key words: CRTs, Electron Microscopes, Accelerators, Positive Ion Beams etc.. Consider the field along the axis of a hollow, positively charged cylinder of inner diameter, 2r, of length, s, and closed at one end, separated at its open end, by a gap distance, d, from the following: A second hollow cylinder of the same diameter, 2r, and length, s', open at both ends and at ground or at a lower positive potential than the first cylinder. The thickness of each cylinder is the same and denoted, h. What is a good intuitive analytical approach for determining the field in the gap between the two cylinders and in the hollow regions eg along the axis, of each cylinder 1)as the voltage difference between the cylinders is increased or 2) the gap between the two cylinders is decreased or 3)the length of either cylinder is increased.? One approach is to think of each cylinder as a series of rings at greater and greater distances from the other cylinder and to consider lines between a point on one cylinder ring to all of a finite set of points on a ring of the other cylinder and so to determine the static induction at all of these points due to the single point of the other ring etc. But all of these forces are like the forces between two parallel strips of lengths (2pi)r and distance apart d+d(j) and all of these forces are additive. Does this approach make sense and is it helpful in giving approximate answers to the three part question?
Seems about right. You just need to find the electric field acting at an arbitrary point from one of the cylinders with an arbitrary charge on it. Once you have that you can easily find the answer for any number of cylinders in any orientation by just adding up each one's contribution(which you found first).
I'm not quite sure if I follow your geometry of the cylinder since I don't understand what you mean by its open at one end and closed at the other... are these solid cylinders what?
obviously the cross sectional area of one of the cylinders is two concentric circles but I'm not sure if one of the regions are solid or not..
In any case you just have to work with the cross section and its pretty easy to do..
The electric field for a uniformly charged ring along its axis is
E = k*Qx/(x^2+a^2)^(3/2)
where a is the radius of the ring and x is the the distance along its axis.
by integrating over a and x in whatever way you can get your cylinder(you might have to add the end configurations)...
Then you have a formula that represents one cylinder and you just add it to itself to add in another cylinder(with the proper translation and rotation).
It should give you the total electric field along the axis.
If you want the general case you'll need to work directly with the integrals in 3 dimensions... this can be done quite easily numerically as your integral is just a volumn integral over the potential(then you can take the gradient to get the field)
i.e.
V = int(k/r dq)
where q is the charge at (x,y,z)... you configure this for your cylinders i.e., define the formula properly.
Lets say the charge is uniform across a cylinder with charge density p then your formula would be p*I(x,y,z)
where I(x,y,z) = 1 if the point (x,y,z) is a point in/on your cylinder and else its 0. (using this characteristic function you can actually do any configuration you want).
then its just a matter of computing the integral numerically to get
V = int(int(int(k/sqrt(((x'-x)^2+(y'-y)^2+(z'-z)^2)*p(x,y,z)*dx)*dy)*dz)
where, ofcourse, p(x,y,z) is your charge density(which I simplified above using a characteristic function).
You could, if you wanted, say, create the cylinders in a 3D cad program and to define your geometry and charge density(but "color" I guess) then import it into a numerics program that will compute the integral.
You probably know all that stuff anyways but its an option... the idea about the rings and stuff is simply to give an analytic answer by using some easy to compute geometry(notice that the cylinder is cylindrically symmetric so really you are just dealing in 2 dimensions but it is also, for the most part, symmetric along its axis so ultimately you are just dealing in 1 dimension).
to go a little farther notice that the potential at a point x along a rings axis, with radius a, with constant charge density is given by
V = kQ/sqrt(x^2+a^2)
for a cylinder with "height" L then we just have to integrate
if you need to add "thickness" to the cylinder then you need to integrate over a.
once you get that formula then since potentials will add you can use it to add in another cylinder and then take the gradient to get the electric field.
(note that the above is only valid along the axis so you will need to do the "math"(which is probably numerically) for the general case)
Appreciate very much your discussion here. I should of said 'hollow cylinders'
Right. I am concerned about determining, Q from the voltage etc., and the static inductive effect of say a ring from each cylinder on the ring in the other cylinder. Say we have a 10cm diameter hollow cylinder on the left whose leftmost end is closed and whose rightmost end is open and after a gap of 5mm to the right there is a hollow cylinder open at both ends. The length of the leftmost cylinder is 1 meter and the rightmost cylinder is 10cm.The closed cylinder on the left is connected to +1kV port on a power supply and the open cylinder on the right is connected to the ground of the power supply. Now perhaps we can think of each cylinder as a series of t= 5mm thick,2r=10cm diameter charged rings with 200 rings plus a solid charged disc of 10cm diameter in the left cylinder and 20 charged rings, 5mm thick and 10cm diameter in the right cylinder. So consider first the pair of rings at the ends of each cylinder that face each other across a gap of, g= 5mm. This is like parallel plate capacitors of area 2(3.1416)(10cm) times g is equal to A. The charge on the plates is CV=Q where C = .885(10^-11)A/g Note that since g=t, it follows that A/g= 2(3.1416)(10cm) and CV= [.885(10^-11)](2(3.1416)(10cm))10^3 =5.56(10^-9) Coulombs and if the gap was 2t then the charge Q would be half of this etc. I suppose then that the more positive ring has this excess of positive charge as electrons are drawn to the power supply 1kV port and the grounded ring has this relative deficiency of positive charge as electrons are drawn from the ground to this ring. We have assumed that the other rings could be ignored and that a similar analysis is possible for say the second ring in the left cylinder and for the first ring in the right cylinder with a gap of
10mm etc.. How would you figure the charge on the solid disc except to say that this can be thought of as the 201st ring but that the charge induced on the center of the disc compared to the ringlike rim facing the first ring of the other cylinder is slightly less than that induced on the rim.? Now to these values of Q(i,1),i=1,...201 we might add Q(i,j), j=2,3...20 and to obtain the charge on each of the i =1 to 201 rings of the left cylinder. But wouldn't this require that we change the value that we calculated initially for the charge on the first ring(closest to left cylinder) in the cylinder on the right? I'm deleting some of the other helpful points you mentioned especially about a program to calculate the necessary integral approximations so as to make it easier to read these other questions and hope to discuss them later. Allen
You could try to use Gauss's law but I think that it would be somewhat complicated. Note that, if I got your geometry right then your problem reduces to the 2D as follows:
The assumption you have to make to make this easy is to assume the surface charge is evenly distributed over the surface and the charge is in equilibrium... else it becomes extremly difficult to do analytically.
Note that
C = Q/V for any Q and V. (C may depend on time and other factors though).
You could measure C and V using a meter and get Q... C will probably be constant at equillibrium but you could also use the fact that
This atleast offers an experimential result that you can use to "verify" your theoretical one.
In most ideal cases that I know(from my limited knowledge) C will only depend on the geometry therefore Q will cancel out from both the numerator and denominator else C will depend on the charge and/or location.
yep. This is the method of integration. This is basicaly what it means to integrate over a geometry. You pick some way to break the geometry up into "pieces" that are easy to work with and do the math on them then add up all the contribution for each piece. By breaking up the geometry into easy pieces then the problem will be easier.
Say you know V(x,y) for this one piece... then the total potential for these
6 pieces would be
V(x,y+a) + V(x,y-a) + V(x+L,y+a) + V(x+L,y-a) +
V(x+2L,y+a) + V(x+2L,y-a)
Where a is the "radius" and L is the distance from the first piece to the second piece + the length of the piece itself.
Then you just have to add in your V for the two disks(but they are the same just translated so you only have to do the math for one).
i.e., really there are only 2 problems you have here. You have to compute V for
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and
||
(which is a cylinder and disk in 3 dimensions. )
Both are pretty easy to compute when dealing with the potential on the axis but is impossible(AFAIK) to find the analytical solution for arbitrary position.
Yes, this is how any two conductors work for any geometry. It comes from the fact that like charges attrack and unlike charges repel and that charge will distribute itself across the whole surface to minimize energy.
If you have two arbitrary objects and one with a charge on it... it will induce a non-zero surface charge density. (though, I guess, for inductors its much harder to do).
e.g. Suppose we have a conductor of arbitrary geometry connected to the + side of a battery and another conductor of arbitrary geometry connected to the - side of the object. The conductors are not connected(so there is no closed circuit for current to flow.
But current will "flow" for a short time. What will happen is the electrons in the batteries - terminal will move down the wire and distribute themselfs along the conductor(and the batter terminal and wire) to minimize the total energy. Its kinda like if you have a large crowd of people stuffed in an elevator with the doors closed(the - side not attached to the conductor) but when the doors open(the terminal is connected to a conductor) the people will see more space and move out of the elevator... if that makes sense. i.e., as far as the electrons are concerned the added conductor to its terminal is just part of the terminal and it has enlarged so they will re-position themselfs to minimize the total energy... these movement though is electron flow....
Now if the other conductor is brought close to the conductor with the - charge it will experience forces from all those electrons... the electrons on the second conductor(hooked to the + side) will want to move farther away from the electrons on the first conductor(since like charges repel).... hence there will be a + charge on the second conductor. Its not necessarily equal(I think) because the geometries differ but for identical geometries it should be the same but opposite sign.
yes... this assumes that we are in equilibrium... if not then I'm not sure since I don't know much about electrodynamics... one could easily simulate the situation but you would have to define dq(x,y,t)... or you could define dq(x,y,0) and use columbs law(to a first approximation) to figure out dq(x,y,t) and then can use that to find out what the potential is.
It doesn't matter because we are dealing with potential. Take any small volumn of space(whatever.. could be empty space or part of the cylinder) and compute dV = kdq/r
now the small volumn has a charge in it given by dq(x,y,t) (i.e., we look at the point x,y at time t and dq will tell us the charge there(we either define dq mathematically or measure it experimentially).
so we have
dV = kdq(x,y,t)/r
where r is the distance from the point in space we are looking at and the point in space that we are trying to find the potential V...
written out,
dV(a,b,t) = kdq(x,y,t)/sqrt((a-x)^2 + (b-y)^2)
This basicaly just tells us the potential between two points in space for all time t.
if you have a charge of Q at 0,0 for all time t and want to know what the potential is at 1,1 for all time t then you just plug it in the formual above... you get
V(0,0,t) = kQ/sqrt(2)
if we add in another charge, say at point 1,3 with charge Q' then we get
note that we could have just computed it seperately then added both contributions at the end(which is ultimately what I did anyways).
The whole point is that potential adds.
If you want to know the potential for some extremly complicated goemetry then all you have to do is add up the contributions to each small piece of geometry that makes up the whole part(and you can choose how you want to break it up).
Anyways, from above
dV = kdq(x,y,t)/r
if we want to know the potential for all x,y,t we just have to integrade(i.e., add up all the small dq contributions from all the little pieces)
we get
V(x,y,t) = int(k/r*dq(x,y,t))
This is basicaly what you are doing when you sliced the cylinders up.
I can tell you the solution along the axis for your "complicated" geometry pretty easy cause its just 3 cylinders which I know the formula for and 2 disks which I also know the formula for.
i.e.,
the Potential along the axis for a cylinder of length L with inner radius a and outer radius b is given by.
To make it slightly more complicated but hopefully more instructive I can also break down a solid cylinder into your hollow cylinder. (I could also just use the geometry that I used above where it was just a rectangle and then compute a surface by revolution)
i.e.
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is the same as the potential from the cylinder given by(which means that you revolve the 2D object by its axis
(the lhs is the larger cylinder and the rhs is a smaller cylinder that is used to remove the part in the larger one when subtracted from it)
which is also juust many disks stacked up on each other
Each disk has a potential of
V = 2pi*k*s*(x/|x| - x/(x^2+R^2)^(1/2))
So the total potential due to one ring(large disk - the smaller inner disk = a washer like disk)
V = 2pi*k*s*(x/|x| - x/(x^2+b^2)^(1/2)) - 2pi*k*s*(x/|x| - x/(x^2+a^2)^(1/2))
to get the cylinder we have to add up all the disks as we move each one a little to the left... i.e., integrate w.r.t to x... if the length of the cylinder is L then we integrate x from 0 to L.
so the potential of the hollow cylinder is now given by
This can be computed exactly but I will leave that up for a CAS. Just note that we have V(x,a,b) in closed form for any arbitrary cylinder with inner radius a and outer radius b.
Hence we now just have to add up the 2 other cylinders to get the potential for all 3 cylinders...
All this comes from the potential for a disk. We were able to break up that geometry into nothing but disks.
e.g. A hollow cylinder is like a solid cylinder minus another smaller solid concentric cylinder.
a solid cylinder is a "stack" of solid disks.
So all that mess gives you the answer for the potential but only along the axis. If you want it for some arbitrary point then its best done with a computer.
No, because its potential ;) if I understand what you mean... If you happen to be adding two things that overlapp then yes you have to worry about it but thats just cause you broke up the geometry wrong.
I will try to do a simulation of the potential for your geometry if I get some type. Its quite easy to do(abit slow using the program I have).. If you have access to a numerics program you could easily write the code compute the potential for all x,y,t for some pre-determined charge density(you can chose... you can make current flow around it any way you want and see how the potential changes).
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