Question

---Urgent question: would appreciate help.---

A ring of radius R has charge Q.

a) at what distance along the z-axis is the electric field strength a maximum?
b) what is the electric field strength at this point?

Answer 1

We need a drawing to see which is your coordinates

Answer 2

Assume the axis to be through the center of the ring.

Answer 3

Let me refresh my memory if I can...

Answer 4

I know part a is R/sqrt2, but I am completely lost on how to get it.

Answer 5

\[dQ= \lambda ds\]Where landa is the amount of charge divided by the length of the ring. We know by symmetry that every point on the z-axis will be in a net electric field in the "z" direction. So: \[dE _{z}=\frac{ dQ }{ 4 \pi \epsilon _{0}r ^{3}}h=\frac{ \lambda ds }{ 4 \pi \epsilon _{0}(h ^{2}+R ^{2}) ^{3}}h\]Any ideas? @theEric

Answer 6

I'm pretty stumped..

Answer 7

would i differentiate z? I've been working at this a while and am getting very frustrated.

Answer 8

Okay I think I have it:

Answer 9

Cool, @ivancsc1996 !

Answer 10

ds: A little piece of arclength; dQ: Little piece of charge; Lambda: Charge density; Ke: Electric constante; I'm going to use the ceil brackets for vectors. We need to do a number of observations. First, using pythagora's theorem:\[r=\sqrt{h ^{2}+R ^{2}}\]Then:\[dQ=\lambda ds\]\lambda ds\]and\[ds=Rd \theta\]so\[dQ=\lambda\ Rd \theta\]We need to keep in mind that we get Q when we integrate the previous equation from 0 to 2pi:\[Q=\int\limits_{0}^{2 \pi}\lambda R d \theta=2\pi \lambda R\]Now we need to find an equation that describes the electric field at every point along the z-axis.\[d\lceil E \rceil=k _{e}\frac{ dQ }{ r ^{3} }\lceil r \rceil=k _{e}\frac{ dQ }{ r ^{3} }(\lceil h \rceil+\lceil R \rceil) \]Since it is a ring, we know that there can only be a electric field in the z-axis. So:\[dE _{z}=k _{e}\frac{ dQ }{ r ^{3} }h=k _{e}\frac{ \lambda R d \theta }{ (h ^{2}+R ^{2}) ^{3/2} }h\]Now we integrate from 0 to 2pi to get the net electric field at any pont in the z-axis:\[E _{z}=k _{e}h\frac{ Q }{ r ^{3} }=k _{e}h\frac{ Q }{ (h ^{2}+R ^{2}) ^{3/2} }\]Now, to tackle the first problem, we need to analyze how this equation changes as "h" changes. To make it easier, can anybody see how to make this equation simpler?