Electric field due to a charged disk

[Noctisdark]

Homework Statement

I'm reading Griffith-Introduction to electrodynamics, In chapter 2 about electrostatics, I've encounter few problems that I've managed, to solve (luck !), I'm asked to calculate the electric field due to a charged disk of radius R in a point P above the center (Pic)

Homework Equations

Electric field due to uniform continuous charge distrubition
$$\vec E = \frac {1}{4 \pi \epsilon _0} \int \int \frac {\sigma \hat{r}}{r^2}dS$$

The Attempt at a Solution

I've tried to use the usual approach, that is cutting the disk into little circles, $\lambda = \frac {Q}{L} = \frac{\sigma A }{L} = \frac{\sigma 2 \pi r dr}{2 \pi r} = \sigma \cdot dr$
The electric field due to that circle is $|\vec E_z| = \frac {\lambda}{4\pi\epsilon_0} \int \frac{dArc}{(R^2 + z^2)^\frac{3}{2}} = \frac {\lambda}{4\pi\epsilon_0} \int_0^{2\pi}\frac {zR \cdot d\theta}{(R^2 + z^2)^\frac{3}{2}}$,You just multiply by 2 $\pi$ and sub the $\lambda$ , $|\vec E_z| = \frac{\sigma zR \cdot dr}{2\pi\epsilon_0}\frac{1}{(R^2 + z^2)^\frac{3}{2}}$, I should have written$d\vec E_z$ because we'll integrate that from 0 up to R, so $$\vec E_z = \frac{\sigma}{2\pi\epsilon_0(R^2 + z^2)^\frac{3}{2}} \int_0^zR\cdot dr = \frac{\sigma zR^2}{2\pi\epsilon_0(R^2 + z^2)^\frac{3}{2}}$$, I know that this is wrong, so can someone tell me where I've messed things up ?
One more Request, where can I find more problems about electrostatics, I need some practice Thanks !

 

[blue_leaf77]

First, in deriving the E field due to the circle, why is there $r$ when you have already defined the circle radius to be $R$? Second you should check again the correct expression for the integrand of the E field due to a circle, for instance I guess you know that all E field components perpendicular to the circle's axis cancel out for any point on the axis. This gives us a cosine of the half-angle of the cone formed by the observation point and the circle, what's the expression for this cosine?

 

[Noctisdark]

I I've missed that, thanks ! I will Edit that right away, I've included the expression of the cosine in $\frac{1}{(R^2 + z^2)^{\frac{3}{2}}}$

 

[blue_leaf77]

I I've missed that, thanks ! I will Edit that right away, I've included the expression of the cosine in $\frac{1}{(R^2 + z^2)^{\frac{3}{2}}}$

Yes I know that but don't you think you missed one thing in the numerator?

 

[Noctisdark]

Yes I know that but don't you think you missed one thing in the numerator?

Aw there a z missing,

 

[blue_leaf77]

And finally in the last integral, the integration element should be $dR$ right, which means any $R$ must be inside the integral and to be integrated along.

 

[Noctisdark]

I think it should be $dr$, because I've followed the approach that cuts the circle into little parts, from $r \text{ to } r+dr$,

 

[blue_leaf77]

I think it should be $dr$, because I've followed the approach that cuts the circle into little parts, from $r \text{ to } r+dr$,

Nope, remember in your approach initially you wanted to divide the disk into co-centric rings and calculate the electric field due to a single ring with radius $R$, let's denote this circle's E field $E_{ring}$. Having found this (hopefully you did it correctly), you still have to add contributions due to the other rings constituting the original disk. Which means you have to add up rings with varying radius from zero out to the radius of the original disk, let's denote it by $R_{disk}$.
May be what keeps confusing you is that you haven't changed the $r$ in the expression of $\lambda$.

 

[Noctisdark]

Aw, I've missed that too, In fact I intended to make $r$ the radius of the rings but I wrote $R$ and sticked with it, Okey I'm going to edit that two, I think I wrote a wrong $\lambda$, it's dR not dr too

 

[Noctisdark]

$\lambda = \sigma dR$
$\vec E= \frac{\sigma z}{2\pi\epsilon_0} \int_0^r \frac{RdR}{(R^2 + z^2)^{\frac{3}{2}}} = \frac{-1}{\sqrt{R^2 + z^2}}$ evaluate from 0 to r and I get $|\vec E| = \frac{\sigma z}{2\pi\epsilon_0}\cdot(\frac{1}{\sqrt{r^2 + z^2}} - \frac{1}{z})$

 

[blue_leaf77]

Only some small mistakes left there, first you should check again whether $\pi$ should be present and why the magnitude of the E field is negative.

 

[Noctisdark]

$|\vec E| = \frac{\sigma}{2\epsilon_0}\cdot(1 - \frac{z}{\sqrt{r^2 + z^2}})$ Waow thank you very for your assistance :)

 

[blue_leaf77]

You are welcome.