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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 d

**a**as above, what do I do with

**E**, which is in spherical coordinates?

I hope you can help me.