Transform the spherical system B = (10/r)i + (10cos θ)j + k into cylindrical form at (5, π/2, -2)

To transform the spherical system ( B = left( frac{10}{r} right)i + (10costheta)j + k ) into cylindrical form, we will utilize the relationships between spherical and cylindrical coordinates.

In spherical coordinates:

– ( r = sqrt{x^2 + y^2 + z^2} )

– ( theta = tan^{-1}left( frac{y}{x} right) )

– ( phi = cos^{-1}left( frac{z}{r} right) )

In cylindrical coordinates:

– ( r_c = sqrt{x^2 + y^2} )

– ( theta_c = tan^{-1}left( frac{y}{x} right) )

– ( z = z )

Given the spherical coordinates ( (r, theta, z) = (5, frac{pi}{2}, -2) ):

– ( r = 5 )

– ( theta = frac{pi}{2} )

– ( z = -2 )

We can convert to Cartesian coordinates:

– ( x = r sin(theta) cos(phi) = 5 sinleft( frac{pi}{2} right) cos(0) = 5 cdot 1 cdot 1 = 5 )

– ( y