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<title>Cylindrical and Spherical Coordinate Systems</title>
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<p class=3DMsoTitle>Cylindrical and Spherical Coordinate Systems<o:p></o:p>=
</p>

<p class=3DMsoTitle><o:p>&nbsp;</o:p></p>

<p class=3DMsoTitle align=3Dleft style=3D'text-align:left'>Cylindrical Syst=
em:<o:p></o:p></p>

<p class=3DMsoNormal><span style=3D'font-size:10.0pt;mso-bidi-font-size:12.=
0pt;
font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p>=
</span></p>

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0pt;
font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p>=
</span></p>

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<p class=3DMsoNormal><span style=3D'font-size:10.0pt;mso-bidi-font-size:12.=
0pt;
font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p>=
</span></p>

<p class=3DMsoNormal><span style=3D'font-size:10.0pt;mso-bidi-font-size:12.=
0pt;
font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p>=
</span></p>

<p class=3DMsoNormal><span style=3D'font-size:10.0pt;mso-bidi-font-size:12.=
0pt;
font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p>=
</span></p>

<p class=3DMsoNormal><span style=3D'font-size:10.0pt;mso-bidi-font-size:12.=
0pt;
font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p>=
</span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'>The<span style=3D'color:navy'> gradient operator</span> =
in the
cylindrical coordinates<o:p></o:p></span></b></p>

<p class=3DMsoNormal><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.=
0pt;
font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p>=
</span></p>

<p class=3DMsoNormal><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.=
0pt'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>&Nt=
ilde;</span></b><span
class=3DSpellE><b style=3D'mso-bidi-font-weight:normal'><sub><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>cyl</span></sub></b></=
span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>=3D <span class=3DGram=
E><b
style=3D'mso-bidi-font-weight:normal'><i style=3D'mso-bidi-font-style:norma=
l'>r</i></b><sub><span
style=3D'mso-spacerun:yes'>&nbsp; </span></sub>(</span></span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>&pa=
ra;</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'> /</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>&pa=
ra;</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>r) + </span><b
style=3D'mso-bidi-font-weight:normal'><i style=3D'mso-bidi-font-style:norma=
l'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>q</=
span></i></b><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp; </span>(1/r)<sub> </sub>(</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>&pa=
ra;</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'> /</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>&pa=
ra;q</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>) + </span><b
style=3D'mso-bidi-font-weight:normal'><i style=3D'mso-bidi-font-style:norma=
l'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'>z</span></i></b><span style=3D'font-size:14.0pt;mso-bidi=
-font-size:
12.0pt'> (</span><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;
font-family:Symbol'>&para;</span><span style=3D'font-size:14.0pt;mso-bidi-f=
ont-size:
12.0pt'> /</span><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;
font-family:Symbol'>&para;</span><span style=3D'font-size:14.0pt;mso-bidi-f=
ont-size:
12.0pt'>z)<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.=
0pt'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal>.<o:p></o:p></p>

<h1><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:A=
rial;
mso-bidi-font-family:"Times New Roman"'>The<span style=3D'color:navy'> line
element</span> in the cylindrical coordinates <o:p></o:p></span></h1>

<h1><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:A=
rial;
mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></h1>

<h1><span class=3DGramE><span style=3D'font-size:14.0pt;mso-bidi-font-size:=
12.0pt;
font-family:Arial;mso-bidi-font-family:"Times New Roman"'>dl</span></span><=
span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'> </span><span style=3D'font-size:14.0pt;mso-bidi-font-si=
ze:
12.0pt;font-family:Arial;mso-bidi-font-family:"Times New Roman";font-weight:
normal;mso-bidi-font-weight:bold'>=3D</span><span style=3D'font-size:14.0pt;
mso-bidi-font-size:12.0pt;font-family:Arial;mso-bidi-font-family:"Times New=
 Roman"'>
r </span><span class=3DSpellE><span style=3D'font-size:14.0pt;mso-bidi-font=
-size:
12.0pt;font-family:Arial;mso-bidi-font-family:"Times New Roman";font-weight:
normal;mso-bidi-font-weight:bold'>dr</span></span><span style=3D'font-size:=
14.0pt;
mso-bidi-font-size:12.0pt;font-family:Arial;mso-bidi-font-family:"Times New=
 Roman";
font-weight:normal;mso-bidi-font-weight:bold'> + (</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>q</=
span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-b=
idi-font-family:
"Times New Roman";font-weight:normal;mso-bidi-font-weight:bold'>/r) d</span=
><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol;
font-weight:normal;mso-bidi-font-weight:bold'>q</span><span style=3D'font-s=
ize:
14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-bidi-font-family:"Ti=
mes New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp; </span></span><span style=3D'font-size:14=
.0pt;
mso-bidi-font-size:12.0pt;font-family:Arial;mso-bidi-font-family:"Times New=
 Roman";
font-weight:normal;mso-bidi-font-weight:bold'>+</span><span style=3D'font-s=
ize:
14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-bidi-font-family:"Ti=
mes New Roman"'>
z </span><span class=3DSpellE><span style=3D'font-size:14.0pt;mso-bidi-font=
-size:
12.0pt;font-family:Arial;mso-bidi-font-family:"Times New Roman";font-weight:
normal;mso-bidi-font-weight:bold'>dz</span></span><span style=3D'font-size:=
14.0pt;
mso-bidi-font-size:12.0pt;font-family:Arial;mso-bidi-font-family:"Times New=
 Roman";
font-weight:normal;mso-bidi-font-weight:bold'><o:p></o:p></span></h1>

<h1><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:A=
rial;
mso-bidi-font-family:"Times New Roman";font-weight:normal;mso-bidi-font-wei=
ght:
bold'><o:p>&nbsp;</o:p></span></h1>

<h1><i style=3D'mso-bidi-font-style:normal'><u><span style=3D'font-size:14.=
0pt;
mso-bidi-font-size:12.0pt'>Note</span></u></i><span style=3D'font-size:14.0=
pt;
mso-bidi-font-size:12.0pt'>: r, </span><span style=3D'font-size:14.0pt;
mso-bidi-font-size:12.0pt;font-family:Symbol'>q</span><span style=3D'font-s=
ize:
14.0pt;mso-bidi-font-size:12.0pt'>, <span class=3DGramE>z</span> </span><sp=
an
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-weight:normal;
mso-bidi-font-weight:bold'>are the associated unit vectors</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'> <br clear=3Dall style=3D'page-break-before:always'>
Spherical System:<o:p></o:p></span></h1>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><!--[if gte=
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 <v:imagedata src=3D"CylindricalandSphericalCoordinateSystems_files/image00=
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  o:title=3D"cyl2"/>
</v:shape><![endif]--><![if !vml]><img width=3D290 height=3D257
src=3D"CylindricalandSphericalCoordinateSystems_files/image003.gif" v:shape=
s=3D"_x0000_i1026"><![endif]><o:p></o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.=
0pt;
font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p>=
</span></p>

<p class=3DMsoNormal><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.=
0pt;
font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p>=
</span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'>The<span style=3D'color:navy'> gradient operator</span> =
in the
spherical coordinates</span><o:p></o:p></b></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'><o:=
p>&nbsp;</o:p></span></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>&Nt=
ilde;</span></b><span
class=3DSpellE><b style=3D'mso-bidi-font-weight:normal'><sub><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>sph</span></sub></b></=
span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>=3D <b style=3D'mso-bi=
di-font-weight:
normal'><i style=3D'mso-bidi-font-style:normal'>r</i></b><sub> </sub>(</spa=
n><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>&pa=
ra;</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'> /</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>&pa=
ra;</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>r) + </span><b
style=3D'mso-bidi-font-weight:normal'><i style=3D'mso-bidi-font-style:norma=
l'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>f</=
span></i></b><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'> (1/r)<sub> </sub>(</s=
pan><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>&pa=
ra;</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'> /</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>&pa=
ra;f</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>) + </span><b
style=3D'mso-bidi-font-weight:normal'><i style=3D'mso-bidi-font-style:norma=
l'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>q</=
span></i></b><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'> (1/[r<sub> </sub>sin<=
/span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>f</=
span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>])<sub> </sub>(</span>=
<span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>&pa=
ra;</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'> /</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>&pa=
ra;q</span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>)<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.=
0pt'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'><o:p>&nbsp;</o:p></span></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'>The<span style=3D'color:navy'> line element</span> in the
spherical coordinates<o:p></o:p></span></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'><o:p>&nbsp;</o:p></span></b></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<h1><span class=3DGramE><span style=3D'font-size:14.0pt;mso-bidi-font-size:=
12.0pt;
font-family:Arial;mso-bidi-font-family:"Times New Roman"'>dl</span></span><=
span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'> </span><span style=3D'font-size:14.0pt;mso-bidi-font-si=
ze:
12.0pt;font-family:Arial;mso-bidi-font-family:"Times New Roman";font-weight:
normal;mso-bidi-font-weight:bold'>=3D</span><span style=3D'font-size:14.0pt;
mso-bidi-font-size:12.0pt;font-family:Arial;mso-bidi-font-family:"Times New=
 Roman"'>
r </span><span class=3DSpellE><span style=3D'font-size:14.0pt;mso-bidi-font=
-size:
12.0pt;font-family:Arial;mso-bidi-font-family:"Times New Roman";font-weight:
normal;mso-bidi-font-weight:bold'>dr</span></span><span style=3D'font-size:=
14.0pt;
mso-bidi-font-size:12.0pt;font-family:Arial;mso-bidi-font-family:"Times New=
 Roman";
font-weight:normal;mso-bidi-font-weight:bold'> +<span
style=3D'mso-spacerun:yes'>&nbsp; </span>r </span><span style=3D'font-size:=
14.0pt;
mso-bidi-font-size:12.0pt;font-family:Symbol'>f</span><span style=3D'font-s=
ize:
14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-bidi-font-family:"Ti=
mes New Roman";
font-weight:normal;mso-bidi-font-weight:bold'> d</span><span style=3D'font-=
size:
14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol;font-weight:normal;
mso-bidi-font-weight:bold'>f</span><span style=3D'font-size:14.0pt;mso-bidi=
-font-size:
12.0pt;font-family:Arial;mso-bidi-font-family:"Times New Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp; </span></span><span style=3D'font-size:14=
.0pt;
mso-bidi-font-size:12.0pt;font-family:Arial;mso-bidi-font-family:"Times New=
 Roman";
font-weight:normal;mso-bidi-font-weight:bold'>+</span><span style=3D'font-s=
ize:
14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-bidi-font-family:"Ti=
mes New Roman"'>
</span><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-famil=
y:
Symbol'>q </span><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;
font-family:Arial;mso-bidi-font-family:"Times New Roman";font-weight:normal;
mso-bidi-font-weight:bold'>r <span class=3DSpellE>sin<span style=3D'font-fa=
mily:
Symbol'>f </span>d</span></span><span style=3D'font-size:14.0pt;mso-bidi-fo=
nt-size:
12.0pt;font-family:Symbol;font-weight:normal;mso-bidi-font-weight:bold'>q</=
span><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Arial;mso-b=
idi-font-family:
"Times New Roman";font-weight:normal;mso-bidi-font-weight:bold'><o:p></o:p>=
</span></h1>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'><o:p>&nbsp;</o:p></spa=
n></b></p>

<p class=3DMsoNormal><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.=
0pt'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal><i style=3D'mso-bidi-font-style:normal'><u><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>Note</span></u></i><sp=
an
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>: <b style=3D'mso-bidi=
-font-weight:
normal'>r, </b></span><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>q</=
span></b><b
style=3D'mso-bidi-font-weight:normal'><span style=3D'font-size:14.0pt;mso-b=
idi-font-size:
12.0pt'>, </span></b><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;font-family:Symbol'>f</=
span></b><sub><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'> </span></sub><span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>are the associated unit
vectors</span>. <span style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'>=
Also
note the change in the </span><span style=3D'font-size:14.0pt;mso-bidi-font=
-size:
12.0pt;font-family:Symbol'>q </span><span style=3D'font-size:14.0pt;mso-bid=
i-font-size:
12.0pt'>and</span><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt;
font-family:Symbol'> <span style=3D'mso-spacerun:yes'>&nbsp;</span>f</span>=
<span
style=3D'font-size:14.0pt;mso-bidi-font-size:12.0pt'> directions for the
spherical system, as compared with the cylindrical system.<o:p></o:p></span=
></p>

<span style=3D'font-size:14.0pt;mso-bidi-font-size:10.0pt;font-family:Arial;
mso-fareast-font-family:"Times New Roman";mso-ansi-language:EN-US;mso-farea=
st-language:
EN-US;mso-bidi-language:AR-SA'><br clear=3Dall style=3D'page-break-before:a=
lways'>
</span>

<p class=3DMsoTitle>Practice Problems<o:p></o:p></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><span class=
=3DGramE><b
style=3D'mso-bidi-font-weight:normal'><i style=3D'mso-bidi-font-style:norma=
l'><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>Cylindri=
cal
and Spherical Coordinate Systems.</span></i></b></span><b style=3D'mso-bidi=
-font-weight:
normal'><i style=3D'mso-bidi-font-style:normal'><span style=3D'font-family:=
Arial;
mso-bidi-font-family:"Times New Roman"'><o:p></o:p></span></i></b></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><b style=3D=
'mso-bidi-font-weight:
normal'><i style=3D'mso-bidi-font-style:normal'><span style=3D'font-family:=
Arial;
mso-bidi-font-family:"Times New Roman"'><o:p>&nbsp;</o:p></span></i></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>Problem =
#1</span></u></b><span
class=3DGramE><i style=3D'mso-bidi-font-style:normal'><span style=3D'font-f=
amily:
Arial;mso-bidi-font-family:"Times New Roman"'>:</span></i><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>Determin=
e</span></span><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'> the pot=
ential
difference in moving a 5-microcoulomb charge from the origin (r, </span><sp=
an
style=3D'font-family:Symbol'>q, f</span><span style=3D'font-family:Arial;
mso-bidi-font-family:"Times New Roman"'>) =3D (0,0,0) to a point (2m, </spa=
n><span
style=3D'font-family:Symbol'>p</span><span style=3D'font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'>/2, </span><span style=3D'font-family:Symbol'>p</span><s=
pan
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>/4) agai=
nst
the electric field<b style=3D'mso-bidi-font-weight:normal'> <o:p></o:p></b>=
</span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>E</span>=
</b><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'> =3D -16=
r<sup>2</sup><b
style=3D'mso-bidi-font-weight:normal'>r</b> + (10/rsin </span><span
style=3D'font-family:Symbol'>f</span><span style=3D'font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'>) </span><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-family:Symbol'>q</span></b><span style=3D'font-family:Arial;
mso-bidi-font-family:"Times New Roman"'> (V/m).<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-family:Arial;mso-bidi-font-family:=
"Times New Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>Solution=
:<o:p></o:p></span></u></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p><sp=
an
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p><sp=
an
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p><sp=
an
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>Problem =
#2:</span></u></b><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'> For a l=
ine
charge with linear charge density </span><span style=3D'font-family:Symbol'=
>l</span><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'> =3D (10=
<sup>-9</sup>/2)
C/m on the z-axis, determine the potential difference V<sub>AB</sub>, where=
 A
is (r, </span><span style=3D'font-family:Symbol'>q</span><span style=3D'fon=
t-family:
Arial;mso-bidi-font-family:"Times New Roman"'>, z) =3D (2m, </span><span
style=3D'font-family:Symbol'>p</span><span style=3D'font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'>/2<span class=3DGramE>,0</span>) and B is (4m,</span><sp=
an
style=3D'font-family:Symbol'>p</span><span style=3D'font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'>, 5m).<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-family:Arial;mso-bidi-font-family:=
"Times New Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>Solution=
:<o:p></o:p></span></u></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p><sp=
an
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p><sp=
an
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p><sp=
an
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>Problem =
#3:</span></u></b><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'> Given t=
he
potential function V (r, </span><span style=3D'font-family:Symbol'>q</span>=
<span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>, z) =3D=
 6r<sup>3</sup>z<sup>2</sup>cos</span><span
style=3D'font-family:Symbol'>q</span><span style=3D'font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'>, determine the <b style=3D'mso-bidi-font-weight:normal'=
>E </b>field
at the point in space (r, </span><span style=3D'font-family:Symbol'>q</span=
><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>, z) =3D=
 (1m, </span><span
style=3D'font-family:Symbol'>p</span><span style=3D'font-family:Arial;mso-b=
idi-font-family:
"Times New Roman"'>/6, 1m).<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-family:Arial;mso-bidi-font-family:=
"Times New Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>Solution=
:<o:p></o:p></span></u></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p><sp=
an
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p><sp=
an
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p><sp=
an
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>Problem =
#4:</span></u></b><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'> A spher=
ical conducting
shell of radius a, centered at the origin, has a potential field<o:p></o:p>=
</span></p>

<p class=3DMsoNormal><span style=3D'font-family:Arial;mso-bidi-font-family:=
"Times New Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><sub><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'><!--[if =
gte vml 1]><v:shape
 id=3D"_x0000_i1027" type=3D"#_x0000_t75" style=3D'width:111pt;height:90pt'
 fillcolor=3D"window">
 <v:imagedata src=3D"CylindricalandSphericalCoordinateSystems_files/image00=
4.wmz"
  o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img width=3D148 height=3D120
src=3D"CylindricalandSphericalCoordinateSystems_files/image005.gif" v:shape=
s=3D"_x0000_i1027"><![endif]></span></sub><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p></o=
:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'><o:p>&nb=
sp;</o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-family:Arial;mso-bidi-font-family:=
"Times New Roman"'>Determine
the electric field both inside and outside the shell.<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-family:Arial;mso-bidi-font-family:=
"Times New Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-family:Arial;mso-bidi-font-family:"Times New Roman"'>Solution=
:<o:p></o:p></span></u></b></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-size:14=
.0pt;
mso-bidi-font-size:12.0pt;line-height:200%'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-size:14=
.0pt;
mso-bidi-font-size:12.0pt;line-height:200%'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-size:14=
.0pt;
mso-bidi-font-size:12.0pt;line-height:200%'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-size:14=
.0pt;
mso-bidi-font-size:12.0pt;line-height:200%'>Additional information on
coordinate systems may be found at:<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.=
0pt'><a
href=3D"http://mathworld.wolfram.com/topics/CoordinateGeometry.html">http:/=
/mathworld.wolfram.com/topics/CoordinateGeometry.html</a><o:p></o:p></span>=
</p>

<p class=3DMsoNormal><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.=
0pt'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:14.0pt;mso-bidi-font-size:12.=
0pt'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

</div>

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