Distance Between Venus and Earth in Speed of Light Units

To find the distance between Venus and Earth in terms of the new unit of length $\alpha$, we follow these steps:

1. Definition of the unit $\alpha$: The new unit of length $\alpha$ is chosen such that its magnitude corresponds to the distance traveled by light in unit time ($1\text{ s}$). Thus: $1\alpha = c \times 1\text{ s}$ where $c$ is the speed of light in vacuum.

2. Calculate the time taken by light: The time $t$ taken by light to travel from Venus to Earth is given as $6\text{ min. } 40\text{ s}$. Converting this time into seconds: $t = 6\text{ min } 40\text{ s} = (6 \times 60 + 40)\text{ s} = 360\text{ s} + 40\text{ s} = 400\text{ s}$

3. Calculate the distance: The distance $d$ covered by light in time $t$ is: $d = c \times t = c \times 400\text{ s}$

4. Express distance in terms of $\alpha$: Substituting $1\alpha = c \times 1\text{ s}$ into the distance equation: $d = 400 \times (c \times 1\text{ s}) = 400\alpha$

Thus, the distance between Venus and Earth is $400\alpha$.

Correct Option: B ($400 \alpha$)