Cylindrical and Spherical Coordinate Systems

Transcription

1 Clindical and Spheical Coodinate Sstems APPENDIX A In Section 1.2, we leaned that the Catesian coodinate sstem is deined b a set o thee mtall othogonal saces, all o which ae planes. The clindical and spheical coodinate sstems also involve sets o thee mtall othogonal saces. Fo the clindical coodinate sstem, the thee saces ae a clinde and two planes, as shown in Fige A.1(a). One o these planes is the same as the = constant plane in the Catesian coodinate sstem. The second plane contains the -ais and makes an angle with a eeence plane, convenientl chosen to be the -plane o the Catesian coodinate sstem. This plane is theeoe deined b = constant. The clindical sace has the -ais as its ais. Since the adial distance om the -ais to points on the clindical sace is a constant, this sace is deined b = constant. Ths, the thee othogonal saces deining the clindical coodinates o a point ae = constant, = constant, and = constant. Onl two o these coodinates ( and ) ae distances; the thid coodinate ( ) is an angle. We note that the entie space is spanned b vaing om 0 to q, om 0 to 2p, and om - q to q. The oigin is given b = 0, = 0, and = 0. An othe point in space is given b the intesection o thee mtall othogonal saces obtained b incementing the coodinates b appopiate amonts. Fo eample, the intesection o the thee saces = 2, = p>4, and = 3 deines the point A(2, p>4, 3), as shown in Fige A.1(a). These thee othogonal saces deine thee cves that ae mtall pependicla. Two o these ae staight lines and the thid is a cicle. We daw nit vectos, a,, and a tangential to these cves at the point A and diected towad inceasing vales o,,and,espectivel.thesetheenitvectosomasetomtallothogonalnit vectos in tems o which vectos dawn at A can be descibed. In a simila manne, we can daw nit vectos at an othe point in the clindical coodinate sstem, as shown, o eample, o point B(1, 3p>4, 5) in Fige A.1(a). It can now be seen that the nit vectos a and at point B ae not paallel to the coesponding nit vectos at point A.Ths,nlike inthecatesiancoodinatesstem,thenitvectosa and in the clindical coodinate sstem do not have the same diections evewhee, that is, the ae not niom. Onl the nit vecto a, which is the same as in the Catesian coodinate sstem, is niom. Finall, we note that o the choice o as in Fige A.l(a), 413

2 414 Appendi A Clindical and Spheical Coodinate Sstems a B a a 3 Q d A a 2 d P d p 4 d d d (a) (b) FIGURE A.1 Clindical coodinate sstem. (a) Othogonal saces and nit vectos. (b) Dieential volme omed b incementing the coodinates. that is, inceasing om the positive -ais towad the positive -ais, the coodinate sstem is ight-handed, that is, a : = a. To obtain epessions o the dieential lengths, saces, and volme in the clindical coodinate sstem, we now conside two points P(,, ) and Q( + d, + d, + d), whee Q is obtained b incementing ininitesimall each coodinate om its vale at P, as shown in Fige A.1(b). The thee othogonal saces intesecting at P and the thee othogonal saces intesecting at Q deine a bo which can be consideed to be ectangla, since d, d, and d ae ininitesimall small. The thee dieential length elements, oming the contigos sides o this bo ae d a, d,and d a.the dieential length vecto dl om P to Q is ths given b dl = d a + d + d a (A.1) The dieential saces omed b pais o the dieential length elements ae ; ds a = ; (d)( d)a = ; d a : d (A.2a) ; ds a = ; ( d)(d)a = ; d : d a (A.2b) ; ds = ; (d)(d) = ; d a : d a (A.2c) Finall, the dieential volme dv omed b the thee dieential lengths is simpl the volme o the bo, that is, dv = (d)( d)(d) = d d d (A.3)

3 Appendi A 415 Fo the spheical coodinate sstem, the thee mtall othogonal saces ae asphee,a cone,and a plane,as shown in Fige A.2(a).The plane is the same as the = constant plane in the clindical coodinate sstem. The sphee has the oigin as its cente. Since the adial distance om the oigin to points on the spheical sace is a constant, this sace is deined b = constant. The spheical coodinate shold not be consed with the clindical coodinate. When these two coodinates appea in the same epession, we shall se the sbscipts c and s to distingish between clindical and spheical. The cone has its vete at the oigin and its sace is smmetical abot the -ais. Since the angle is the angle that the conical sace makes with the -ais, this sace is deined b = constant. Ths, the thee othogonal saces deining the spheical coodinates o a point ae = constant, = constant, and = constant. Onl one o these coodinates () is distance; the othe two coodinates ( and ) ae angles. We note that the entie space is spanned b vaing om 0 to q, om 0 to p, and om 0 to 2p. p 6 A a a B p 3 3 sin d d P d d d d d Q (a) (b) FIGURE A.2 Spheical coodinate sstem. (a) Othogonal saces and nit vectos. (b) Dieential volme omed b incementing the coodinates. The oigin is given b = 0, = 0, and = 0. An othe point in space is given b the intesection o thee mtall othogonal saces obtained b incementing the coodinates b appopiate amonts. Fo eample, the intesection o the thee saces = 3, = p>6, and = p>3 deines the point A(3, p>6, p>3) as shown in Fige A.2(a). These thee othogonal saces deine thee cves that ae mtall pependicla. One o these is a staight line and the othe two ae cicles.we daw nit vectos a,, and tangential to these cves at point A and diected towad inceasing vales o,, and, espectivel. These thee nit vectos om a set o mtall

4 416 Appendi A Clindical and Spheical Coodinate Sstems othogonal nit vectos in tems o which vectos dawn at A can be descibed. In a simila manne, we can daw nit vectos at an othe point in the spheical coodinate sstem, as shown, o eample, o point B(1, p>2, 0) in Fige A.2(a). It can now be seen that these nit vectos at point B ae not paallel to the coesponding nit vectos at point A. Ths, in the spheical coodinate sstem, all thee nit vectos a,, and do not have the same diections evewhee, that is, the ae not niom. Finall, we note that o the choice o as in Fige A.2(a), that is, inceasing om the positive -ais towad the -plane, the coodinate sstem is ight-handed, that is, a : =. To obtain epessions o the dieential lengths, saces, and volme in the spheical coodinate sstem, we now conside two points P(,, ) and Q( + d, + d, + d), whee Q is obtained b incementing ininitesimall each coodinate om its vale at P, as shown in Fige A.2(b). The thee othogonal saces intesecting at P and the thee othogonal saces intesecting at Q deine a bo that can be consideed to be ectangla, since d, d, and d ae ininitesimall small. The thee dieential length elements oming the contigos sides o this bo ae d a, d, and sin d.the dieential length vecto dl om P to Q is ths given b dl = d a + d + sin d (A.4) The dieential saces omed b pais o the dieential length elements ae ; ds = ; (d)( d) = ; d a : d ; ds a = ; ( d)( sin d)a = ; d : sin d ; ds = ; ( sin d)(d) = ; sin d : d a (A.5a) (A.5b) (A.5c) Finall, the dieential volme dv omed b the thee dieential lengths is simpl the volme o the bo, that is, dv = (d)( d)( sin d) = 2 sin d d d (A.6) In the std o electomagnetics it is sometimes sel to be able to convet the coodinates o a point and vectos dawn at a point om one coodinate sstem to anothe, paticlal om the Catesian sstem to the clindical sstem and vice vesa, and om the Catesian sstem to the spheical sstem and vice vesa.to deive ist the elationships o the convesion o the coodinates, let s conside Fige A.3(a), which illstates the geomet petinent to the coodinates o a point P in the thee dieent coodinate sstems. Ths, om simple geometical consideations, we have = c cos = c sin = = s sin cos = s sin sin = s cos (A.7) (A.8) Convesel, we have c = = tan - 1 = (A.9) s = = tan = tan - 1 (A.10)

5 Appendi A 417 a s P a a a s a c O c a c a (b) (c) (a) FIGURE A.3 (a) Fo convesion o coodinates o a point om one coodinate sstem to anothe. (b) and (c) Fo epessing nit vectos in clindical and spheical coodinate sstems, espectivel, in tems o nit vectos in the Catesian coodinate sstem. Relationships (A.7) and (A.9) coespond to convesion om clindical coodinates to Catesian coodinates and vice vesa. Relationships (A.8) and (A.10) coespond to convesion om spheical coodinates to Catesian coodinates and vice vesa. Consideing net the convesion o vectos om one coodinate sstem to anothe, we note that in ode to do this, we need to epess each o the nit vectos o the ist coodinate sstem in tems o its components along the nit vectos in the second coodinate sstem. Fom the deinition o the dot podct o two vectos, the component o a nit vecto along anothe nit vecto, that is, the cosine o the angle between the nit vectos, is simpl the dot podct o the two nit vectos. Ths, consideing the sets o nit vectos in the clindical and Catesian coodinate sstems, we have with the aid o Fige A.3(b), a c # a = cos a c # a = sin a c # a = 0 # a = - sin # a = cos # a = 0 a # a = 0 a # a = 0 a # a = 1 (A.11a) (A.11b) (A.11c) Similal, o the sets o nit vectos in the spheical and Catesian coodinate sstems, we obtain with the aid o Fige A.3(c) and Fige A.3(b), a s # a = sin cos a s # a = sin sin a s # a = cos # a = cos cos # a = cos sin # a = - sin # a = - sin # a = cos # a = 0 (A.12a) (A.12b) (A.12c) We shall now illstate the se o these elationships b means o an eample.

6 418 Appendi A Clindical and Spheical Coodinate Sstems Eample A.1 Let s conside the vecto 3a + 4a + 5a at the point (3, 4, 5) and convet the vecto to one in spheical coodinates. Fist, om the elationships (A.10), we obtain the spheical coodinates o the point (3, 4, 5) to be s = = 522 = tan = tan = 45 5 = tan = Then noting om the elationships (A.12) that at the point nde consideation, a = sin cos a s + cos cos - sin = 0.322a s a = sin sin a s + cos sin + cos = 0.422a s a = cos a s - sin = 0.522a s we obtain 3a + 4a + 5a = ( )a s + ( ) + ( ) = 522a s This eslt is to be epected since the given vecto has components eqal to the coodinates o the point at which it is speciied. Hence, its magnitde is eqal to the distance o the point om the oigin, that is, the spheical coodinate o the point, and its diection is along the line dawn om the oigin to the point, that is, along the nit vecto a s at that point. In act, the given vecto is a paticla case o the vecto a + a + a = s a s, known as the position vecto,since it is the same as the vecto dawn om the oigin to the point (,, ). 3a a REVIEW QUESTIONS A.1. What ae the thee othogonal saces deining the clindical coodinate sstem? A.2. What ae the limits o vaiation o the clindical coodinates? A.3. Which o the nit vectos in the clindical coodinate sstem ae not niom? at the point (2, p>2, 3) ae eqal o not. A.4. State whethe the vecto 3a a at the point (1, 0, 2) and the vecto A.5. What ae the dieential length vectos in clindical coodinates? A.6. What ae the thee othogonal saces deining the spheical coodinate sstem? A.7. What ae the limits o vaiation o the spheical coodinates?

7 Appendi A 419 A.8. Which o the nit vectos in the spheical coodinate sstem ae not niom? A.9. State i the vecto 3a + 4 at the point (1, p>2, 0) and the vecto 3a + 4 at the point (2, 0, p>2) ae eqal o not. A.10. What ae the dieential length vectos in spheical coodinates? A.11. Otline the pocede o conveting a vecto at a point om one coodinate sstem to anothe. A.12. What is the epession o the position vecto in the clindical coodinate sstem? PROBLEMS (1, p>4, 2) (2, p>3, p>6) p>4, 0) A.1. Epess in tems o Catesian coodinates the vecto dawn om the point P(2, p>3, 1) to the point Q(4, 2p>3, 2) in clindical coodinates. A.2. Epess in tems o Catesian coodinates the vecto dawn om the point P(1, p>3, p>4) to the point Q(2, 2p>3, 3p>4) in spheical coodinates. A.3. Detemine i the vecto a + + 2a at the point and the vecto 22a + 2a at the point (2, p>2, 3) ae eqal o not. A.4. Detemine i the vecto 3a at the point and the vecto a at the point (1, p>6, p>3) ae eqal o not. A.5. Find the dot and coss podcts o the nit vecto a at the point (1, 0, 0) and the nit vecto at the point (2, p>4, 1) in clindical coodinates. A.6. Find the dot and coss podcts o the nit vecto a at the point (1, and the nit vecto at the point (2, p>2, p>2) in spheical coodinates. A.7. Convet the vecto 5a + 12a + 6a at the point (5, 12, 4) to one in clindical coodinates. A.8. Convet the vecto 3a + 4a - 5a at the point (3, 4, 5) to one in spheical coodinates.