Use the model of the electric field represented by 100 field lines. Use protractor to measure the angle between the area vector of the surface and the direction of ...

The electric potential in a region that is within 2.00 m of the origin of a rectangular coordinate system is given by $V = Ax^l + By^m + Cz^n + D$, where $A, B, C, D, l, m$, and $n$ are constants. The units of $A, B, C,$ and $D$ are such that if $x, y,$ and $z$ are in meters, then $V$ is in volts. You measure $V$ and each component of the electric field at four points and obtain these results:
(a) Use the data in the table to calculate $A, B, C, D, l, m$, and $n$. (b) What are $V$ and the magnitude of $E$ at the points $(0, 0, 0), \space(0.50 \space m, 0.50\space m, 0.50\space m)$, and $(1.00\space m, 1.00 \space m, 1.00\space m)$?

My friends. As in the figure, there are two metric field even in you too. We have to find the net Electric here even is given MM 1200 nuke in particular. And he too is given 1700 million particular. We have to find the total electric field peak so he can be obtained by the formula. He one squared. Plus it was squared. Plus two. You want me to cause of theater? Theater is the angry between this and Tita will be 55 minus 35. That is Yeah. Continually putting the value even. Is 1200 1700 square true Mhm. 1200 to 1700 because of 20. Yeah. So, little baby. Sorry, that electric field having the magnitude. Yeah. 43 900. You can part hold up. And the angle The resultant seed mix Never 50 50 Mix alpha angle. But even so And what will be 10 universe of he to sign Optical forward even less people because of theater. Yeah, So he too is 1700. Sign of 30 fight 1200 plus 1700 course of 35. So it is to me very quickly And data with X access building 35 bliss 12. 47 degree then xx is that son, Thanks for watching

Okay, so we're given that our function is okay over X squared plus y squared. And we want to find ingredient of this sort of you can find the electric fuel and the electric field is the negative ingredient. A terrible negative sign. Negative, radiant of you. So what we do is let's take the partials. No, take the partial of V with respect to X and the one. So what is this? And actually, let's make those negative. So we're going to have negative one half times negative one from here times. Okay, times the derivative of the inside. And then this whole thing X square, it was Y squared is going to be to the three hubs. And so why is the same thing? Mhm. Let's just not really quick that those all cancel. So we're going to get X X squared plus y squared to the three house. Okay. And, uh, notice that why is also under this and to the same power. So the same things were going to happen with the power rule and the change wall, except the derivative of the insight will simply be different. So you're going of why over it's raining over your action and let's not particular k there. So we're going of que wine over X squared plus y squared to the three hubs and this vector here you can write it in transposed form. It's just easier at the moment. Right in this form is our ingredients. And, uh so next we should, uh, realize that the electric field, since this is the electric field points renew yet words everywhere. So for positive ax que is also positive. So for positive X Hawaii, we have a vector pointing in that direction somewhat positive y negative X in that direction negative y negative X in that direction. And the same thing applies for positive xnegative one. So it's not hard to reason that that should be pointed outward. Okay, so, no, but we need to show Is that the electric field S O. B? So this has a constant radial component for some fixed radius. We want to show that. So what we're going to deal is this is the component. This is the white component. Let's just square those. But let's also realize we could rewrite this denominator. Uh, you could rewrite this whole vector s Let's move down a little bit. You could rewrite this whole vector as que multiplied by X over r cubed. Why over our cube? And so if we want to find the length of the high part news of this factor which would be the radial component, it's just gonna be k times x squared plus y squared, which is just gonna be r squared all over our two the sixth. This actually shouldn't bien vector brackets because we're talking about the magnitude here. Okay, so this the magnitude we'll be way also take the square root. So to minus six is negative four. So that's going to give us one over or to the fore, and that's gonna come out to be k over or square. All right, so next it wants us to show that our ingredients vector here is orthogonal to the curve. Everywhere are orthogonal to the level curves. So let's just move down a little bit more. And we had that v sk over, uh X squared plus y squared square rooted. And so if we have some fixed Z equal to K over the square root of expert was y squared, what does this imply? Okay, So what does this? She probably we're trying. Well, you can bring the constants over the that side, and let's just raise it to the negative one. Power, actually. So we get following that. Is the Samos seeing okay or is he is equal to the square root of X where it was wise. Weird. Yeah, this is just some circle with radius Ko Brazilian. Okay, so we know what our potential curves look like. The potential three expert ECU. It potential curves. Sorry about that. Look. Wait. All right. Ah, But in order to prove that the ingredients is perpendicular to the curve, let's find the tangent to this. So it's again, Let's just square both sides. This goes away. This is where and then let's different. This is a concept that zero to x plus, let's assume why is a function of X so we can see the wind yanks and then we're gonna minus two x to the site. Divide by two wise, you'll get negative. X over white is equal to deride the X, using the fact that the tangent vector is then one negative x over y or actually, it's a probably going to be easier if we work with vectors like such. We can multiply that by negative white and just have X over here. And then let's just start that with the ingredient which waas x. Why? Yeah, over or cute. So we're going to get negative y x plus positive Y X over or cube, which is zero. So we've shown that they're worth agonal everywhere. And, uh, yeah, since the tension along some curve Europe is Warsaw Agonal Thio, our unit vector or sorry, our ingredient vector poor drawing. But the tension vector is orthogonal to the greedy in Victor And since we wrote in this form, we've proven that for all

Hi. The given problem there is a square. This is the square whose side is giving us A. If you consider it to be A. That is given as 0.300 Nieto. Then the two charged particles are kept At the two lower corners of this square. Both the charges are having same magnitude, suppose this point is B. And this is supposed to be S. Then there is a point and the upper side of the square and exactly in the middle. And this point is named as D. So we have to find net electric field at point B. As these two charged particles are positive, this charge is having a magnitude Of 7.00 Micro Cola. So first of all we join this observation point with the positions were we are having the charged particles. So the first is this T. V. And another is this P. S. S. B. Sorry. So both of these distances will be equal then as we know the direction of electrical, it always goes away from the positive charge. So there are two elective fields at this point B. One because of Q. Put at P. Going away like this and another because of Q again. But which is put at S. So again going away from Q. Like this. So these are the two electric fields, who's the resultant we have to find. This is electric field E. P. Due to the charge which is put at P. And this is E. S. Electric field into the charge which is put at point S. Okay, no, these two distances as this point B is the midpoint. So here this distance we become half of the Side. Land means a by two. This distance will also be half means a by two. And these two triangles are the right angled triangles. So whether this is A P. B. Or this is S. B. Both of them are the hyper tennis's. So using particle storm, they will be given us a square means this site of the square which is A. So this is a square by a by two to the whole square. So it will come out to be five A square by four. Or we can say that this is a buy to route five. As the distances are saying the charge is also having same magnitude. Hence we can see the two electric bills, E. P. And yes, these two will also be same because the expression for the electric field is scared into Q by our square mr distance and distance. The same pd whether it is PV or CSP. So if we take their component along X axis, these two angles will also be same. Using the symmetry. So the horizontal component E p X will be equal to E. S. X. And directed opposite to each other. So the horizontal components of these two electric fields reduce E. S. X. Here towards left and E. B exhale towards right. So these tools can sell each other. But when we will look at the vertical components, both of them are in the same direction E. B. White and E. As why. So both of them will be added to give the net electric field at point B. His horizontal components are canceling each other so the neglected field at Point B will be given by we have shown it in the figure E. B. Why? Let's E. S. Y. All you can see as both of the electric fields are seen two times of e. p. y. So here it will become two times off. Okay. Into cube divided by distance square of distance. Which we have found to be a by route to a by two. Into Route five to the whole square. Now for this science theater because this component will be given by E. Twice of E. P. Into sine theta your theater. This data will be this angle and that angle will be equal to this angle here in this right angle triangle. So first of all we should find this angle here and that angle as we have to use science to. So in this right angle triangle, scientists will be perpendicular which is a divided by hyper tania's which is a road five by two. So we will put it here for science theater. It will be a means the perpendicular divided by A route five x 2 means the hypothesis, so canceling this air. Finally it will come out to be here. This will be swear of two means four and 4 to 8. And then eight. We go up here, it will become 16. So this is 16 K. Q. Divided by This is a square off Route five which will become five, 5.5 into a square. Finally, we will plug in all the values here. So this is 16 in two K, which is 19 to 10 days to 2.9 into charge, which is seven micro Coolum or 7 to 10 days per minus six cola divided by five. Route five into 0.3 to the whole square for the side. So this electric field and net electric field at point B comes out to be 1.0 into 10 takes about six newton park column. It's direction is upward in the plane of paper. So it becomes the answer for the given problem. Thank you.

Question e states that find a the direction and be the magnitude of the Net. Electric fuel at the center of the ecological triangle, showing the figure here. Give your answers in terms of the angle feta, as defined again in the bottom left image there and e the manager electric you of bruised by any one of the churches were saying that, um, we're gonna find the electric eel at the center point of being called P whichever. And just as a kind of description, our answer has to be interred with the angle an electric field value so we don't have enough initially. Would cute meet? You don't have a value for Q, but it's a theft. Any one of these charges produces an electric field of eat two. Basic What's a magnitude produced? But it was Magic Magic Field produced from Egypt. The net result, right? So first thing we'll do is we'll indicate the direction of the electric field lines from the point charges towards that towards or away from that object. So you know, the positive charge comes towards it early ice. It leaves the charging in. This scenario points directly towards the point p and the negative charge points directly to be, um, negative charge the bottom rate. So what we'll do? We need to look at season roaming over the management. The direction of this in that electric field the person will do is we look at the electric fume that point being each of the X directions do that because charge one of the top is solely in the Y direction. That's zero. And for the other two, they're both pointing towards or away from the center. So this is an equal and triangle. Each of these angles here would be 30 degrees. It was half that means for Q two. Now it's point the positive direction, right? Electric kill is pointing the positive X direction this turns positive. So you're k times queuing and most looking the magnitude over the distance. Well, we don't actually, you see, because we're just seeing Don't you know, don't We don't need to knock. You were saying that it's due to let your field whenever it is times angle, which is here again. This is a state. Uh, is it the coastline of 30? That's the charge to in the bottom left. But if we notice Charge to be in the bottom left has the same one of the same component. Exactly. So it is most place by two. So this too. So this term is ribs and, um, the extraction of both Charge two and three. Those are maybe two times e coastline 30. So again, we only use it to the city because that's what the answer says. We need it to be in. Um, otherwise, we would have foot and take you time over the distance squared. But we don't do that. Uh, for the why Component for Q one is pointing solely negatively downward. It's what is called negative e. We should say this is positive. X hat just for completeness. Now if we note for the Charge two and three why they did it the same x component because they have a state magnitude and the same distance away, we see that due to charge three has a positive like important at the same angle as a negative way component of charge to the white components of Charge two and three cancel, and only it's consider is the magnitude of the electric field due to charge of one sewing negatively downward. Excuse me. So that means if you want to find the magnitude of our electric field, we can simply just take our terms and find the result of that's e x squared plus y squared all square Rudin coastline. 30. If we remember, I think those 30 well they will do. Khost 30 30 degrees, of course, is Route 3/2. So if two times were 3/2 in their first term is simply just e times Route three squared plus eastward because it is too cancels out this to enough of even three whole terms squared. Can square rooted some things? We have three eastward plus the square just 40 squared, which is simply to eat. So you have the magnitude Osceola's E prime. Perhaps I should say magnitude isn't the two times the electric field contribution to any from any one of these charges, and we can find the angle to be inverse tangent or arc tangent of E y over E X. When you look at this ratio, divine ey over e x. We simply get 1/3, which shouldn't come out to Mercer. Spies would be exactly 30 degrees. We're seeing all that it makes with both charged three in charge do so here. Our final answers are total magnitude and our angle relationship.