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Ifthe potential Ina certain region Euven DY field (In Vlm) at point p (1.2) IsThe magnltude of the electric...

Question

Ifthe potential Ina certain region Euven DY field (In Vlm) at point p (1.2) IsThe magnltude of the electric

Ifthe potential Ina certain region Euven DY field (In Vlm) at point p (1.2) Is The magnltude of the electric



Answers

The potential in a region is given by $V=a x y,$ where $a$ is a constant. (a) Determine the electric field in the region. (b) Sketch some equipotentials and field lines.

To talk about this question. We are given the electric potential over here and we need to find the magnet during the direction of the electric field at this point. Executive to and why were 2.4. So we know that electric potential is given by negative of change and potential ratings. So that's gonna be the partial derivative with respect to X. Towards I. And the partial derivative of Y in the direction of Jacob. So let's do that. The partial derivative with respect to X. We're gonna take Y as a constant. So that's going to be a Y two x minus B Y square. That's I. Cap. And now we are going to differentiate with respect to Y. So that's a X squared minus to B X Y. So that's going to be Jacob. So this is going to be equal to X Y a minus B Y square. This is icap plus a X square minus to be expired. This is towards Jacob. So let's substitute values here, we have negative of uh two X Y. S or two X Y A uh is for minus B. Why square. This is one. And the next days we are just opening up the parentheses as well. Five X square minus to B X. Why? And this is an Jacob. Uh so let's uh let's solve this one. Um we have four times four is 16, 16 times 160.46 point four. So we have 6.4 minus .4 square times it is going to be 1.2 I minus. This will be 20 Minour that will be four times eight times 0.4. So that's going to be 12.8 cheap. So if you can provide us further, This becomes a uh -7.2. J This part and This part becomes 6.4 -1 point away. So that's gonna be negative five point 12, I think it a 5.12. Hi cap. So this is the required electric field. Uh now this is in the vector form, but we need the magnitude and the direction. So the magnitude is going to be a square root of 5.12 square plus 7.2 Square. So that's going to be equal to the square plus 7.2 Square and square out of this, there's going to be eight 83. Uh Newton for cooler. So that is the magnitude of the electric field and we need to find that direction as well. Uh so clearly this is both the corners and negative. It means that it is pointing somewhere over here. So we can find a direction by finding this angle alpha and then uh adding uh perhaps 1 80 degrees to it. So we can get the angle with respect to the positive x axis. So, the anger tunnel for it's going to be uh the y coordinate or the X coordinate. So that's gonna be negative 7.2 over negative 5.12. So a figure of a calculator, that's going to be 7.2 divided by twice 0.1 to 10. And worse of this comes at 54.58 degrees. So the total angle it's called tita is going to be 1 80 degrees plus alpha and uh valuable phobia. Well, we already have, so that's gonna be 2 34 .58°. Uh We're gonna say this is counter clockwise with respect to positive X. Axis. This is the direction. Thank you.

So the potential is given by something like that. This is the potential as a function of old on and that is 30 voltage. This is one. This is two and here is three. And here is four and this is X in Sandy Reader. So we have to do DVD X native of that with he was e as a function of X. Because of the numbers, we have to just do it once. Generally, the numbers will be 30 volt, divided by one centimeter equals 30 word per centimeter. And then we have to understand that here the slope is positive. Negative, native positive. So yet because invited when you withdraw the the electric field. So there's right, like 12 three, 45 This is actually five year. So from 0 to 1 DVD excess positive. So the electric field this negative then it is positive. Then it is Ito. Then it is positive again. And then it is native and all these values are This is 30. The unities would bar centimeters on. Here is native 30. This is X in centimeter

Okay, so we're in Chapter 23 Problem 51. So we have in a certain region of space. The electric potential was given by be equaling y squared, plus 2.5 x y plus or minus 3.5 x y z and it says, determine the electric field vector. Okay, well, he is given by the negative Grady Inspector off potential so we can find E X as negative, deep partial The partial X that comes out to being 2.5 Live minus 3.5 y z. Let's now find you why that's made a partial. The partial. Why, which comes out of being too? Why plus 2.5 x minus 3.5 x z. And lastly, we have easy, of course. Negative partial. Be partial Z miss. It comes out the negative 3.5 x y and that's everything. So just write this out. We have 2.5. Why? Minus 3.5 y Z. That's all the ex components. That's kind of the I, Huh? Plus, too I plus 2.5 x minus 3.5 x Z. That's the J component. It must leave. What's at Argo que component Negative 3.5 x y. This is all in the K hat direction. There is our concert.

Until the electric people can be defined by the following equation. So E is equal to the derivative of the potential over the derogative Off the position far. So now if you substitute the the potential that were given, which is the musical positive A explain why must be x squared b y squared. So if we substitute that let me hold on Let me just get that down on paper first p is equal to X squared. Why minus b x y squared So now you see that and bring it into the potential into this equation. So then you enter in back so be the are That's a really long line every lunch And now go back to Penn now A Let us be x y squared tonight he compute this, uh, so now we need to come computed for each of the components. So if you computed for E X and B now the whole for the position that X he x who would respect to X and now the mistress being straight down here. All right, Uh, we take this year. I believe that, uh, let me just make it easier. So if we can do this quicker. So you move this down here and, uh, differentiated with respect to X. Beginning at that belly is equal to two times exp i minus. And since this excellent power one was the coefficients of the White Square And now we know the conference teacher just given to us in equation in the and the question. So if you substitute that like, I don't think I need to write it. But if you plug it straight into a calculator, substitute the values that getting the question we're going to get that yeah, is equal to negative 6.72 both for meter. Now, that's potentially so I mean to caucus potential. Why? So if you do the same thing, so the differentiated. But this time with this respected why interior cooperating percent feel that way. Oh, she bring this year. Bring it down here. So now if you differentiated with respect the wire, you're gonna get the equal distances y to the power line here it just becomes a coefficient. So x squared. And since is what part two? Here it just be bring to theaters and then subtract power. So it's minus to be a why Now I see so city the values that we get from the question straight into this, uh, into the corn that we can get that it's equal to negative 7.2. Okay, go screw meter square number. Always remember the unit on finally, the potential, etc. I actually don't think I need to read it out. We can just look at the about the equation. And our remember we're different to you than respect, is he. But just by looking at this, we can notice that there is no variable for them to differentiate with respect. Disease just equals zero to the potential. That easy sequence there. So now, to find the net magnitudes off the electric field, we can use this foreign formula. So let me let me write this in the different colors. Easy quote, too. So this is the net magnitude of Flex Field, which is equal to X squared. Plus he y squared plus e c. Ever. So now we substitute the values that we just computer to above. We can get that this is equal to negative. Six point 72 squares negative 7.2 squared. And since it's just zero for a concert Easy. We don't eat right the You know, if you compute this and I call clear, we're gonna get that meant potential. The Net electric field is a put 9.8 over 9.8 both from here. So not defined the direction off the electric field that you can just use this falling for me. So let me get that in between. So the pita You mean like a better feed it there. The angle. The sequel to the pan and worse uh, e y over e x. This is justified direction. Since these two are the heads of the triangle, If you could imagine like, So now, if you just substitute the values that we got from the previous equation, it's time negative one that's equal to negative. This is the very, very calculated up here in black. So now it's over. Think it is six point seven to now. If you compute this, you're gonna find that is equal to 47 degrees. So now we're cooking with the magnitude in that naked huge here and direction here, and that's it for this question. Thank you for


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