Home / Natural Sciences & Medicine /QUESTION 1 r? b. Given point P1 = (1,-1, 2) and a vector, C = x + 24, +y? + y2 i. Express point P, and vector C in spherical coordinate system. (10 marks) ii. Evaluate vector Cat Pı. (3 marks) iii. Find divergence of vector C. (8 marks)

Question QUESTION 1 r? b. Given point P1 = (1,-1, 2) and a vector, C = x + 24, +y? + y2 i. Express point P, and vector C in spherical coordinate system. (10 marks) ii. Evaluate vector Cat Pı. (3 marks) iii. Find divergence of vector C. (8 marks)

H1ILAV The Asker · Physics

Transcribed Image Text: QUESTION 1 r? b. Given point P1 = (1,-1, 2) and a vector, C = x + 24, +y? + y2 i. Express point P, and vector C in spherical coordinate system. (10 marks) ii. Evaluate vector Cat Pı. (3 marks) iii. Find divergence of vector C. (8 marks)
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Transcribed Image Text: QUESTION 1 r? b. Given point P1 = (1,-1, 2) and a vector, C = x + 24, +y? + y2 i. Express point P, and vector C in spherical coordinate system. (10 marks) ii. Evaluate vector Cat Pı. (3 marks) iii. Find divergence of vector C. (8 marks)
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088UZ3

The relations between cartesian and spherical coordin - ates are{:[x=r sin theta cos phi],[y=r sin theta sin phi],[z=r cos theta]:}Where, r is the distance of the point from arigin, theta is the angle made by r with z axis and phi is the angle made by the component n sin theta with x axis.(i) We have to express the point (1,-1,2) in sphericas coordinates.From equations (1) we get{:[r=sqrt(x^(2)+y^(2)+z^(2))=sqrt(1^(2)+(-1)^(2)+2^(2))=sqrt6],[phi=tan^(-1)(y//x)=tan^(-1)(-1//1)=-pi//4],[theta=cos^(-1)(z//r)=cos^(-1)((2)/(sqrt6))=35.26^(@)],[(1","-1","2)longrightarrow^(" spherical ")(sqrt6","quad","-pi//4)]:}(11) vec(c)= hat(x)((y^(2))/(x^(2)+y^(2)))- hat(y)((x^(2))/(x^(2)+y^(2)))+4 hat(z)we have, to express vec(c) in sphenical coordinate.First, we will find out the relations between the unit rectars of the tro systems.{:[ vec(r)=x hat(i)+y hat(j)+2 hat(k)quad[" position recter in contesian] "],[ vec(r)=(r sin theta cos phi) hat(i)+(n sin theta sin phi) hat(j)+(n cos theta) hat(k)],[[[" Position rector in "],[" spherical "]]:}],[:. hat(r)=(del( vec(n))∣del n)/(|(del( vec(n)))/(del n)|)=(sin theta cos phi) hat(i)+(sin theta sin phi) hat(j)+(cos theta) hat(x)],[ hat(theta)=((del( vec(r)))/(del theta))/(|(del( vec(x)))/(del theta)|)=(1)/(b)[(r cos theta cos phi) hat(i)+(n cos theta sin phi) hat(j)],[ hat(phi)=((del( vec(r)))/(del phi))/(|(del( vec(n)))/(del q)|)=(1)/(r sin theta)[(-r sin theta sin phi) hat(i)+(b sin theta cos phi) hat(j)]],[ hat(phi)=-(cos theta cos phi) hat(i)+(cos theta sin phi) hat(j)-(sin theta) hat(x)+(cos phi) hat(j)]:}Now, a vector can be expressed in any co-ardinale system with theip coordinates and unit reetors in each system.vec(c)=c_(3) hat(i)+c_(y) hat(ȷ)+c_(z) hat(x) it can be expressed in spherical as vec(c)=c_(r) hat(n)+c_(theta) hat(theta)+c_(phi) hat(phi) mere,{:[c_(n)= vec(c)* hat(b)],[c_(theta)= vec(c)* hat(theta)],[c_(phi)= vec(c)* hat(phi)]:}{:[" Here, " vec(c)=[((y^(2))/(x^(2)+y^(2)))^( hat(L))-((x^(2))/(x^(2)+y^(2)))( hat(j))+4( hat(z))]],[:.c_(n)= vec(c_(1)) hat(n)],[=[((r^(2)sin^(2)thetasin^(2)phi)/(r^(2)sin 2theta))( hat(i))-((r^(2)sin ... See the full answer

6L81ZN

The relations between cartesian and spherical coordin - ates are{:[x=r sin theta cos phi],[y=r sin theta sin phi],[z=r cos theta]:}Where, r is the distance of the point from arigin, theta is the angle made by r with z axis and phi is the angle made by the component n sin theta with x axis.(i) We have to express the point (1,-1,2) in sphericas coordinates.From equations (1) we get{:[r=sqrt(x^(2)+y^(2)+z^(2))=sqrt(1^(2)+(-1)^(2)+2^(2))=sqrt6],[phi=tan^(-1)(y//x)=tan^(-1)(-1//1)=-pi//4],[theta=cos^(-1)(z//r)=cos^(-1)((2)/(sqrt6))=35.26^(@)],[(1","-1","2)longrightarrow^(" spherical ")(sqrt6","quad","-pi//4)]:}(11) vec(c)= hat(x)((y^(2))/(x^(2)+y^(2)))- hat(y)((x^(2))/(x^(2)+y^(2)))+4 hat(z)we have, to express vec(c) in sphenical coordinate.First, we will find out the relations between the unit rectars of the tro systems.{:[ vec(r)=x hat(i)+y hat(j)+2 hat(k)quad[" position recter in contesian] "],[ vec(r)=(r sin theta cos phi) hat(i)+(n sin theta sin phi) hat(j)+(n cos theta) hat(k)],[[[" Position rector in "],[" spherical "]]:}],[:. hat(r)=(del( vec(n))∣del n)/(|(del( vec(n)))/(del n)|)=(sin theta cos phi) hat(i)+(sin theta sin phi) hat(j)+(cos theta) hat(x)],[ hat(theta)=((del( vec(r)))/(del theta))/(|(del( vec(x)))/(del theta)|)=(1)/(b)[(r cos theta cos phi) hat(i)+(n cos theta sin phi) hat(j)],[ hat(phi)=((del( vec(r)))/(del phi))/(|(del( vec(n)))/(del q)|)=(1)/(r sin theta)[(-r sin theta sin phi) hat(i)+(b sin theta cos phi) hat(j)]],[ hat(phi)=-(cos theta cos phi) hat(i)+(cos theta sin phi) hat(j)-(sin theta) hat(x)+(cos phi) hat(j)]:}Now, a vector can be expressed in any co-ardinale system with theip coordinates and unit reetors in each system.vec(c)=c_(3) hat(i)+c_(y) hat(ȷ)+c_(z) hat(x) it can be expressed in spherical as vec(c)=c_(r) hat(n)+c_(theta) hat(theta)+c_(phi) hat(phi) mere,{:[c_(n)= vec(c)* hat(b)],[c_(theta)= vec(c)* hat(theta)],[c_(phi)= vec(c)* hat(phi)]:}{:[" Here, " vec(c)=[((y^(2))/(x^(2)+y^(2)))^( hat(L))-((x^(2))/(x^(2)+y^(2)))( hat(j))+4( hat(z))]],[:.c_(n)= vec(c_(1)) hat(n)],[=[((r^(2)sin^(2)thetasin^(2)phi)/(r^(2)sin 2theta))( hat(i))-((r^(2)sin ... See the full answer

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