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Explain Spherical co-ordinate system. State the transformation relation between Cartesian and Spherical co-ordinates.
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• For spherical coordinate system x,y,z are used for reference. Imagine a sphere of radius 'r' with center at origin. Any point on the sphere is at the same distance 'r' from origin, therefore the spherical surface is defined as r = constant surface.
• Now consider a line from origin making angle θ with z-axis. Rotate this line about z-axis fixing the end at the origin. This forms a cone with angle θ. This conical surface is defined as θ= constant surface.
• When a sphere with center at origin intersects with a vertical cone with vertex at origin the intersection is a horizontal circle with radius equal to rsinθ.
• To locate a point in spherical co-ordinate system. Imagine Φ=constant plane. A horizontal circle with center on z-axis intersects Φ=constant plane. the intersection is a point.

Point in spherical system = (r, θ ,Φ)

In spherical system variation of angle θ is from 0 to 180 deg and variation of Φ is from 0 to 360 deg.

Cartesian from spherical

x = rsinθcosΦ

y = rsinθsinΦ

z = rcosθ

Spherical from Cartesian coordinates