In standard spherical polar coordinates (r, θ, φ), where θ is the polar angle measured from the positive z axis and φ is the azimuthal angle measured from the positive x axis in the x y plane, the y component of a vector P of magnitude P is given by which of the following expressions?

Introduction / Context:

In physics, especially in electromagnetism and quantum mechanics, spherical polar coordinates are frequently used to exploit symmetry in problems involving spheres or central potentials. It is essential to know how to resolve a vector into its Cartesian components using spherical coordinates. This question asks specifically for the y component of a vector P expressed in terms of its magnitude P and the spherical angles θ and φ.

Given Data / Assumptions:

• Spherical coordinates (r, θ, φ) are used.
• θ is the polar angle measured from the positive z axis.
• φ is the azimuthal angle measured in the x y plane from the positive x axis.
• The vector P has magnitude P and direction specified by (θ, φ).

Concept / Approach:

In spherical polar coordinates, a point in space is represented by (r, θ, φ). The relationships between Cartesian coordinates (x, y, z) and spherical coordinates are standard and widely used. For a point at distance r from the origin and angles θ and φ defined as above, the Cartesian coordinates are x = r sinθ cosφ, y = r sinθ sinφ and z = r cosθ. For a vector of magnitude P pointing in the same direction, the components scale in the same way, giving P_x = P sinθ cosφ, P_y = P sinθ sinφ and P_z = P cosθ. Therefore, the y component must involve both sinθ and sinφ.

Step-by-Step Solution:

Verification / Alternative check:

If φ = 0, the vector lies in the x z plane, so its y component should be zero. Substituting φ = 0 into P sinθ sinφ gives P sinθ * 0 = 0, which is correct. If φ = 90 degree (or π / 2 radians), the vector lies in the y z plane and its x component should be zero, while y and z remain. Substituting φ = 90 degree into P sinθ sinφ gives P sinθ * 1 = P sinθ, which is non zero in general, consistent with a non zero y component in that plane.

Why Other Options Are Wrong:

Option B: P sinθ cosφ corresponds to the x component, not the y component.

Option C: P cosθ sinφ does not match any standard Cartesian component for the given angular definitions and would give incorrect behaviour, for example non zero y at θ = 0 where the vector should point along z.

Option D: P cosθ cosφ similarly fails to vanish when expected and does not correspond to a correct Cartesian component.

Common Pitfalls:

Many students confuse the roles of sinθ and cosθ or interchange the positions of sinφ and cosφ. A helpful way to remember is that the projection in the x y plane has factor sinθ, while the distribution between x and y involves cosφ for x and sinφ for y. Visualising the geometry in two steps projection down from the z axis and then projection into the x y plane helps prevent errors.

Final Answer:

The y component of P is P sinθ sinφ.