What is the Electric Field of a Ring with Uniform Surface Charge Density?

[vorcil]

Homework Statement

A thin, circular ring of inner radius a and outer radius b carries a uniform surface charge density \sigma

(i) Find an expression for E at a point on an axis perpendicular to the plane of the disk, the axis passes through the centre of the disk.

(ii) Keeping the surface charge density the same as that in part (i) find the electric field, E, as a->o and b->infinity

Homework Equations

The usual ones, e.g \frac{1}{4\pi \epsilon_o} \frac{q}{r^2}\hat{r}

The Attempt at a Solution

I can't really show you what I've drawn for a picture,
but imagine a circular disk, with the normal being the z axis, with a hole in the middle,( like a dvd disk)
where the distance from the center of the disk to the inner part is radius a, and to the edge of the disk, radius b

solving for the Electric field:
where sigma is the charge surface density of the disk
and R is the radius

and varsigma \varsigma \sqrt{R^2 + z^2 }
and cos \theta = \frac{R}{\varsigma}dE = \frac{1}{4\pi \epsilon_o} \sigma \frac{1}{\varsigma^2} cos \theta \hat{z}

I think I also have to multiply the equation by 2\pi R but I'm not too sure why, it's just intuition,

making it

dE = \frac{2\pi R}{4\pi \epsilon_o} \sigma \frac{1}{\varsigma^2} cos \theta \hat{z}

substituting in the varsigma and the cos theta,

dE = \frac{2\pi R }{4\pi \epsilon_o} \sigma \frac{1}{R^2+z^2} \frac{R}{\sqrt{R^2+z^2}}\hat{z}

integrating(and taking out the constant)

\bf{E} = \frac{2\pi R }{4\pi \epsilon_o} \sigma \int \frac{1}{R^2+z^2} \frac{z}{\sqrt{R^2+z^2}}\hat{z}

\bf{E} = \frac{2\pi R }{4\pi \epsilon_o} \sigma \int \frac{1}{R^2+z^2} \frac{R}{\sqrt{R^2+z^2}}\hat{z}

\bf{E} = \frac{2\pi R }{4\pi \epsilon_o} \sigma \int \frac{1}{(R^2+z^2)^{\frac{3}{2}}}

\bf{E} = \frac{2\pi R }{4\pi \epsilon_o} \sigma \frac{R}{z^2\sqrt{R^2+z^2}} |_a^b

now I'm pretty sure this is the general result for a disk, not a disk with a hole,

but I've been given the inner and outer radii, so I think I can just take the two disks and subtract the smaller disk from the larger disk to get the electric field of the ring (because electric fields and potential can be super imposed)

giving me the final expression for the electric field strength E, at a point on the Z axis (in the normal direction)

\bf{E} = \frac{R \sigma}{2\epsilon_o} \frac{R}{z^2\sqrt{R^2+z^2}}|_a^b

=

\bf{E} = \frac{\sigma}{2\epsilon_o} (b-a) \frac{b-a}{z^2\sqrt{(b-a)^2+z^2}}<br /> <br /> can someone please tell me if that&#039;s an accurate expression for the electric field of a ring?<br /> <br /> cheers<br /> <br /> for part(b)<br /> letting a-&gt;0 and b-&gt;infinity,<br /> in the limits, it becomes a singular disk without a hole and an infinite radius<br /> <br /> \bf{E} = \frac{\sigma}{2\epsilon_o} \frac{1}{z^2} since all the r&#039;s cross out, this is the equation for a disk of infinite radius assuming my part(A) was correct

 

[vorcil]

Just looking at what I've done above I can see I've made a huge mess

i'm writing it out for a third time now

 

[vorcil]

[PLAIN]http://a.yfrog.com/img838/1428/diskw.jpg

by super position, all of the horizontal components cancel out

and \eta ^2 = r^2 + z^2 by Pythagoras also cos\theta = \frac{z}{\eta}

and from the definition for the electric field of a surface:
\bf{E} = \frac{1}{4 \pi \epsilon_o } { \int \frac{\lambda d\bf{l} }{\eta^2}cos\theta } \hat{z}

where the radius r, height z, charge density lambda, \bf{\eta} and cos(theta) are all constants!

the only thing that changes is the infinitesimally small increments of dl(strips of length 2pi r (small circles) ) that make up the disk and the integration

and that integral, \int d\bf{l} = 2\pi r

making the equation similar to the one I had in my first attempt above,

\frac{1}{4\pi \epsilon_o} \lambda \frac{z}{(r^2 + z^2)^3/2 } \int dl

making the final equation for a disk of radius r, the electric field a distance z, on the z axis is

\bf{E} = \frac{1}{4 \pi \epsilon_o} \frac{\lambda (2\pi r) z }{(r^2 + z^2)^{3/2} }

however this isn't the answer they were looking for BUT
if I just take two disks, one with radius A and radius B,
and subtract the smaller disk (A) from the disk B, I should get the equation for the electric field of the ring the question was asking for,

\bf{E} = \frac{1}{4 \pi \epsilon_o} \frac{\lambda (2\pi B) z }{(B^2 + z^2)^{3/2} } - \frac{1}{4 \pi \epsilon_o} \frac{\lambda (2\pi A) z }{(A^2 + z^2)^{3/2} }

and simplifying it gives me

\frac{1}{2 \epsilon_o} \left{ \frac{\lambda B z }{(B^2 + z^2)^{3/2} } - \frac{\lambda A z}{(A^2+z^2)^{3/2}} \right}

pulling out like factors i get\frac{\lambda z}{2 \epsilon_o} \left[ \frac{B}{(B^2 + z^2)^{3/2} } - \frac{A}{(A^2+z^2)^{3/2}} \right]

and I think that is the correct equation for the electric field of a disk, with inner radius A and outer radius B

 

[gabbagabbahey]

\frac{\lambda z}{2 \epsilon_o} \left[ \frac{B}{(B^2 + z^2)^{3/2} } - \frac{A}{(A^2+z^2)^{3/2}} \right]

and I think that is the correct equation for the electric field of a disk, with inner radius A and outer radius B

No, that looks more like the field of two concentric rings carrying linear charge density \lambda. Rings have infinitesimal extent in the radial direction, while disks have some finite extent in the radial direction.

When you have some surface carrying a surface charge density of \sigma, then the amount of charge on a "small" patch of that surface is \sigma da...Use that to find dE and then integrate.

 

[vorcil]

<br /> \frac{\sigma z}{2 \epsilon_o} \left[ \frac{B}{(B^2 + z^2)^{3/2} } - \frac{A}{(A^2+z^2)^{3/2}} \right] <br />

 

[gabbagabbahey]

<br /> \frac{\sigma z}{2 \epsilon_o} \left[ \frac{B}{(B^2 + z^2)^{3/2} } - \frac{A}{(A^2+z^2)^{3/2}} \right] <br />

No, that doesn't even have the correct units; \sigma has units of charge per unit area. Start by expressing the separation vector \mathbf{\eta}=\textbf{r}-\textbf{r}&#039; (script-r in your diagram) from a small patch of surface charge on the disk (say at \textbf{r}&#039;=x&#039;\hat{x}+y&#039;\hat{y} ) in cylindrical coordinates to a field point along the z-axis ( \textbf{r}=z\hat{z} )...what do you get for that? What does that make d\textbf{E}?

 

[vorcil]

[PLAIN]http://desmond.yfrog.com/Himg836/scaled.php?tn=0&server=836&filename=diskw.jpg&xsize=640&ysize=640

From the principle of super position the Electric field components E(x), E(y) all cancel out, and the only factor that remains is E(z)

dE = \frac{1}{4\pi\epsilon_o} \int_S \frac{\sigma (r&#039;)}{\eta^2} \hat{z} d\tau&#039;

then I use \sigma da ?

 

[gabbagabbahey]

[PLAIN]http://desmond.yfrog.com/Himg836/scaled.php?tn=0&server=836&filename=diskw.jpg&xsize=640&ysize=640

From the principle of super position the Electric field components E(x), E(y) all cancel out, and the only factor that remains is E(z)

dE = \frac{1}{4\pi\epsilon_o} \int_S \frac{\sigma (r&#039;)}{\eta^2} \hat{z} d\tau&#039;

then I use \sigma da ?

You need to be careful, Coulomb's law tells you

d\textbf{E}=\frac{1}{4\pi\epsilon_0}\frac{dq&#039;}{\eta^2}\hat{\mathbf{\eta}}=\frac{1}{4\pi\epsilon_0}\frac{dq&#039;}{\eta^3}\mathbf{\eta}

In this case, you have a surface charge distribution, so dq&#039;=\sigma da&#039;. What is \mathbf{\eta} in this case? What is da&#039; in cylindrical coordinates?

 

[vorcil]

Whoops I wrote it as a volume charge density

I should've done a surface charge density

<br /> d\bf{E}=\frac{1}{4\pi\epsilon_0}\frac{dq&#039;}{\eta^2}\hat{\bf{\eta}}=<br /> \frac{1}{4\pi\epsilon_o} \frac{\sigma da&#039;}{\eta^2} \hat{\bf{\eta}}

where eta is the curly R I've drawn on the graph,
I don't know what the greek symbol for it is so can't write it in latex

\eta = r - r&#039;

da' in cylindrical coordinates
I know r is specified by (s, thi, z) of a point P

x = s cos thi
y = s sin thi
z = z

i'm not sure how to get an area of a disk with a hole,
is it not just \pi R^2 - \pi r^2
where R = B(the outer radius of the disk) and r = A(the inner radius of the disk)

 

[vorcil]

<br /> <br /> d\bf{E}=\frac{1}{4\pi\epsilon_0}\frac{dq&#039;}{\eta^2} \hat{\bf{\eta}}=<br /> \frac{1}{4\pi\epsilon_o} \frac{\sigma da&#039;}{\eta^2} \hat{\bf{\eta}} <br />

integrating
also noting that
\eta^2 = r^2 + z^2

<br /> <br /> \int dE = \int \frac{1}{4\pi\epsilon_o} \frac{\sigma da&#039;}{r^2+z^2} \hat{\eta}

=

E = \frac{1}{4\pi\epsilon_o} \sigma \right[ \frac{arctan(r/z)}{z} \pi r^2 \left]_A^B
- here I've used the integral of da' to be pi r^2,

this correct?

I don't know If i should've multiplied by cos(theta) also

- if I did multiply by cos theta, before integration the equation would've become:

\int dE = \int \frac{1}{4\pi\epsilon_o} \frac{\sigma da&#039;}{(r^2+z^2)^{3/2}} \hat{\eta}E = \frac{1}{4\pi\epsilon_o} \sigma \int \frac{da&#039;}{(r^2 + z^2)^{3/2}}

E = \frac{1}{4\pi\epsilon_o} \sigma \left[\frac{\pi r^2 r}{z^2\sqrt{r^2+z^2}} \right]_A^B

E = \frac{\sigma}{4\pi\epsilon_o} \left[\frac{ \pi r^3}{z^2\sqrt{r^2+z^2}} \right]_A^B

E = \frac{\sigma \pi}{4\pi\epsilon_o z^2} \left[\frac{r^3}{\sqrt{r^2+z^2}} \right]_A^B

 

[vorcil]

Could someone please check my results?

I've come to the result:
for part a)

\bf{E} = \frac{\sigma}{4\epsilon_o} \left[ \frac{R^3}{z\sqrt{R^2+z^2}} \right]_A^B

for the electric field of a flat ring i.e a flat disk with a hole in it,
with inner radius A and outer radius B

for part b) (find the electric field in the limit that a->0 and b->infinity)

I could see that the ring just becomes a disk of infinite radius

and the electric field at a point Z above the disk is,

\frac{\sigma}{4\epsilon_o} \frac{1}{z} (i got this result after canceling out some terms and stuff)

this makes sense, because the disk has an infinite radius, so the only thing affecting the electric field strength is the height z above the disk

 

[vela]

\int dE = \int \frac{1}{4\pi\epsilon_o} \frac{\sigma da&#039;}{(r^2+z^2)^{3/2}} \hat{\eta}

E = \frac{1}{4\pi\epsilon_o} \sigma \int \frac{da&#039;}{(r^2 + z^2)^{3/2}}

In the first integral, you should have \hat{z} as the unit vector since you're only looking at the vertical component, and the electric field on the LHS should be a vector quantity. You're also missing a factor of z (from the cosine) in the numerator. For just calculating the magnitude of E, the second integral is fine except, again, the missing factor of z. At this point, you want to set da&#039;=r\,dr\,d\theta and use the appropriate limits.

I've come to the result:
for part a)

\bf{E} = \frac{\sigma}{4\epsilon_o} \left[ \frac{R^3}{z\sqrt{R^2+z^2}} \right]_A^B

for the electric field of a flat ring i.e a flat disk with a hole in it,
with inner radius A and outer radius B

If you check the units on your expression, you'll find they're not correct. You need to go back and recheck your work.

for part b) (find the electric field in the limit that a->0 and b->infinity)

I could see that the ring just becomes a disk of infinite radius

and the electric field at a point Z above the disk is,

\frac{\sigma}{4\epsilon_o} \frac{1}{z} (i got this result after canceling out some terms and stuff)

this makes sense, because the disk has an infinite radius, so the only thing affecting the electric field strength is the height z above the disk

You didn't take the limit correctly. If you let B go to infinity, the expression you derived diverges. Also, the quantity in the brackets in your answer to part (a) has units of length, yet when you took the limit, somehow you got a quantity that has units of 1/length. So something obviously went wrong.

Your physical intuition is also off, but I think if you get the correct expression and find the actual limit, you'll recognize the answer and it'll make sense.

 

[gabbagabbahey]

Whoops I wrote it as a volume charge density

I should've done a surface charge density

<br /> d\bf{E}=\frac{1}{4\pi\epsilon_0}\frac{dq&#039;}{\eta^2}\hat{\bf{\eta}}=<br /> \frac{1}{4\pi\epsilon_o} \frac{\sigma da&#039;}{\eta^2} \hat{\bf{\eta}}

Good.

where eta is the curly R I've drawn on the graph,
I don't know what the greek symbol for it is so can't write it in latex

Yeah, the Curly R that Griffiths uses in his text isn't a Greek letter. I think the \LaTeX code for it is \script{r}, but there needs to be a certain package installed for it to render properly. Let's just continue to use \eta.

\eta = r - r&#039;

You should really use \mathbf{} to make clear that a quantity is a vector.

da' in cylindrical coordinates
I know r is specified by (s, thi, z) of a point P

x = s cos thi
y = s sin thi
z = z

i'm not sure how to get an area of a disk with a hole,
is it not just \pi R^2 - \pi r^2
where R = B(the outer radius of the disk) and r = A(the inner radius of the disk)

First, you know your field point is on the z-axis, so \textbf{r}=z\hat{z}. Second, da&#039; is the differential area element of your surface. In the case of disk, that is da&#039;=s&#039;ds&#039;d\phi&#039;, since that is the area subtended when you vary s&#039; and \phi&#039; by an infinitesimal amount ds&#039; and d\phi&#039; respectively.

Thirdly, what is \textbf{r}&#039; in cylindrical coordinates (remember, your disk lies in the x-y plane, so only the x and y-components should be non-zero.)? What does that make \mathbf{\eta}.

The best way to ensure you get the correct factors and components in your integrand is always to just write out \mathbf{\eta} in terms of Cartesian unit vectors in whatever coordinate system you like.