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Consider tne following Laplace' equavon(y)-uy(*Y) =0<*<2 D<y<2subject to the given boundary conditionsu(o,y) = u(2y)-Y(2-v), Vy (K,0) = Uy (x,2)=0&...

Question

Consider tne following Laplace' equavon(y)-uy(*Y) =0<*<2 D<y<2subject to the given boundary conditionsu(o,y) = u(2y)-Y(2-v), Vy (K,0) = Uy (x,2)=0<Y < 2 0<y < 2 0 <* <2 <<2Given that the general solution of the Laplace' equationMX QX cosh _ B, sinhnaku(xy) = Ag Bo* +Detemmine the values of Aj, Bo A, and B, Hence, vilta tho speclic solullon ulxy) (10 marks}Consider circular nng conditions areinner radiusand outer radius where the boundary0)-100 50cos

Consider tne following Laplace' equavon (y)-uy(*Y) = 0<*<2 D<y<2 subject to the given boundary conditions u(o,y) = u(2y)-Y(2-v), Vy (K,0) = Uy (x,2)= 0<Y < 2 0<y < 2 0 <* <2 <<2 Given that the general solution of the Laplace' equation MX QX cosh _ B, sinh nak u(xy) = Ag Bo* + Detemmine the values of Aj, Bo A, and B, Hence, vilta tho speclic solullon ulxy) (10 marks} Consider circular nng conditions are inner radius and outer radius where the boundary 0)-100 50cos 8 ult 8) = 200, 0<8<21 0 <644 If the gonoral solution E ulc, 8) = Aa Bo Inr + (Ar" B,r-= cosng detemmlne the specfic _ solubon "lcar" Dor sinne_



Answers

A slab of current-carrying conductor extends infinitely in the x-y plane, and has thickness d along the z-axis, and is centered on the origin.
The slab carries a uniform current density~J=Jˆx pointing out of the page.
Find the magnetic field (both magnitude and direction) inside (?d/2?z?d/2)and outside (|z| ?d/2) the slab,
as functions of the distance z from the center plane.
Sketch a graph ofB(z) vs z with B(z), choosing a sensible sign convention for which direction is “positive.”Hint:
Imagine
that
the
slab
is
composed
of
a
large
number
of
parallel
wires
and
imagine
what the superposition of their fields must look like above and below the slab (recall our superposition worksheet/exercise).
Use the symmetry of the problem and Ampere’s Law.

Hello. We have a question in which parametric questions are given as X. Equal to cause to T. And Y equal to 10 costly. Where it is between zero to okay one or zero to anything that is not visible here. So we have to find the body X. And the esquire whereby the X. Squared. So for parametric equation do you I buy the X. Equal to divide by D. T. Divide by dx by DT. So first of all we need to find the Ui by D. T. So this is minus 10 cost and the X. By deity that is minus two. Okay this is I'm sorry because this will be minus two and scientific as cost. He has differentiation Scientific and the X. By DTs minus two scientists. So do you have a. D. X. Will be cool too minus 10 sign D. It will be because it has two T. So let us right to tear. Okay divide by minus to sign duty. Okay this will be five sign T. by signed to T. Now from double angle formulas I in two days to scientific or stay to sign T. Cost so scientists scientific that canceled out. So we have finally divided by D. X. Equal to five by two. And cost is in denominators or by reciprocal identity. It will be safety. So this is disability. DX Now the Squire whereby the excess choir. This is quite whereby the excess choir. We have to just all right leg. Okay five by two this is safety. So this will be safety and then T. Mhm. So these are the answers. Thank you so much.

In this problem. Two equations and in X and Y coordinate system is given. And both of these equations represents two different circles. So what are the equations here? Let us say first equation is circle C one and its equation is ex esquire bless Why minus half? Holy Square as equals to one by four. So if we write this equation in a standard form this will be x minus zero. Holy squared plus y minus half. Holy square is equals 21 by two. Holy square. So this is the equation for circle one. So what we can extract from this equation is center of, the circle is at zero comma half, and radius of the circle is half. This is the radius. Okay no question of the second circle is given as x minus half. Holy Square Bliss Why is quite is equals to one by four. So again, if we write this equation in standard form so it will reduce us x minus half. Police. Quiet Bliss, Y minus zero. Holy square is equals two half. Holy square. So what we can extract from this equation, that center will be half comma zero. So the center of the second circle is half comma zero and radius is half. So if he go for and to plot the plot a plot these equations or the circles. So it will be, let us say this is why you exist and this one is X axis. So we have to draw two circles. Like this is so this circle will be tangent to X axis and the second circle here it will be tangent to buy access. Now if you look at the centers so this is center for the first circle. So this was curve seven and this was for C. Two. So this is for seven and this is C. Two. And center here is for first circle is zero comma half And for the second circle is zero comma sorry, half comma zero. Since it is on the X axis. So this is X. Axis. And this is why you exist. All right? We have to find area of intersection. So clearly we have to find this area. So how we will write this this area. So we can choose either horizontal strip or vertical strip. So, before proceeding for calculation of area. With the help of double integration method. First of all, we need to find points of intersection. One point of intersection is clearly on the origin. So, we need to find the second point of intersection here. Okay, for that we will solve the two equations of the circle. So let us say this is equation one and this is equation too. So question one is ex esquire bliss. Why minus half. Holy Square is equals to one by four. So this is equation one. No we can expand this bracket. So this will be X square plus Y squared minus two times one by two into Y plus one by two. Holy Square is equals to one by four. Now this one by two holy Square is actually one by four. So this one by 41 by four can be canceled here to gets canceled with two. So the remaining in terms are x squared plus y squared minus Y is equals to zero. So let us say uh no we can more simplify it as X squared plus y squared. It's equals to hawaii let us say this is equation two. Do we have already mentioned? This is a question three. No this was a question too. So we can simplify a question to also. So from a question too, what was equation two, X minus half Holy square plus why is Choir is equals to one by four. So if we expand this bracket this will be x squared minus two times X into one by two plus one by two. Holy Squire plus Y squared as equals to one by full. So from here one by two leagues queries one by four. So this one by 41 by two is square and one by four can be cancelled and here to gets cancelled with the two. That is in multiplication. So the remaining terms will be X. Esquire minus X plus why square is equals to zero? Now we can put the value of X squared plus y squared from equation three. So as in equation three, X squared plus y square was it was Y. So putting the value of X squared plus y squared from equation three. So this equation will reduce to why minus X is equal to zero. Therefore why is equals two. X. Let us say this is equation number, what was the last number? It was three. So this is the question number four. Now we can put the value of why it goes to X in equation three. So working, why equals two X. And equation three. What was the question? Three, equation three was X squared plus Y squared equals two. Why? So if you put the value of Y. Cultura X or X by Y. So this will be Y squared plus Y squared is equals two. Why? So this becomes too wide square is equals two. Y. So we can take a Y. In left side of the equality. So this will be too wise. Square minus Y is equal to zero. Now why can't we take uncommon? So this will be Y into two. Y minus one is equals to zero. So from here we get to solutions. First days, either this wife must be zero or two. Y minus ones would be zero. Then then from here we get to values of Y. First is why is equals to zero and second is to why management equals to zero. So why is equals 21 by two. So we have got to values of Y. Now for these two values of why we can get the value of X corresponding to its value of Y from equation for So what was the question for? It is y is equal to X. So from equation four for why is equals to zero, X equals to zero. For wise equals to have X is also equals to half. Therefore points of intersection. Uh huh. Zero comma zero and one by two comma one by two. Okay. No, let us move to the uh diagram that we had drawn initially. So this was the diagram. Now I'm copying it since we need it more than one times in this problem, so I'm copying it here. Okay, so this was the diagram and now we have got the value of points of intersection. So this point is one by two comma one by two and the second point is zero comma zero. Now we can take we can choose either horizontal strip or particular strip. So if he is considered here, a particular street in this region whose area we have to calculate. So limit of this particular strict will go from lower limit is lower limit of the circle of this strip. We can see here that it is on circle C. One, while the upper limit is on circle C. Two. Therefore area for this particular strip can be written as in double integration form so area is equals two. Oh DVD X. Yeah know what will be the value of Y. So value of why will be why is ranging from seven curve to see tucker from seven curb to see tucker. Now we have to write these values of seven and C two a value here, we have to put value of Y as a function of X. And what is the range of values of X. So X is going from 0 to 1 by two. G 021 by two. So instead of writing seven institute since we have already used this these two terms to indicate the circles. So I'm writing here some other things like FX and GX so this is Fx and G X. So we need we have to find the value of fx So fx will be equation are effects will be value of Y in terms of x. Further curve Shivan. So what was Stephen? So steven was a question of Stephen was x squared plus y minus half. Holy square is equals to one by four. So if we expand this, this becomes x squared plus Y squared minus two way plus minus. Sorry, this will be minus two into one by two and two, Y plus one by four is equals to one by four. So here when by four gets canceled and to also gets canceled. So the remaining terms are excess squad plus y squared minus Y is equal to zero. So from here we have to get the value of Y in terms of X, how we will get that. So this is a question of and this is equation in Y of order two of degree two. So this is a quadratic equation and why? So how we solve here for X? So why is equals two minus B. That is minus of minus one plus minus under root of be square, so be square can be minus one. Holy square minus four into A. That is one in to see that is see excess while upon the way. So why is equals to one plus minus under root of one minus four. X esquire divided by two. So why is equals two. Now here we have to choose. We have to choose to values since this reason is lying in first quadrant. Therefore we will go for the positive value of this expression so it will be one plus under root of one minus four X esquire all divided by two. So this is the function what This was the equation of curve Shivan. Therefore this is for the lower limit and it is the value of F of X. So ffx is equals to one plus under root of one minus four. X esquire divided by two. Similarly we will go forward we will go to find the value of G. X. That is the equation of Y as a function of X. Forker. She too. So what was he to C two was x minus half fully square plus. Y squared is equal to one by four. So we can simplify this equation as we already know that here. Why? Square turn with isolated so we can directly get the value of why. In terms of X as y square will be won by four minus x minus half. Holy square. So from here if we square root both the side. So this will be wise equals to plus minus under root of one by four minus x minus half. Holy square. So from here again we get to values of why but our region is lying in first quarter and therefore value of why will be positive so we will choose positive value here. So it will be under root of one by four minus x minus half. Holy square so how the value of G X will be here and the root of one by four minus x minus half. Holy square. Now we've got the value of fx and gx so we can put these values in double integration here for expression of 84 area. So area will be integration from 0 to 10 to half. This is limit fall X. And limit for what will be. We have got the value of fx what it was one plus and the root of one minus four. X. Esquire whole divided by two. This is the lower limit of Y and upper limited G. X. So what is the X under root of one by four minus x minus half. Holy squared. And what is written here? Dy dx Why is written inside Because we are going to integrate first with respect to Y. Since these two functions are the limit of white. So this is the expression of area in uh in the form of double integration in X. Y system. So this will be the answer for part A of the problem. As we can see here that these expressions will be very lengthy while we will putting the value of limit of Y and then evaluating the integration with respect to X will be a difficult task. So we have to switch the coordinate system from coordinate system from polar coordinate system from Cartesian coordinate system to the polar coordinate system. Why we are choosing polar coordinate system? Because does diagram or the areas have circular symmetry? So since they are circular, therefore choosing the polar Gardner system will be very beneficial for us. So again, I'm going to hear right rather paste the picture so this is the picture. And now we have to write polar form of this equation. So first we should recall that how to how to write polar form of a circle. So for a circle polar farm is written as to a cost peter less to be a scientist to with equal maybe is that center? So from here, what will be the equation for seven for seven? As we can see here that he is zero here we can see that 80 and B is half. Therefore, expression of our will be two in 20 cost theater Bliss two and two half scientific. Therefore equation of C. One will be in polar coordinates system R is equal to this first time will be zero. Therefore the remaining terms are these two can be canceled. So R. Is equal to sine theta. No, if you go for to write the equation of C. Two similarly. So our will be equals to hear A value of A is half and value of B is zero. So this will be two and two half cost later Bliss two in 20 scientist to So expression of our reduces to the second term of this expression will be zero. So the remaining term here too can be canceled. So it is our as equals to cost hitter. Therefore here the curve she tube this girl has equation are is equals to cost heater and the second curbs C one has equations. Our is equals two. Scientific. No, we knew the point of intersection of these two circles where half comma half. So we will get the value of theater for this point of intersection as we know that our is equals to cost theater. So what is the value of our here? So r. Is the distance of the point from autism. So how we will get the distance? So distance in Cartesian coordinate system, we know that under root of excellent minus X two. Holy square or is equals two. Excellent minus X two. Holy square bless. Yeah. Why one minus Y. To holy square. So this is equals two. Mhm. And the root of X. How what is the value of X. One here? So this line segment is from zero comma zero to half comma half. I'm talking about this line segment. So distance of this line segment will be one by two minus zero. Holy square plus one by two minus zero. Holy squared. So this is equals two and the root of one by four plus one by four. This is equals two under root of one by two. So this is one by route to. So we have to put this value of R. S. Equals to one by route to in either equation of seven or equation of C. Two. Therefore one by route to is equals too costly to therefore tita is equals to buy buy food. It means that this line of intersection or this common line that I've grown here by a black color. So this angle angle for this, the line is theta is equal to buy buy food. No, if you draw of infinitesimally small strip on this polar cornered system, let us say this is stripped. So for this trip, uh, here I'm saying it again. So I'm talking about this strip. So this strip has lower limit at origin and upper limit on the curve. Seven and seven was described in polar coordinate by the equation R is equal to sine theta. Therefore this for this history value of our will range from zero to scientists. Um, and this uh, and the curve if you integrate this area. So this strip follows the curve. She won up to the value of theta equals two pi by four. Right? Therefore for the region that I'm going to show here with a different color. Let us I'm using green color. So for this green color region for this portion of the graph of our, for the area of intersection, the graph is going to follow the CIA bunker and for the rest of the part that I'm going to use here orange color. So for this interesting for this area that infinitesimally small element is going to follow the curve. She too. So if if I draw here both the curve. So basically we had to find this this area and I'm calculating this area separately and the upper area this one separately because these two curves are different. The lower curve is of equation C one and the upper one is for equation C. Two and the seven is described by Rs equals two. R. S. Equals two scientists and the C two is described by the equation are as equals to cause theta. Therefore, if you date idea so area will be equals to even plus eight to what is even even. Is the double integration of D. R. D. Theta and value of art is ranging from zero to scientist to and tita is ranging from zero to pi by food. And what is the value of a two? A two is the upper portion of this area. So area will be indeed level integration of D. R. D. Theater and what is the range of art. So art is art is going from and now this time zero to cost heater. So this is zero tube cost hitter and tita is going from five by 42 by by two. So for this range of tita mm the radius is as a function radius is ranging from zero to cost theater. So basically to find this area we have to evaluate this simple integration. So first we will integrate with respect to our here. So this is 0 to 5 by four. No integration of DRS are and limit of art. We have to put from zero to scientist to and to the theater plus integration from zero to from by by four to highway two. Now again integration of D. R. S. R. And the limit of our is from zero to scientist to deter. So if you evaluate this integration and put the limit so this reduces to integration from zero to pi by four. Mhm. So instead of art we can put scientist to scientist. Um and zero will be simply science theater. So this is scientific to the theater plus integration from by by four too. Bye bye to now again. Okay I think I have made some mistake here. All right here a limit of integration is going to be from zero to cost theater. So this will be from here in second term a second integration. It will be caused theater and the theater. Okay now we have to evaluate the simple integration. Now we know that integration of science theater is minus course theater. Therefore this is equals two minus cost hitter and limit of integration is from zero to buy buy food bless integration of course Tita is simply Science theater And limit of integration is from by by 42 by by two. Now we can put these values here. So the first term in first integration our first area will be minus cause of five by four minus minus cost zero plus Now if you put the limits in second term so this will be signed by way too minus signed by by for So we can evaluate these values here. So caused by by two is basically zero minus into minus is plus because zero is one. So this is the first term and in second term signed power to is one and signed power for is one by route to so this is equals to one plus one. Is two minus one by route to. So we can get this value here, we can solve this expression so two minus this. If you multiply here by route to buy route to. So this is route two divided by two. What is the value of route to 1.414 So we can put the value 1.414 divided by two. So this will be equals two two minus 0.707 So if it's abstract this, this will be a point three and nine and 972 Okay. And 1.273 Therefore area of the area of the common intersection portion is one point 293 So this is the answer for the second part. So if you if you know summarize the results here. So what was the answer for first part? So this was the answer, So I'm copying it. So this was the answer for part A and for part B answer is area is equals to one point 293 All right. So these are the final answers of this problem.

In this problem. Two equations and in X and Y coordinate system is given. And both of these equations represents two different circles. So what are the equations here? Let us say first equation is circle C one and its equation is ex esquire bless Why minus half? Holy Square as equals to one by four. So if we write this equation in a standard form this will be x minus zero. Holy squared plus y minus half. Holy square is equals 21 by two. Holy square. So this is the equation for circle one. So what we can extract from this equation is center of, the circle is at zero comma half, and radius of the circle is half. This is the radius. Okay no question of the second circle is given as x minus half. Holy Square Bliss Why is quite is equals to one by four. So again, if we write this equation in standard form so it will reduce us x minus half. Police. Quiet Bliss, Y minus zero. Holy square is equals two half. Holy square. So what we can extract from this equation, that center will be half comma zero. So the center of the second circle is half comma zero and radius is half. So if he go for and to plot the plot a plot these equations or the circles. So it will be, let us say this is why you exist and this one is X axis. So we have to draw two circles. Like this is so this circle will be tangent to X axis and the second circle here it will be tangent to buy access. Now if you look at the centers so this is center for the first circle. So this was curve seven and this was for C. Two. So this is for seven and this is C. Two. And center here is for first circle is zero comma half And for the second circle is zero comma sorry, half comma zero. Since it is on the X axis. So this is X. Axis. And this is why you exist. All right? We have to find area of intersection. So clearly we have to find this area. So how we will write this this area. So we can choose either horizontal strip or vertical strip. So, before proceeding for calculation of area. With the help of double integration method. First of all, we need to find points of intersection. One point of intersection is clearly on the origin. So, we need to find the second point of intersection here. Okay, for that we will solve the two equations of the circle. So let us say this is equation one and this is equation too. So question one is ex esquire bliss. Why minus half. Holy Square is equals to one by four. So this is equation one. No we can expand this bracket. So this will be X square plus Y squared minus two times one by two into Y plus one by two. Holy Square is equals to one by four. Now this one by two holy Square is actually one by four. So this one by 41 by four can be canceled here to gets canceled with two. So the remaining in terms are x squared plus y squared minus Y is equals to zero. So let us say uh no we can more simplify it as X squared plus y squared. It's equals to hawaii let us say this is equation two. Do we have already mentioned? This is a question three. No this was a question too. So we can simplify a question to also. So from a question too, what was equation two, X minus half Holy square plus why is Choir is equals to one by four. So if we expand this bracket this will be x squared minus two times X into one by two plus one by two. Holy Squire plus Y squared as equals to one by full. So from here one by two leagues queries one by four. So this one by 41 by two is square and one by four can be cancelled and here to gets cancelled with the two. That is in multiplication. So the remaining terms will be X. Esquire minus X plus why square is equals to zero? Now we can put the value of X squared plus y squared from equation three. So as in equation three, X squared plus y square was it was Y. So putting the value of X squared plus y squared from equation three. So this equation will reduce to why minus X is equal to zero. Therefore why is equals two. X. Let us say this is equation number, what was the last number? It was three. So this is the question number four. Now we can put the value of why it goes to X in equation three. So working, why equals two X. And equation three. What was the question? Three, equation three was X squared plus Y squared equals two. Why? So if you put the value of Y. Cultura X or X by Y. So this will be Y squared plus Y squared is equals two. Why? So this becomes too wide square is equals two. Y. So we can take a Y. In left side of the equality. So this will be too wise. Square minus Y is equal to zero. Now why can't we take uncommon? So this will be Y into two. Y minus one is equals to zero. So from here we get to solutions. First days, either this wife must be zero or two. Y minus ones would be zero. Then then from here we get to values of Y. First is why is equals to zero and second is to why management equals to zero. So why is equals 21 by two. So we have got to values of Y. Now for these two values of why we can get the value of X corresponding to its value of Y from equation for So what was the question for? It is y is equal to X. So from equation four for why is equals to zero, X equals to zero. For wise equals to have X is also equals to half. Therefore points of intersection. Uh huh. Zero comma zero and one by two comma one by two. Okay. No, let us move to the uh diagram that we had drawn initially. So this was the diagram. Now I'm copying it since we need it more than one times in this problem, so I'm copying it here. Okay, so this was the diagram and now we have got the value of points of intersection. So this point is one by two comma one by two and the second point is zero comma zero. Now we can take we can choose either horizontal strip or particular strip. So if he is considered here, a particular street in this region whose area we have to calculate. So limit of this particular strict will go from lower limit is lower limit of the circle of this strip. We can see here that it is on circle C. One, while the upper limit is on circle C. Two. Therefore area for this particular strip can be written as in double integration form so area is equals two. Oh DVD X. Yeah know what will be the value of Y. So value of why will be why is ranging from seven curve to see tucker from seven curb to see tucker. Now we have to write these values of seven and C two a value here, we have to put value of Y as a function of X. And what is the range of values of X. So X is going from 0 to 1 by two. G 021 by two. So instead of writing seven institute since we have already used this these two terms to indicate the circles. So I'm writing here some other things like FX and GX so this is Fx and G X. So we need we have to find the value of fx So fx will be equation are effects will be value of Y in terms of x. Further curve Shivan. So what was Stephen? So steven was a question of Stephen was x squared plus y minus half. Holy square is equals to one by four. So if we expand this, this becomes x squared plus Y squared minus two way plus minus. Sorry, this will be minus two into one by two and two, Y plus one by four is equals to one by four. So here when by four gets canceled and to also gets canceled. So the remaining terms are excess squad plus y squared minus Y is equal to zero. So from here we have to get the value of Y in terms of X, how we will get that. So this is a question of and this is equation in Y of order two of degree two. So this is a quadratic equation and why? So how we solve here for X? So why is equals two minus B. That is minus of minus one plus minus under root of be square, so be square can be minus one. Holy square minus four into A. That is one in to see that is see excess while upon the way. So why is equals to one plus minus under root of one minus four. X esquire divided by two. So why is equals two. Now here we have to choose. We have to choose to values since this reason is lying in first quadrant. Therefore we will go for the positive value of this expression so it will be one plus under root of one minus four X esquire all divided by two. So this is the function what This was the equation of curve Shivan. Therefore this is for the lower limit and it is the value of F of X. So ffx is equals to one plus under root of one minus four. X esquire divided by two. Similarly we will go forward we will go to find the value of G. X. That is the equation of Y as a function of X. Forker. She too. So what was he to C two was x minus half fully square plus. Y squared is equal to one by four. So we can simplify this equation as we already know that here. Why? Square turn with isolated so we can directly get the value of why. In terms of X as y square will be won by four minus x minus half. Holy square. So from here if we square root both the side. So this will be wise equals to plus minus under root of one by four minus x minus half. Holy square. So from here again we get to values of why but our region is lying in first quarter and therefore value of why will be positive so we will choose positive value here. So it will be under root of one by four minus x minus half. Holy square so how the value of G X will be here and the root of one by four minus x minus half. Holy square. Now we've got the value of fx and gx so we can put these values in double integration here for expression of 84 area. So area will be integration from 0 to 10 to half. This is limit fall X. And limit for what will be. We have got the value of fx what it was one plus and the root of one minus four. X. Esquire whole divided by two. This is the lower limit of Y and upper limited G. X. So what is the X under root of one by four minus x minus half. Holy squared. And what is written here? Dy dx Why is written inside Because we are going to integrate first with respect to Y. Since these two functions are the limit of white. So this is the expression of area in uh in the form of double integration in X. Y system. So this will be the answer for part A of the problem. As we can see here that these expressions will be very lengthy while we will putting the value of limit of Y and then evaluating the integration with respect to X will be a difficult task. So we have to switch the coordinate system from coordinate system from polar coordinate system from Cartesian coordinate system to the polar coordinate system. Why we are choosing polar coordinate system? Because does diagram or the areas have circular symmetry? So since they are circular, therefore choosing the polar Gardner system will be very beneficial for us. So again, I'm going to hear right rather paste the picture so this is the picture. And now we have to write polar form of this equation. So first we should recall that how to how to write polar form of a circle. So for a circle polar farm is written as to a cost peter less to be a scientist to with equal maybe is that center? So from here, what will be the equation for seven for seven? As we can see here that he is zero here we can see that 80 and B is half. Therefore, expression of our will be two in 20 cost theater Bliss two and two half scientific. Therefore equation of C. One will be in polar coordinates system R is equal to this first time will be zero. Therefore the remaining terms are these two can be canceled. So R. Is equal to sine theta. No, if you go for to write the equation of C. Two similarly. So our will be equals to hear A value of A is half and value of B is zero. So this will be two and two half cost later Bliss two in 20 scientist to So expression of our reduces to the second term of this expression will be zero. So the remaining term here too can be canceled. So it is our as equals to cost hitter. Therefore here the curve she tube this girl has equation are is equals to cost heater and the second curbs C one has equations. Our is equals two. Scientific. No, we knew the point of intersection of these two circles where half comma half. So we will get the value of theater for this point of intersection as we know that our is equals to cost theater. So what is the value of our here? So r. Is the distance of the point from autism. So how we will get the distance? So distance in Cartesian coordinate system, we know that under root of excellent minus X two. Holy square or is equals two. Excellent minus X two. Holy square bless. Yeah. Why one minus Y. To holy square. So this is equals two. Mhm. And the root of X. How what is the value of X. One here? So this line segment is from zero comma zero to half comma half. I'm talking about this line segment. So distance of this line segment will be one by two minus zero. Holy square plus one by two minus zero. Holy squared. So this is equals two and the root of one by four plus one by four. This is equals two under root of one by two. So this is one by route to. So we have to put this value of R. S. Equals to one by route to in either equation of seven or equation of C. Two. Therefore one by route to is equals too costly to therefore tita is equals to buy buy food. It means that this line of intersection or this common line that I've grown here by a black color. So this angle angle for this, the line is theta is equal to buy buy food. No, if you draw of infinitesimally small strip on this polar cornered system, let us say this is stripped. So for this trip, uh, here I'm saying it again. So I'm talking about this strip. So this strip has lower limit at origin and upper limit on the curve. Seven and seven was described in polar coordinate by the equation R is equal to sine theta. Therefore this for this history value of our will range from zero to scientists. Um, and this uh, and the curve if you integrate this area. So this strip follows the curve. She won up to the value of theta equals two pi by four. Right? Therefore for the region that I'm going to show here with a different color. Let us I'm using green color. So for this green color region for this portion of the graph of our, for the area of intersection, the graph is going to follow the CIA bunker and for the rest of the part that I'm going to use here orange color. So for this interesting for this area that infinitesimally small element is going to follow the curve. She too. So if if I draw here both the curve. So basically we had to find this this area and I'm calculating this area separately and the upper area this one separately because these two curves are different. The lower curve is of equation C one and the upper one is for equation C. Two and the seven is described by Rs equals two. R. S. Equals two scientists and the C two is described by the equation are as equals to cause theta. Therefore, if you date idea so area will be equals to even plus eight to what is even even. Is the double integration of D. R. D. Theta and value of art is ranging from zero to scientist to and tita is ranging from zero to pi by food. And what is the value of a two? A two is the upper portion of this area. So area will be indeed level integration of D. R. D. Theater and what is the range of art. So art is art is going from and now this time zero to cost heater. So this is zero tube cost hitter and tita is going from five by 42 by by two. So for this range of tita mm the radius is as a function radius is ranging from zero to cost theater. So basically to find this area we have to evaluate this simple integration. So first we will integrate with respect to our here. So this is 0 to 5 by four. No integration of DRS are and limit of art. We have to put from zero to scientist to and to the theater plus integration from zero to from by by four to highway two. Now again integration of D. R. S. R. And the limit of our is from zero to scientist to deter. So if you evaluate this integration and put the limit so this reduces to integration from zero to pi by four. Mhm. So instead of art we can put scientist to scientist. Um and zero will be simply science theater. So this is scientific to the theater plus integration from by by four too. Bye bye to now again. Okay I think I have made some mistake here. All right here a limit of integration is going to be from zero to cost theater. So this will be from here in second term a second integration. It will be caused theater and the theater. Okay now we have to evaluate the simple integration. Now we know that integration of science theater is minus course theater. Therefore this is equals two minus cost hitter and limit of integration is from zero to buy buy food bless integration of course Tita is simply Science theater And limit of integration is from by by 42 by by two. Now we can put these values here. So the first term in first integration our first area will be minus cause of five by four minus minus cost zero plus Now if you put the limits in second term so this will be signed by way too minus signed by by for So we can evaluate these values here. So caused by by two is basically zero minus into minus is plus because zero is one. So this is the first term and in second term signed power to is one and signed power for is one by route to so this is equals to one plus one. Is two minus one by route to. So we can get this value here, we can solve this expression so two minus this. If you multiply here by route to buy route to. So this is route two divided by two. What is the value of route to 1.414 So we can put the value 1.414 divided by two. So this will be equals two two minus 0.707 So if it's abstract this, this will be a point three and nine and 972 Okay. And 1.273 Therefore area of the area of the common intersection portion is one point 293 So this is the answer for the second part. So if you if you know summarize the results here. So what was the answer for first part? So this was the answer, So I'm copying it. So this was the answer for part A and for part B answer is area is equals to one point 293 All right. So these are the final answers of this problem.

Yeah. We're going to look at the magnetic field of a conducting slab that's carrying a current density in the X. Direction. And we're going to use NPR's law to find the magnetic field both inside and outside the slab to understand how this works. It pays to think about the current density as made up of individual liars which I've shown inside our blue slab down below. And this will kind of help us determine the symmetry that's involved. So like usual you can use the right hand rule to figure out the way the magnetic field loops around a wire. Um And I'm going to do that for a couple of the wires in that slab. Those imaginary wires that are representing our current density. And the way I have it drawn the magnetic field circulates counter clockwise around all those little wires. And I just want to do a couple of adjacent ones so that we can see what happens Yeah. When we start super posing those little wires. So if you look you can see that wires adjacent to each other, cancel the Z. Component of the magnetic field. So she is up and down like usual. And that will happen all the way across the board. However if we look at the top of the wires um those are pointing all in the negative Y. Direction. And those are going to conspire to make a strong magnetic field. So what we can now say is that the magnetic field is going to have a very strong why component? Um No matter where you are looking at the magnetic field. The other thing we can notice is that uh huh. Where the two wires imaginary butt up against each other at the origin. Now they are quite at the origin. But you can imagine if they were. What's happening. Is that the why components actually get a chance to cancel? So that also tells us that be why of zero has to equal zero. So we know a couple things already just by drawing a nice diagram. So the next thing to do is we're going to find the magnetic field inside by using amperes Law. And so what we'll have to do is draw an ampere and loop. Um And I'll make that loop loop all the way around um in the direction that we expect those magnetic fields to be circulating. And what I see, we'll give that loop a length C. And it extends up, it's supposed to be symmetric. Um See down from minus C. All the way up to see. So it's an arbitrary loop just somewhere inside. Uh It stops at some arbitrary coordinate inside but we have B. And then we know it's a. Y. Components. And we'll go ahead and do that dotted onto that loop has to places where there's a contribution, the top and the bottom. So that's twice times L. Is equal to you. Not jay times the area cross sectional area of that loop, which is L. Times two Z. Mhm. And I'm going to be careful when I finally write it down. But we see that the the two's cancel. That's nice and the arbitrary L goes away. That's nice. And so we're left with here. I'm going to be very careful when I write my component down. B. Y. Is minus you, not J. Time. See because that magnetic field points to the minus Y when he's positive and it flips around to the positive direction down below. So this is what the magnetic field looks like inside. And that's all very well and good. It goes to zero at the origin and we're happy. Okay, now let's take a look outside and we're going to look above the plane and it'll be something very similar below the plane. But we'll do that fairly quickly. Again, I'll draw my ampere ian loop for above the plane will let the loop extend outside the material. And uh it's supposed to be symmetric about the origin. But I kind of missed up some of those wires and it really doesn't matter. Mhm. But again, we have B. Y. And there's our loop, but the only place the magnetic field contributes is up the top. Um So B. Y. Times L. Is equal to you're not J times the cross sectional area is ill times A. Now why I started not A. D. Over to. Why did we stop at the over to? The reason we stopped at the over to is because we stopped enclosing current when we ran out of material so jane closed. And so we have to limit that that area. Yeah. Yes. And again the els cancel. And we're going to be very careful writing down um The why? Mhm. Yes. We're not going to smear it with that marker. Um B. Y. Sorry I can't get rid of the marker here. Okay, so B. Y. Is equal to minus mu naught jay. Do you over to Why? Negative? Because the right hand rule gives us that direction above the plane and it's going to be a very similar situation for below the plane. The loop that we're going to draw will surround some of the material but it will also go outside and we're going to loop it around in the direction of the right hand rule. The magnetic field. So be dot Dll is going to give us just B. Y. L. And that's mu not times the I. Enclosed, which is just J. Times again the area inside the material. Again, the els cancel. And I'm left with B. Y. Equals you're not J. Do you over to And that is going to be a positive component simply because that's the direction the magnetic field points in that part of the world. And what last thing to do. We're going to draw a graph of what B. Y. Looks like as a function of Z. Yeah, so it's zero on the inside and it grows to a constant negative value up to D over to and then it flattens out and it's negative whereas he is positive and it does exactly the opposite in the other direction. And that's typical is that the magnetic field will grow inside of a conductor. And then once you're outside it stops growing. And oftentimes actually diminishes here because our planes are conducting slab is infinite. We don't have that drop off


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