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Uniformly charged solid slab Consider an infinite plane with charge per unit area 0. What is the electric field & distance h above the surface of the plane? Con...

Question

Uniformly charged solid slab Consider an infinite plane with charge per unit area 0. What is the electric field & distance h above the surface of the plane? Consider an infinite slab of width W in the y direction and charge density p. The slab extends to infinity in the € and z directions What is the electric field above the slab (aka where y 2)?LzConsider an infinite slab of width w in the y direction and charge density What is the electric field inside the slab at & distance y above

Uniformly charged solid slab Consider an infinite plane with charge per unit area 0. What is the electric field & distance h above the surface of the plane? Consider an infinite slab of width W in the y direction and charge density p. The slab extends to infinity in the € and z directions What is the electric field above the slab (aka where y 2)? Lz Consider an infinite slab of width w in the y direction and charge density What is the electric field inside the slab at & distance y above the center of the slab where y 2 Make & plot of Ey as & function of y that spans from y ~W to y = W Note that y = w is distance Zw outside of the slab- What is the potential difference between the center of the slab and its top surface?



Answers

A Uniformly Charged Slab. A slab of insulating material has thickness 2$d$ and is oriented so that its faces are parallel to the $y z$ -plane and given by the planes $x=d$ and $x=-d$ . The $y$ -and $z$ -dimensions of the slab are very large compared to $d$ and may be treated as essentially infinite. The slab has a uniform positive charge density $\rho .$ (a) Explain why the electric field due to the slab is zero at the center of the slab $(x=0)$ . (b) Using Gauss's law,find the electric field due to the slab (magnitude and direction) at all points in space.

So in this question, we're looking Teoh drive the electric field inside and outside of a charge slab so and extends infinitely as a thickness d. So it's kind of draw that out. And what else? Um and then the volume trends density is roe. So let's kind of put an access through here. And then our goal is to find, um, the electric field within this and then also, um, on the outside of it. So we want to use Gousis Law. Ah, and so for a we can say, Well, I was, In general, we can say the integral of e dot d. A. Is Q and closed over absolute night. And then to make these arguments, we're gonna need to think about the other dimensions so we can say so we have this one. Let's say that this is a dimension now. I guess it should actually be infinite in both ways, but obviously I can't really draw that. So let's say we're talking about an l bile portion. Um and then you can just do this in the limit that Al goes to infinity. But for to make the argument a little easier to make. Let's think about just think about, ah, a dimensional. So the Q enclosed, Um, we first need to make our galaxy and surface. So the Gaussian service it has the proper symmetry is gonna be another infinite slab. Um, and this is supposed to be an infinite dimension. Um, so I guess I should have drawn it sort of out here, um, and its infinite in both directions. Um And so and it goes through some point acts were accessed the point inside that we're concerned with. So the Q and close there is gonna be road times the volume and the volume of this is going to be l squared acts. So that's our cue enclosed. And we can divide that by epsilon Not now. Eat a d A. So the only surfaces that are gonna be intercepting flux are going to be the top and bottom surfaces. Because by symmetry, we know the electric field is has to be straight up or straight down, so sort of important, um, and then a box of shows, this symmetry. And so, um, the only surfaces intercept flux. We're gonna be the top of the box in the bottom of the box So let's just draw us for our surface. It's gonna go through like, plus X on this side. And then, um, like, the top is a distance x above the access and its bottom is the distance, um, ex below the axis. Oh, I just realised we're supposed to use D Oh, no. D is just a total that best never mind. Um And then, actually, if that's the case, then this with is gonna be two x So let me go ahead and modify this to be two x and, um, OK, back to eat out A So, um, this in a row simplify. So just EA for the surfaces that are intercepting flux. And so it's gonna be e l Square in times two and then these both have positive flux because the electric field us out in the area vectors air out for the shape. So it's to yell, squared. And then if you simplify this, you got ta is equal. Teoh cancel too. Cancel all squared and then we get row acts over. Absolutely not. And then that's let's say that this is the X dimension. It's an X hot direction. And so that's the answer for a B goes very similarly except for Q. And the left side stays the same. Um, actually, let me remind a little bit. I'll make a new galaxy and surface so the Gaussian Circus is gonna be on the outside, So this is still a distance X from the access. So here's our X. So the left side is gonna be the same you don t a Nothing changes. So that's to yell squared. And then the right side, though the charging closed is capped at a distance D so then this becomes row, not acts, but it becomes row D um, who actually let me be careful. It's actually so Q enclosed in this case would be l squared just times just times d not two d so great. And then we wanted to buy that by absolutely not. My picture is sort of ugly. I hope it makes sense. There's probably some nice ones on the tax, too, if that helps. But yeah, um and so uh oh, yeah, I forgot my l squared here because, right, that's that's the volume of this Gaussian surface, this calc in, um slab. So the L squares cancel again and then we're left with e is equal. Teoh, row over to Absalon at times D. Just a whole check. Yeah, and then rode times. He is actually the surface charge. Density and signal were to absolute not. And I think Sigma over to have salon not is definitely worth remembering. And the notable thing about these is that they don't not depend on at least outside the slab does not depend on how far you are and anything. It's sort of a remarkable result. It's definitely worth from Burma.

We have Eddie a vector. It was two centimeter times three centimeter he had it was six centimeter square. Gay hat sucks. It was e dart A equals as electric field use 50 I had thus 100. Gay had dark area is six centimeters square. So six times 10 to the minus four Major squared he had. So from here we find the flats to be six times 10 to the minus two Newton meters squared.

Hi everyone in this problem it is given lets you charge is located at the opposition. Okay or is curto plus why not jail having divided city? We not hype. We have to find minimum velocity in order to particulate hipster bar to solve this product. Yeah. We need to rotate the coordinate system. Right right 45 daily. So new coordinate would be are not a sculptor. Why not sign of theater? I are kept. Uh huh. Why not cause upped it up Jared. Unit vector are not can be written it's why not buy route to our unit vector plus why not? Why it would too judy you know Director the velocity of the particle is given by we're not his cult of minus do not Because of 45 our gap Let's be not sign off 45 Jacob so you can be written as being not by two are kept plus B. Not by route to Jacob using Newton's second up. Yeah, force is called Touma's into acceleration is called QE acceleration will be QE by AM but electric field is given ro 80 upon to absolute not so acceleration we will get to through E. D. Yeah By two absolute not hemp. Okay, this is the question. But now final velocity can be defined as We are not squared -28 ar minus are not here we are. F must be zero. So we are not square. You will get to a ar minus are not so we are not you will get to it ar minus are not. We are not scared to be not by route to soap. Be not sculptural for a ar minus are not no substitute the value. Yeah. Four judo et upon to absolute not. M Why not buy it would to zero so minimum speed of the particle. Yeah. Mhm. Yes. Should be equal or less than Yeah, ruled off. Yeah Hugh roe A. Why not be upon? Absolutely. And that's thanks for watching it.

The problem estates a slab of insulating material has thickness too deep. Right? It has to take place today and is oriented so that its space our panel to devise it name and given by the planes X equals today and X equals two minus divi NZ dimensions of the slab are very large compared today, treat them as essentially in finite. So this lab has a uniform positive charge density room. So now it is asked for the first part explained why electric field due to slab Is zero at this circle. Oh, so at the center of this lab. Right. I'm using goes law. Finally electric field you do this slow magnitude and direction at all the points in this space. So here we can see the formula for this lab. You can see the charge. It can steep formula for any point eggs great. It would be mm Don't not equals two X by tease. Right? So that is why at this center of this lab equals to zero because it's zero year since access you don't know. So this whole equation becomes zero entire standstill Izzy Ruden. He is also. So now for the the part we are asked to find the electric field due to this lab. So here we need to find the electric field Between you can say it is given zero less than it. Okay, this is Cindy. Right? So for this we can say thank you is given by road E V All you can write it as gone numerous 88's right. These idiots can be written right again. This is the SrV X squared dX. You can see my things. Can we write this? Yes. So this is the basic thing. No Cuban close means the charge enclosed Will be between 02 eggs here. So it will be too Times of 02 eggs. Thank you. Right. And we will just integrated putting the value the queue. So throw a body square is a constant. Who is also constant taken outside only access available. So excess square when uh integrated, it becomes two by three row, not xQ by the square. Okay, so we know that five is equal to cuba Absolutely not by his word flats. So you can say electric field. He is given by actually if I buy you right, we can say since electric fields he is he was too bye bye. So can we right at us by this. Is that this plug sense? You can say E into a right? So that is what we have done here. We have written four instead of five. We have written Yeah to e trade something. Right? So here, when we'll put all the values like you enclose and we will get E. S. So not xQ by three. Absalon not be spread. So this is how we solve this problem. I hope your industry concept. Thank you for watching


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