91Ó°ÊÓ

The spherical coordinates are defined in terms of , where is the distance from origin, is the pole angle and is the azimuthal angle.

The spherical coordinates can be drawn as,

The scalar potentials is $v=\frac{r^{âˆ\S }}{r2}$and the position vector is $\stackrel{â‡\P Ä}{r}=xi+yj+zk$. The unit vector in the direction of $\stackrel{â‡\P Ä}{r}$, is obtained as,

role="math" localid="1650621631899" $r^{âˆ\S }={\frac{\stackrel{â‡\P Ä}{r}}{\left|r\right|}}_{}=\frac{xi+yj+zk}{\left|\sqrt{x^2+y^2+z^2}\right|}$

The spherical coordinates of the system is defined as,

role="math" localid="1650621894963" $$\begin{gathered} x=(r\sin \mathrm{θ})\cos \mathrm{ϕ}) \\ y=(r\sin \mathrm{θ})\sin \mathrm{ϕ}) \\ z=r\cos \mathrm{θ} \end{gathered}$$

Substitute $(r\sin \mathrm{θ})\cos \mathrm{ϕ}$ for x, $(r\sin \mathrm{θ})\sin \mathrm{ϕ}$ for y and $r\cos \mathrm{θ}$ for z into

$$\begin{gathered} \stackrel{â‡\P Ä}{r}=xi+yj+zk \\ . \\ \stackrel{â‡\P Ä}{r}=xi+yj+zk \\ =(r\sin \mathrm{θ})\cos \mathrm{Ï•}+(r\sin \mathrm{θ})\sin \mathrm{Ï•}+r\cos \mathrm{θ}k \end{gathered}$$

The unit vector $r^{âˆ\S }$is obtained as$r^{âˆ\S }=\sin \mathrm{θ}\cos \mathrm{Ï•}{x}^{âˆ\S }+\sin \mathrm{θ}\sin \mathrm{Ï•}{y}^{âˆ\S }+\cos \mathrm{θ}{z}^{âˆ\S }$

The infinitesimal displacement along the direction, is obtained as

$\stackrel{â‡\P Ä}{dI}{}_{{\mathrm{θ}}_{}}=rd\mathrm{θ}{\mathrm{θ}}^{âˆ\S }$ ……. (3)

The infinitesimal displacement along the direction $\mathrm{θ}$, in terms of Cartesian coordinates is written as,

${\stackrel{â‡\P Ä}{dI}}_{\mathrm{θ}}=dx{x}^{âˆ\S }+dy{y}^{âˆ\S }+dz{z}^{âˆ\S }$

As $x=(r\sin \mathrm{θ})\cos \mathrm{ϕ},y=(r\sin \mathrm{θ})\sin \mathrm{ϕ},z=r\cos \mathrm{θ},$infinitesimal displacement along the direction $\mathrm{θ}$, can be written as,

${\stackrel{→}{dI}}_{\mathrm{θ}}=((r\sin \mathrm{θ})\cos \mathrm{Ï•})x^{âˆ\S }+((r\sin \mathrm{θ})\sin \mathrm{Ï•})y^{âˆ\S }+(r\cos \mathrm{θ})z^{âˆ\S }$

From equation (3),${\stackrel{→}{dI}}_{\mathrm{θ}}=rd\mathrm{θ}{\mathrm{θ}}^{âˆ\S }$

The infinitesimal displacement along the direction $\mathrm{θ}$, is obtained as

${\stackrel{→}{dI}}_{\mathrm{θ}}=r\sin \mathrm{θ}d\mathrm{Ï•}{\mathrm{Ï•}}^{âˆ\S }$ ……. (3)

The infinitesimal displacement along the direction $\mathrm{θ}$, in terms of Cartesian coordinates is written as,

${\stackrel{→}{dI}}_{\mathrm{θ}}=dx{x}^{âˆ\S }+dy{y}^{âˆ\S }+dz{z}^{âˆ\S }$

As $x=(r\sin \mathrm{θ})\cos \mathrm{ϕ},y=(r\sin \mathrm{θ})\sin \mathrm{ϕ},z=r\cos \mathrm{θ}$, infinitesimal displacement along the direction $\mathrm{θ}$, can be written as,

${\stackrel{→}{dl}}_{\mathrm{θ}}=((r\sin \mathrm{θ})\cos \mathrm{Ï•})x^{âˆ\S }+((r\sin \mathrm{θ})\sin \mathrm{Ï•})y^{âˆ\S }+(r\cos \mathrm{θ})z^{âˆ\S }$

From equation (3),$$\begin{gathered} {\stackrel{→}{dl}}_{\mathrm{θ}}=r\sin \mathrm{θ}d\mathrm{Ï•}{\mathrm{Ï•}}^{âˆ\S } \\ r\sin \mathrm{θ}d\mathrm{Ï•}{\mathrm{Ï•}}^{âˆ\S }=((r\sin \mathrm{θ})\cos \mathrm{Ï•})x^{âˆ\S }+((r\sin \mathrm{θ})\sin \mathrm{Ï•})y^{âˆ\S } \\ {\mathrm{Ï•}}^{âˆ\S }=-\sin \mathrm{Ï•}{x}^{âˆ\S }+\cos \mathrm{Ï•}{y}^{âˆ\S } \end{gathered}$$

The product of $r^{âˆ\S }.r^{âˆ\S }$, is calculated as,

$$\begin{gathered} r^{âˆ\S }.r^{âˆ\S }={\sin }^2\mathrm{θ}({\cos }^2\mathrm{Ï•}+{\sin }^2\mathrm{Ï•})+{\cos }^2\mathrm{θ} \\ ={\sin }^2\mathrm{θ}+{\cos }^2\mathrm{θ} \\ =1 \end{gathered}$$

Multiply the vectors ${\mathrm{θ}}^{âˆ\S }$and ${\mathrm{Ï•}}^{âˆ\S }$

$$\begin{gathered} {\mathrm{θ}}^{âˆ\S }.{\mathrm{Ï•}}^{âˆ\S }=-\cos \mathrm{θ}\sin \mathrm{Ï•}\cos \mathrm{Ï•}+\cos \mathrm{θ}\sin \mathrm{Ï•}\cos \mathrm{Ï•} \\ =0 \end{gathered}$$

As $x=(r\sin \mathrm{θ})\cos \mathrm{ϕ},y=(r\sin \mathrm{θ})\sin \mathrm{ϕ},z=r\cos \mathrm{θ}$,the position vector

$r^{âˆ\S }=(r\sin \mathrm{θ})\cos x^{âˆ\S }+\sin \mathrm{θ}\sin \mathrm{Ï•}{y}^{âˆ\S }+\cos \mathrm{θ}{z}^{âˆ\S }$

Multiply above equation by $\sin \mathrm{θ}$on both sides,

$\sin \mathrm{θ}{r}^{âˆ\S }={\sin }^2\mathrm{Ï•}\cos x^{âˆ\S }+{\sin }^2\mathrm{θ}\sin \mathrm{Ï•}{y}^{âˆ\S }+\sin \mathrm{θ}\cos \mathrm{θ}{z}^{âˆ\S }$ ……. (1)

Now the theta vector is ${\mathrm{θ}}^{âˆ\S }=\cos \mathrm{θ}\cos \mathrm{Ï•}{x}^{âˆ\S }+\cos \mathrm{θ}\sin \mathrm{Ï•}{y}^{âˆ\S }-\sin \mathrm{θ}{z}^{âˆ\S }$

Multiply above equation by $\cos \mathrm{θ}$on both sides,

$\cos \mathrm{θ}{\mathrm{θ}}^{âˆ\S }={\cos }^2\mathrm{θ}\cos \mathrm{Ï•}{x}^{âˆ\S }+{\cos }^2\mathrm{θ}\sin \mathrm{Ï•}{y}^{âˆ\S }-\sin \mathrm{θ}\cos \mathrm{θ}{z}^{âˆ\S }$ ……. (2)

Add equations (1) and (2) as,

$$\begin{gathered} \sin \mathrm{θ}{r}^{âˆ\S }+\cos \mathrm{θ}{\mathrm{θ}}^{âˆ\S }={\sin }^2\mathrm{θ}\cos \mathrm{Ï•}{x}^{âˆ\S }+{\sin }^2\mathrm{θ}\sin \mathrm{Ï•}{y}^{âˆ\S }+\sin \mathrm{θ}\cos \mathrm{θ}{z}^{âˆ\S }+{\cos }^2\mathrm{θ}\cos \mathrm{Ï•}{x}^{âˆ\S }+{\cos }^2\mathrm{θ}\sin \mathrm{Ï•}{y}^{âˆ\S }-\sin \mathrm{θ}\cos \mathrm{Ï•} \\ ={\sin }^2\mathrm{θ}\cos \mathrm{Ï•}{x}^{âˆ\S }+{\sin }^2\mathrm{θ}\sin \mathrm{Ï•}{y}^{âˆ\S }+{\cos }^2\mathrm{θ}\cos \mathrm{Ï•}{x}^{âˆ\S }+{\cos }^2\mathrm{θ}\sin \mathrm{Ï•}{y}^{âˆ\S } \\ =\left[\begin{array}{cc}{\sin }^2\mathrm{θ}+& {\cos }^2\mathrm{θ}{x}^{âˆ\S }\end{array}\right]\cos \mathrm{θ}{x}^{âˆ\S }+({\sin }^2\mathrm{θ}+co{s}^2\mathrm{θ})\sin \mathrm{θ}{y}^{âˆ\S } \\ =\cos \mathrm{θ}{x}^{âˆ\S }+\sin \mathrm{θ}{y}^{âˆ\S } \end{gathered}$$

solve further as,

${\mathrm{Ï•}}^{âˆ\S }=\sin \mathrm{Ï•}{x}^{âˆ\S }+\cos \mathrm{Ï•}{y}^{âˆ\S }$

Multiply $\sin \mathrm{θ}{r}^{âˆ\S }+\cos \mathrm{θ}{\mathrm{θ}}^{âˆ\S }=\cos \mathrm{Ï•}{x}^{âˆ\S }+\sin \mathrm{Ï•}{y}^{âˆ\S }$by $\sin \mathrm{Ï•}$on both sides, $\sin \mathrm{θ}\cos \mathrm{φ}{r}^{âˆ\S }+\cos \mathrm{θ}\cos \mathrm{Ï•}{\mathrm{θ}}^{âˆ\S }={\cos }^2\mathrm{Ï•}{x}^{âˆ\S }+\sin \mathrm{Ï•}\cos \mathrm{Ï•}{y}^{âˆ\S }$ ……. (3)

Multiply ${\mathrm{Ï•}}^{âˆ\S }=-\sin \mathrm{Ï•}{x}^{âˆ\S }+\cos \mathrm{Ï•}{y}^{âˆ\S }$by $\sin \mathrm{Ï•}$on both sides,

$\sin \mathrm{Ï•}{\mathrm{Ï•}}^{âˆ\S }=-{\sin }^2\mathrm{Ï•}{x}^{âˆ\S }+\cos \mathrm{Ï•}\sin \mathrm{Ï•}{y}^{âˆ\S }$ ……. (4)

Subtract equation (4) from equation (3).

$$\begin{gathered} \sin \mathrm{θ}\cos \mathrm{Ï•}{r}^{âˆ\S }+\cos \mathrm{θ}\cos \mathrm{Ï•}{\mathrm{θ}}^{âˆ\S }-\sin \mathrm{Ï•}{\mathrm{Ï•}}^{âˆ\S }={\cos }^2\mathrm{Ï•}{x}^{âˆ\S }+\sin \mathrm{Ï•}\cos \mathrm{Ï•}{y}^{âˆ\S }-\sin \mathrm{Ï•}\cos \mathrm{Ï•}{y}^{âˆ\S }+{\sin }^2\mathrm{Ï•}{x}^{âˆ\S } \\ ={\cos }^2\mathrm{Ï•}{x}^{âˆ\S }+{\sin }^2\mathrm{Ï•}{x}^{âˆ\S } \\ =x^{âˆ\S } \end{gathered}$$

Thus, $x^{âˆ\S }=\sin \mathrm{θ}\cos \mathrm{Ï•}{r}^{âˆ\S }+\cos \mathrm{θ}\cos \mathrm{Ï•}{\mathrm{θ}}^{âˆ\S }-\sin \mathrm{Ï•}{\mathrm{Ï•}}^{âˆ\S }$

Multiply localid="1650630821221" $\sin \mathrm{θ}{r}^{âˆ\S }+\cos \mathrm{θ}{\mathrm{θ}}^{âˆ\S }=\cos \mathrm{Ï•}{x}^{âˆ\S }+\sin \mathrm{Ï•}{y}^{âˆ\S }$by $\sin \mathrm{Ï•}$on both sides,

$\sin \mathrm{θ}\sin \mathrm{Ï•}{r}^{âˆ\S }+\cos \mathrm{θ}\sin \mathrm{Ï•}{\mathrm{θ}}^{âˆ\S }=\cos \mathrm{Ï•}\sin \mathrm{Ï•}{x}^{âˆ\S }+{\sin }^2\mathrm{Ï•}{y}^{âˆ\S }$ ........(5)

Multiply role="math" localid="1650631342455" ${\mathrm{Ï•}}^{âˆ\S }=-\sin \mathrm{Ï•}\cos \mathrm{Ï•}{x}^{âˆ\S }+{\cos }^2\mathrm{Ï•}{y}^{âˆ\S }$by $\cos \mathrm{Ï•}$on both sides, $\cos \mathrm{Ï•}{\mathrm{Ï•}}^{âˆ\S }=-\sin \mathrm{Ï•}\cos \mathrm{Ï•}{x}^{âˆ\S }+{\cos }^2\mathrm{Ï•}{y}^{âˆ\S }$……. (5)

Add equation (5) and (6).

$$\begin{gathered} \sin \mathrm{θ}\sin \mathrm{Ï•}{r}^{âˆ\S }+\cos \mathrm{θ}\sin \mathrm{Ï•}{\mathrm{θ}}^{âˆ\S }+\cos \mathrm{Ï•}{\mathrm{Ï•}}^{âˆ\S }=\cos \mathrm{Ï•}{\mathrm{Ï•}}^{âˆ\S }\sin \mathrm{Ï•}{x}^{âˆ\S }+{\sin }^2\mathrm{Ï•}{y}^{âˆ\S }-\sin \mathrm{Ï•}\cos \mathrm{Ï•}{x}^{âˆ\S }+{\cos }^2\mathrm{Ï•}{y}^{âˆ\S } \\ ={\sin }^2\mathrm{Ï•}{y}^{âˆ\S }+{\cos }^2\mathrm{Ï•}{y}^{âˆ\S } \\ =y^{âˆ\S } \end{gathered}$$

Thus, $y^{âˆ\S }=\sin \mathrm{θ}\sin \mathrm{Ï•}{r}^{âˆ\S }+\cos \mathrm{θ}\sin \mathrm{Ï•}{\mathrm{θ}}^{âˆ\S }+\cos \mathrm{Ï•}{\mathrm{Ï•}}^{âˆ\S }$

As$x=(r\sin \mathrm{θ})\cos \mathrm{ϕ},y=(r\sin \mathrm{θ})\sin \mathrm{ϕ},z=r\cos \mathrm{θ}$, the position vector is

$r^{âˆ\S }=(r\sin \mathrm{θ})\cos x^{âˆ\S }+\sin \mathrm{θ}\sin \mathrm{Ï•}{y}^{âˆ\S }+\cos \mathrm{θ}{z}^{âˆ\S }$

Multiply above equation by on both sides, ……. (6)