Step 1: Understanding the Concept:
A central force is one that is always directed towards or away from a fixed central point and its magnitude depends only on the distance from that point.
Step 2: Key Formula or Approach:
The approach involves calculating the torque $\vec{\tau}$ acting on the particle.
Torque is given by the cross product of position vector and force:
\[ \vec{\tau} = \vec{r} \times \vec{F} \]
Newton's second law for rotation states that torque equals the rate of change of angular momentum: $\vec{\tau} = \frac{d\vec{L}}{dt}$.
Step 3: Detailed Explanation:
For a central force field, the force vector $\vec{F}$ is purely radial, which means it is parallel to the position vector $\vec{r}$.
We can write $\vec{F} = f(r) \hat{r}$ and $\vec{r} = r \hat{r}$.
The cross product of two parallel vectors is zero:
\[ \vec{\tau} = (r \hat{r}) \times (f(r) \hat{r}) = r f(r) (\hat{r} \times \hat{r}) = 0 \]
Since the net torque is zero ($\vec{\tau} = 0$), the rate of change of angular momentum $\frac{d\vec{L}}{dt}$ must also be zero.
This implies that the angular momentum $\vec{L}$ is a constant vector.
Step 4: Final Answer:
The correct option is (B).