1. Convert the point (3,4,5) from Cartesian to spherical coordinates
a) (7.07,45⁰,53⁰)
b) (0.707,45⁰,53⁰)
c) (7.07,54⁰,63⁰)
d) (0.707,54⁰,63⁰)
Explanation: r = √(x2+y2+z2) = √50 = 7.07
Θ = cos-1(z/r) = cos-1(5/5√2) = 45⁰
Φ = tan-1(y/x) = tan-1(4/3) = 53⁰.
2. Example of spherical system in the following is
a) Charge in space
b) Charge in box
c) Charge in dielectric
d) Uncharged system
Explanation: From a point charge +Q, the electric field spreads in all 360 degrees. The calculation of electric field in this case will be spherical system. Thus it is charge in the space.
3. Spherical systems are employed in waveguides. State True/False
a) True
b) False
Explanation: There is no waveguide designed spherically to avoid absorption, rather than propagation
4. Choose which of following condition is not required for a waveguide to exist.
a) The dimensions should be in accordance with desired frequency
b) Cut-off frequency should be minimum 6GHz
c) The shape should be spherical
d) No specific condition is required for waveguide design
Explanation: A waveguide need not be spherical, it has to be rectangular or circular, as it violates the propagation of the wave.
5. Find the spherical coordinates of A(2,3,-1)
a) (3.74, 105.5⁰, 56.13⁰)
b) (3.74, 105.5⁰, 56.31⁰)
c) (3.74, 106.5⁰, 56.13⁰)
d) (3.74, 106.5⁰, 56.31⁰)
Explanation: r = √(x2+y2+z2) = √14 = 3.74
Θ = cos-1(z/r) = cos-1(-1/3.74) = 105.5⁰
Φ = tan-1(y/x) = tan-1(3/2) = 56.31⁰.
6. Find the Cartesian coordinates of B(4,25⁰,120⁰)
a) (0.845, 1.462, 3.625)
b) (-0.845, 1.462, 3.625)
c) (-8.45, 2.462, 6.325)
d) (8.45, 2.462, 6.325)
Explanation: x = r sin θ cos φ = 4 sin25⁰ cos 120⁰ = -0.845
y = r sin θ sin φ = 4 sin 25⁰ sin 120⁰ = 1.462
z = r cos θ = 4 cos 25⁰ = 3.625.
7. The area of sphere can be computed from the sphere volume. State True/False.
a) True
b) False
Explanation: On double integrating the differential volume, the area can be computed for a sphere.
8. Given B= (10/r)i+( rcos θ) j+k in spherical coordinates. Find Cartesian points at (-3,4,0)
a) -2i + j
b) 2i + k
c) i + 2j
d) –i – 2k
Explanation: r = √(x2+y2+z2) = √25 = 5
Θ = cos-1(z/r) = 1
Φ = tan-1(y/x) = tan-1(-4/3)
Thus, B = -2i + j
9. The scalar factor of spherical coordinates is
a) 1, r, r sin θ
b) 1, r, r
c) r, r, 1
d) r, 1, r
Explanation: The radius varies from unity to infinity, the plane angle from zero to 360 ⁰ and the z plane from (-∞, ∞) .
10. Transform the vector (4,-2,-4) at (1,2,3) into spherical coordinates.
a) 3.197i – 2.393j + 4.472k
b) -3.197i + 2.393j – 4.472k
c) 3.197i + 2.393j + 4.472k
d) -3.197i – 2.393j – 4.472k
Explanation: r = √(x2+y2+z2) = 3.74
Θ = cos-1(z/r) = cos-1(3/3.74) = 36.7⁰
Φ = tan-1(y/x) = tan-1(2/1) = 63.4⁰
A = (4 sin θ cos φ – 2 sin θ sin φ – 4cos θ)i + (4 cos θ cos φ – 2 cos θ sin φ + 4 sin θ)j + (-4 sin φ – 2 cos φ)k
On substituting r, θ, φ, A = -3.197i + 2.393j – 4.472k.