Coordinate Systems and Solid Angle with explained answers for your competitive exam preparation:
1. Which of the following is a 3D coordinate system representing points by (x, y, z)?
- A) Polar
- B) Cylindrical
- C) Cartesian
- D) Spherical
Answer: C) Cartesian
Explanation: Cartesian coordinates represent points in 3D space as (x, y, z) along mutually perpendicular axes.
2. In spherical coordinates, the point is represented as (r, θ, φ). What is θ?
- A) Radial distance
- B) Polar angle (angle with z-axis)
- C) Azimuthal angle (angle in xy-plane)
- D) None of these
Answer: B) Polar angle (angle with z-axis)
Explanation: θ is the polar angle measured from the positive z-axis downward.
3. What is the measure of a solid angle in 3D space?
- A) Steradian
- B) Degree
- C) Radian
- D) Meter
Answer: A) Steradian
Explanation: Solid angle is measured in steradians (sr), representing 3D angular spread.
4. The surface area of a unit sphere is:
- A) 4Ï€ steradians
- B) 2Ï€ steradians
- C) π steradians
- D) 1 steradian
Answer: A) 4Ï€ steradians
Explanation: The total solid angle around a point (surface area of unit sphere) is 4Ï€ sr.
5. How many steradians are there in the whole sphere?
- A) 2Ï€
- B) 4Ï€
- C) 8Ï€
- D) π
Answer: B) 4Ï€
Explanation: Whole spherical surface corresponds to 4Ï€ steradians.
6. Which coordinate system is best suited for problems with cylindrical symmetry?
- A) Cartesian
- B) Polar
- C) Cylindrical
- D) Spherical
Answer: C) Cylindrical
Explanation: Cylindrical coordinates (r, θ, z) adapt well to cylindrical symmetry.
7. In spherical coordinates, the volume element is given by:
- A) $$dr\,d\theta\,d\phi$$
- B) $$r \sin\theta\,dr\,d\theta\,d\phi$$
- C) $$r^2 \sin\theta\,dr\,d\theta\,d\phi$$
- D) $$r^2 dr d\theta$$
Answer: C) $$r^2 \sin\theta\,dr\,d\theta\,d\phi$$
Explanation: The volume element in spherical coordinates accounts for spherical geometry.
8. Which of these is not a valid coordinate transformation?
- A) $$x = r \cos\theta$$
- B) $$y = r \sin\theta$$
- C) $$z = r \cos\phi$$
- D) None of these
Answer: C) $$z = r \cos\phi$$
Explanation: Usually φ is azimuthal angle in the xy-plane; z corresponds to r cosθ.
9. The solid angle subtended by a whole sphere at its center is:
- A) $$2\pi$$ steradian
- B) $$4\pi$$ steradian
- C) $$8\pi$$ steradian
- D) $$0$$ steradian
Answer: B) $$4\pi$$ steradian
Explanation: The entire sphere area corresponds to $$4\pi$$ steradian.
10. What is the relationship between degrees and steradians?
- A) $$1^\circ = \frac{\pi}{180} $$ steradian
- B) $$1$$ steradian = $$\frac{180}{\pi}$$ degrees
- C) Steradians measure solid angle; degrees measure planar angle (not directly convertible)
- D) None
Answer: C) Steradians measure solid angle; degrees measure planar angle (not directly convertible)
Explanation: Steradians are 3D angle units; degrees are 2D and cannot be directly converted.
11. Which coordinate system uses (r, θ, z)?
- A) Cartesian
- B) Polar
- C) Cylindrical
- D) Spherical
Answer: C) Cylindrical
Explanation: Cylindrical coordinates combine polar coordinates in xy-plane with height z.
12. The azimuthal angle in spherical coordinates lies in which plane?
- A) xz-plane
- B) yz-plane
- C) xy-plane
- D) None
Answer: C) xy-plane
Explanation: Azimuthal angle φ measures the angle in the xy-plane.
13. Which of the following is true about solid angle?
- A) It can be negative
- B) It has dimensions of area/length^2
- C) It is dimensionless
- D) None
Answer: C) It is dimensionless
Explanation: Solid angle is the ratio of an area on a sphere to the square of the radius, hence dimensionless.
14. The differential solid angle element $$d\Omega$$ in spherical coordinates is:
- A) $$\sin\theta d\theta d\phi$$
- B) $$r^2 \sin\theta d\theta d\phi$$
- C) $$\cos\theta d\theta d\phi$$
- D) None
Answer: A) $$\sin\theta d\theta d\phi$$
Explanation: The differential solid angle element involves these angular components on a unit sphere.
15. Conversion from Cartesian to Cylindrical coordinates is given by:
- A) $$r = \sqrt{x^2 + y^2}, \theta = \tan^{-1}(y/x), z = z$$
- B) $$r = x + y, \theta = y/x, z = z$$
- C) $$r = \sqrt{x^2 + z^2}, \theta = \tan^{-1}(z/x), y = y$$
- D) None
Answer: A) $$r = \sqrt{x^2 + y^2}, \theta = \tan^{-1}(y/x), z = z$$
Explanation: Cylindrical coordinates relate to the xy-plane projection and height z.
16. What type of angle is a solid angle?
- A) 1D angle
- B) 2D angle
- C) 3D angle
- D) None
Answer: C) 3D angle
Explanation: Solid angle represents angular extent in three dimensions.
17. In spherical coordinates, the radial distance $$r$$ is the distance from:
- A) xy-plane
- B) z-axis
- C) Origin
- D) None
Answer: C) Origin
Explanation: Radial distance is the distance from the origin in 3D space.
18. The integral of $$d\Omega$$ over the sphere is:
- A) $$2\pi$$
- B) $$4\pi$$
- C) $$6\pi$$
- D) $$8\pi$$
Answer: B) $$4\pi$$
Explanation: Integrating the solid angle element over a sphere gives $$4\pi$$.
19. Which coordinate system can be described with radius, polar angle, and azimuth angle?
- A) Polar
- B) Cylindrical
- C) Spherical
- D) None
Answer: C) Spherical
Explanation: Spherical coordinates describe positions with radius, polar, and azimuthal angles.
20. Solid angle subtended by a hemisphere is:
- A) $$2\pi$$
- B) $$4\pi$$
- C) $$\pi$$
- D) $$0$$
Answer: A) $$2\pi$$
Explanation: Solid angle of hemisphere is half the full sphere, thus $$2\pi$$ steradians.
21. The cylindrical coordinate θ is measured from which axis?
- A) x-axis
- B) y-axis
- C) z-axis
- D) xy-plane
Answer: A) x-axis
Explanation: θ in cylindrical coordinates is the azimuthal angle measured from the positive x-axis in the xy-plane.
22. Which of these is NOT true about Cartesian coordinates?
- A) Orthogonal axes
- B) Represent points as (x, y, z)
- C) Use angles instead of distances
- D) Commonly used in physics
Answer: C) Use angles instead of distances
Explanation: Cartesian coordinates are based on distances along axes, not angles.
23. The unit vector in the radial direction in spherical coordinates is directed:
- A) Toward z-axis
- B) Away from origin in radial direction
- C) Tangential to the sphere
- D) Along azimuthal angle
Answer: B) Away from origin in radial direction
Explanation: The radial unit vector points outward from the origin.
24. The angular part of spherical coordinates consists of:
- A) One angle
- B) Two angles
- C) Three angles
- D) None
Answer: B) Two angles
Explanation: Spherical coordinates are defined by two angles: polar (θ) and azimuthal (φ).
25. The volume element in cylindrical coordinates is:
- A) $$r\,dr\,d\theta\,dz$$
- B) $$dr\,d\theta\,dz$$
- C) $$r^2\,dr\,d\theta\,dz$$
- D) $$r \sin\theta\,dr\,d\theta\,dz$$
Answer: A) $$r\,dr\,d\theta\,dz$$
Explanation: The factor $$r$$ accounts for the radius in the cylindrical coordinate volume element.
26. Solid angle related to an area $$A$$ on a sphere of radius $$r$$ is:
- A) $$ \frac{A}{r^2} $$
- B) $$ A \times r^2 $$
- C) $$ A \times r $$
- D) $$ \frac{A}{r} $$
Answer: A) $$ \frac{A}{r^2} $$
Explanation: Solid angle $$\Omega = \frac{A}{r^2}$$ dimensionless.
27. In which coordinate system is the Laplacian simplest for problems with spherical symmetry?
- A) Cartesian
- B) Cylindrical
- C) Spherical
- D) Polar
Answer: C) Spherical
Explanation: Spherical symmetry problems are best handled using spherical coordinates.
28. Which of the following is true for the azimuth angle, φ, in spherical coordinates?
- A) 0 to $$ \pi $$
- B) 0 to $$ 2\pi $$
- C) -$$ \pi $$ to $$ \pi $$
- D) None
Answer: B) 0 to $$ 2\pi $$
Explanation: Azimuthal angle φ ranges from 0 to $$2\pi$$.
29. The conversion of spherical to Cartesian coordinate for z is:
- A) $$z = r \sin \theta \cos \phi$$
- B) $$z = r \cos \theta$$
- C) $$z = r \sin \phi$$
- D) $$z = r \cos \phi$$
Answer: B) $$z = r \cos \theta$$
Explanation: z-coordinate depends on radial distance and polar angle θ.
30. Which coordinate system is best used to describe the motion of a particle moving on a cone?
- A) Cartesian
- B) Cylindrical
- C) Spherical
- D) None
Answer: C) Spherical
Explanation: Spherical coordinates suit problems with angular motion in 3D, like a cone.
31. What is the solid angle subtended by a square of side $$a$$ at a distance $$r$$ when $$a \ll r$$?
- A) $$a^2 / r^2$$
- B) $$a / r$$
- C) $$\sqrt{a}/r$$
- D) None
Answer: A) $$a^2 / r^2$$
Explanation: For small areas, solid angle approximates area over squared distance.
32. What is the Jacobian determinant for transformation from Cartesian to spherical coordinates?
- A) $$r^2 \sin\theta$$
- B) $$r \sin\theta$$
- C) $$r^2 \cos\theta$$
- D) $$r^3 \sin \phi$$
Answer: A) $$r^2 \sin\theta$$
Explanation: Jacobian for spherical coordinates is $$r^2 \sin \theta$$.
33. What is the shape of the surface $$r = \text{constant}$$ in spherical coordinates?
- A) Plane
- B) Sphere
- C) Cylinder
- D) Cone
Answer: B) Sphere
Explanation: Constant radius defines a spherical surface.
34. Which quantity remains constant to define cylindrical coordinates?
- A) Radial distance and height
- B) Radial distance and angle
- C) Angle and height
- D) Height and polar angle
Answer: A) Radial distance and height
Explanation: Cylindrical coordinates specify (r, θ, z), with r and z values.
35. The solid angle subtended by an object depends on:
- A) Its size only
- B) Distance from observer only
- C) Size and distance
- D) None
Answer: C) Size and distance
Explanation: Bigger or closer objects subtend larger solid angles.
36. How is the azimuthal angle measured in cylindrical coordinates?
- A) From y-axis anticlockwise
- B) From x-axis anticlockwise
- C) From z-axis downward
- D) None
Answer: B) From x-axis anticlockwise
Explanation: In cylindrical coords, θ is angle from x-axis in xy-plane.
37. The differential arc length in spherical coordinates for angle θ is proportional to:
- A) $$r d\theta$$
- B) $$r \sin \theta d\theta$$
- C) $$r \cos \theta d\theta$$
- D) $$d\theta$$
Answer: A) $$r d\theta$$
Explanation: Arc length for θ on a circle of radius r is $$r d\theta$$.
38. A solid angle of 1 steradian covers what fraction of a sphere?
- A) $$1/4\pi$$
- B) $$1/2\pi$$
- C) $$1/\pi$$
- D) None
Answer: A) $$1/4\pi$$
Explanation: Since full sphere is $$4\pi$$ sr, 1 sr is $$1/4\pi$$ fraction.
39. In spherical coordinates, what does the volume element look like when $$r=1$$?
- A) $$\sin\theta d\theta d\phi$$
- B) $$d\theta d\phi$$
- C) $$r^2 \sin\theta d\theta d\phi$$
- D) None
Answer: A) $$\sin\theta d\theta d\phi$$
Explanation: At unit radius, volume element reduces to differential solid angle.
40. What is the key advantage of spherical coordinates in physics?
- A) Easy vector calculations
- B) Suited for spherical symmetry problems
- C) Useful in planar geometry
- D) None
Answer: B) Suited for spherical symmetry problems
Explanation: Spherical coords simplify problems with radial symmetry.
