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Niles
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[SOLVED] Electrodynamics and electrical fields
The question is: A long cylinder carries a charge density that is proportional to the distance from the axis ρ =kr, where k is a constant, r is the distance from the axis. Find electric field inside the cylinder.
My attempt: Ok, first of all we know from Gauss' law that [tex]\oint_S {{\bf{E}} \cdot d{\bf{a}}} = \frac{1}{{\varepsilon _0 }}Q[/tex].
We also know that [tex]Q = \int {\rho dV = } \int {k \cdot s \cdot s \cdot dsd\phi dz = \frac{2}{3}\pi kls^3 }[/tex]. Here I have used a Gaussian cylinder of length l and radius s.
Now I must find [tex]\oint_S {{\bf{E}} \cdot d{\bf{a}}} [/tex].
First question: The electric field E is given by [tex]{\bf{E}} = \frac{1}{{4\pi \varepsilon _0 }}\frac{q}{{r^2 }}{\bf{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}
\over r} }}[/tex]. Is the unit vector r in this case the spherical radius-vector?
For this cylinder, I write the parametric and differentiate it with respect to phi and z. Then I find the normal-vector as the cross-products of these two.
Question two: When I find the element da as above, what do I do with E, which is in spherical coordinates?
I hope you can help me.
The question is: A long cylinder carries a charge density that is proportional to the distance from the axis ρ =kr, where k is a constant, r is the distance from the axis. Find electric field inside the cylinder.
My attempt: Ok, first of all we know from Gauss' law that [tex]\oint_S {{\bf{E}} \cdot d{\bf{a}}} = \frac{1}{{\varepsilon _0 }}Q[/tex].
We also know that [tex]Q = \int {\rho dV = } \int {k \cdot s \cdot s \cdot dsd\phi dz = \frac{2}{3}\pi kls^3 }[/tex]. Here I have used a Gaussian cylinder of length l and radius s.
Now I must find [tex]\oint_S {{\bf{E}} \cdot d{\bf{a}}} [/tex].
First question: The electric field E is given by [tex]{\bf{E}} = \frac{1}{{4\pi \varepsilon _0 }}\frac{q}{{r^2 }}{\bf{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}
\over r} }}[/tex]. Is the unit vector r in this case the spherical radius-vector?
For this cylinder, I write the parametric and differentiate it with respect to phi and z. Then I find the normal-vector as the cross-products of these two.
Question two: When I find the element da as above, what do I do with E, which is in spherical coordinates?
I hope you can help me.