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An electric charge is spread over the half-disk $H$ described by $x^{2}+y^{2}=4, y geq 0$. Find the total charge on $H$ if the charge density at any point $(x, y)$ ...

Question

An electric charge is spread over the half-disk $H$ described by $x^{2}+y^{2}=4, y geq 0$. Find the total charge on $H$ if the charge density at any point $(x, y)$ in $H$ (measured in coulombs per square meter) is $sigma(x, y)=sqrt{x^{2}+y^{2}}$

An electric charge is spread over the half-disk $H$ described by $x^{2}+y^{2}=4, y geq 0$. Find the total charge on $H$ if the charge density at any point $(x, y)$ in $H$ (measured in coulombs per square meter) is $sigma(x, y)=sqrt{x^{2}+y^{2}}$



Answers

A uniformly charge disk has a radius $ R $ and surface charge density $ \sigma $ as in the figure. The electric potential $ V $ at a point $ P $ at a distance $ d $ along the perpendicular central axis of the disk is

$ V = 2 \pi k_e \sigma \left( \sqrt {d^2 + R^2} - d \right) $
where $ k_e $ is a constant (called Coulomb's constant). Show that
$ V \approx \frac {\pi k_e R^2 \sigma}{d} $ for large $ d $

To begin the problem. Let's write down the information we hop. We have the rectangular region are which is described as X comma y So start zero less than he called to X Less cynical to five. 10 to listen. You can't tell why the cynical to five That's the rectangular region. We have the density function sickle to two weeks plus four way Klump Bar Squire meter. Want me to explain? So we have to find the total charge on the rectangle. So the total charge is he going to integration over our then Sigma? Uh, just me. Did you justice? So this is going to be sick, Ma off. So Sigma off X comma. Y t x t y. Okay, so no, what's my wife? But the why limit is 2 to 5. Text limited c 025 Then this becomes two weeks plus four way t x followed. But divide salutes. Evaluate this antique road on two x plus four way If I intricate with respect to X so this becomes 2 to 5. Let me go back. So this becomes exists, Claire, and this becomes four. Explain. Okay, So excess Claire plus four x y and then the limiters from zero Fife 0 to 5 and then followed by. So that's 100. That number. So five is going to be 25 on then. Fife times for its 2020 by do you I. And now this becomes 25 x plus 20 White Square, divided by two. The LTD's from 5 to 2 hooking. This becomes 25 5 minus two. It's simply three. The three I can cancel the tend. So this becomes 10 than 25 minus four, which is just 21 So 75 plus totin pretty famous 2 85 clump, So that's the total charge.

To begin the problem. This right on the information we have we kept the charge 10 city Sigma off X comma y Is he going to square root off excess square plus y squared column per square meter And we have the region. It's the disc he is It's karma Wife Aesthetics escort plus twice coitus Less than equal to one. Okay, so now we have to compute the total church. So what is total charge? So the total charges Double integration over tea Sigma off X comma life d x t. What? But the first thing to notice It's these a circular region. So it is easier if we use the polar court, innit? Solutions? The polar recording it so in fuller court Already puller co ordinate than sigma off X comma boy, it's exist carp lous y squared square root. So this become simply are and then remember in polar court in it Do I d x it's triplets by are do your defeat. Okay, so no. So that integration becomes integration. There are limits or the teeter limit is simply from 0 to 2 pi the our limit is 0 to 1 because we're in the desk where our can be a smaller Ciro and atT SLA Just one. Bandon. That's My Sigma, which you simply are. Times are followed by Tear D. Peter Circuit. What is Santa Christian? Off our square? It is simply R cubed, divided by three and 0 to 1, followed by the theater. So Tess becomes Let me go back. So this becomes one Kurt into question from to fight 1/3 dictator on the integration off 0 to 2 pi ti theater. It's simply to pipe, so Don Cities to fight Divided right? Three. That's the total church in club setup I divided by three cool lump.

So let's say we have coordinate plane and here we have two charges and these two charges are having magnitude cube. And they are positioned at one. Let's say it is positioned at L. zero and another Q charge is positioned at minus L zero. So we need to find out the electric feet at any point on the Y axis. So let's say this is that Why distance from the origin? So at this point will be zero. Y. And here if we find out the electric field. So due to this church, the electric field will be in this direction. And due to this charge, electric field will be in this direction. So the magnitude will be safe because the charge and the separation is same. So let us find out one electric feet. So that will be okay Q. By this separation square. So this is L. And this is why. So separation will be root of elsewhere plus y squared by using pythagoras through in this right angled triangle. So this square will be and square plus white square. So that is the electric field. Hm So this will also be eating but due to symmetry here this if we take it as theater, this also will be teeter and this also will be twitter and this also will be twitter to find out the net electric field. Let us project the electric fields. So if you projected this electric field, it will be he signed peter in this direction and e coast hitter in this direction. Similarly the projection of this electric field will be he signed peter in this direction and Ecos data. So that will make it to Ecosoc to So these two electric fields will be cancelled. And we get the net electric field coming out to be to e cost data. So this is the net electric feed. So let's put the value of Egypt. So two evils cuba L squared plus Y square. And for cost you to let us take the youth of the strangle. So it will be base. That is why over the hypothalamus which is blue top L square plus white square. So if we simplify, we are getting the neck electric field as two K. Q. Why buy elsewhere plus Y squared race too three by two. So this is the magnitude of the net electric field. And we are observing that this net electric field is along right direction. So this is let us take this complete value as E dash. So we can see that net electric field is E dash if Y is positive and it is. So let's write the direction is because this is a vector and minus E. Dash Jacob if y value this negative, that means if the point is lying below the origin. So let us find that point where the electric field is maximum. So what we do here we take the expression of electric feed and differentiate this D. Of impact with respect to why And we put it at zero. So we do D by D. Y. Off the electric field expression to take you by by and square. Bliss Y squared raised to three by two. So here we don't deal with the denominators because anyways if it is equated to zero then it will be cancelled. So let us just take that numerator because we if we have you buy we and if we differentiate we're gonna get you dash minus UV dash by the square so will not care about this. We'll just frame the numerator and equated to zero anyways. So you'd Ashby. So if we differentiate numerator we get to take you Why differentiation is one into L squared plus Y squared, raised to three by two and then minus two Kq by times the differentiation of denominator That will be three x 2 times and squared plus y squared phrase to one x 2. Into the differentiation of inner part. That is to buy. And if it is equated to zero that means one time we'll go to the right and let us cancel the terms. So yeah, we can cancel this Route as well. So from three x 2, route of L squared plus y square will be cut. So we'll have only the paris one. So we'll get elsewhere plus Y squared is equal to three x 2 into two, y squared. So this makes elsewhere plus y squares three Y square. And this gives us elsewhere is to buy square. So from here we can figure out why square? That is elsewhere by two. So why will be plus minus L by photo? So this is the point on Y axis from the origin where the electric field intensity is maximum.

Okay in this problem, we have a thin flat disc with a radius or not in a truck. Charge to it has a hole that's half its radius. We were asked to find much potential on the X axis. So for this problem, we're gonna have to find Viva integration. So probably first steps to try to find a chart to get a charge density and integrate that up over the area of this disc. So are charged in our charge. Density Sigma, which represents a service charge density, is going to simply be the truck total charge Q over the every remaining area. Factoring in the hole that's just the area of a circle of the whole circle are not squared. Fires spread minus pi. 1/2 are not squared, which will simplify down to four Q over three pie are not squared. And that's the first step. Now we're gonna wanna integrate up small little pieces of charge. We couldn't express a little D Q element to be charged density times an area little area you can plug in the four Q over three pie are not squared. Time's a two pi r d r. So these they're gonna be little rains of charge for small, for a little area. Oh, this is a r and we can integrate these up So the total potential on the X axis is going to be the end of a roll. It is simply one over for pie. Absolute. Not any girl of little bees. The charge over the distance to that said piece of charge, which for little brings a charge. What's kind of nice is that our is consistently the same on axis, so we can do it with this integral in the radius of our disk from are not over to you or not. And we have four Q over three. Pie are not squared. Times are two pi r d r element. This is our small charge men and for the distance from the charge to our point of interest on the X axis the distance between them, which should simply be extra plus r squared. So it's simple. Find some stuff down. We can pull some constants out of this integral. We have de tu tu que over three absolute nod pie are not squared in our integral becomes this integral which weaken d'oh! Same Constance is before the integral is evaluated when evaluated, will become X squared plus R squared with 1/2. We evaluate this, of course, from our at our limits, and it turns out to you to Q over three. Absolute nod pie are not squared, multiplied by roots. Exported, plus are not squared minus root X squared plus 1/4 are not squared box. That that is the electric potential on the access due to a ring, rather a disc with a hole in it, and that's it.


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