric field is maximum, and find the value of thatmaximum field.

Question
Asked Sep 14, 2019
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(Problems in images below.)

Both problems require a volume charge integration of the given charge density functions. Technically, a volume integration is a triple integral. However, since the charge density is spherically symmetric, this reduces to a one-dimensional integration.

22.53 CALC A nonuniform, but spherically symmetric, dis-
tribution of charge has a charge density p(r) given as follows:
Ho-(1-2
r
p(r)= Po
for r R
R
for r R
P(r) 0
where po 30/TR is a positive constant. (a) Show that the total
charge contained in the charge distribution is Q. (b) Show that the
electric field in the region r 2 R is identical to that produced by a
point charge Q at r = 0. (c) Obtain an expression for the electric
field in the region r = R. (d) Graph the electric-field magnitude E
(e) Find the value of r at which the electric field
is maximum, and find the value of that maximum field.
as a function of r.
slab of insulating
nar-
22.5
help_outline

Image Transcriptionclose

22.53 CALC A nonuniform, but spherically symmetric, dis- tribution of charge has a charge density p(r) given as follows: Ho-(1-2 r p(r)= Po for r R R for r R P(r) 0 where po 30/TR is a positive constant. (a) Show that the total charge contained in the charge distribution is Q. (b) Show that the electric field in the region r 2 R is identical to that produced by a point charge Q at r = 0. (c) Obtain an expression for the electric field in the region r = R. (d) Graph the electric-field magnitude E (e) Find the value of r at which the electric field is maximum, and find the value of that maximum field. as a function of r. slab of insulating nar- 22.5

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22.56 CALC A nonuniform, but spherically symmetric, distri-
bution of charge has a charge density p(r) given as follows:
4r
P(r) = A
for r R
3R,
P(r)= 0
for r R
where po is a positive constant. (a) Find the total charge contained
in the charge distribution. Obtain an expression for the electric
field in the region (b) r z R; (c) rs R. (d) Graph the electric-
field magnitude E as a function of r. (e) Find the value of r at
which the electric field is maximum, and find the value of that
maximum field.
help_outline

Image Transcriptionclose

22.56 CALC A nonuniform, but spherically symmetric, distri- bution of charge has a charge density p(r) given as follows: 4r P(r) = A for r R 3R, P(r)= 0 for r R where po is a positive constant. (a) Find the total charge contained in the charge distribution. Obtain an expression for the electric field in the region (b) r z R; (c) rs R. (d) Graph the electric- field magnitude E as a function of r. (e) Find the value of r at which the electric field is maximum, and find the value of that maximum field.

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Expert Answer

Step 1

a) Total charge is,

Qu
(r)dV
--()
4 T03
= 4,7 30|R3 3
TR
3
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Qu (r)dV --() 4 T03 = 4,7 30|R3 3 TR 3

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Step 2

b) Using Gauss law,

E-dA=
E(4m2=2
EO
1
E =
4,Te0
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Image Transcriptionclose

E-dA= E(4m2=2 EO 1 E = 4,Te0

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Step 3

c) Total char...

О.в.
(r)dV
4 TO
—4 З0 )Гз
IR
4R
3r
R
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Image Transcriptionclose

О.в. (r)dV 4 TO —4 З0 )Гз IR 4R 3r R

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