Final answer:
To an observer moving at 90% of the speed of light, the electron in a Hydrogen atom will stay in the excited state for approximately 2.294 times longer than in the stationary frame, resulting in a time duration close to 2.22 × 10⁻⁸ seconds.
Step-by-step explanation:
The duration an electron stays in an excited state in a Hydrogen atom, according to an observer moving at a significant fraction of the speed of light, can be calculated using the principles of time dilation in special relativity. Time dilation is a phenomenon in which time, as measured by a moving observer, occurs at a slower pace when compared to the time measured by a stationary observer. In this case, for an observer moving at 90% the speed of light (c), the time dilation factor (γ) can be calculated using the formula γ = 1 / √(1 - v²/c²), where v is the velocity of the moving frame of reference.
Applying this formula, γ = 1 / √(1 - (0.9c)²/c²) = 1 / √(1 - 0.81) = 1 / √(0.19) = 1 / (0.435889) ≈ 2.294. Thus, the time the electron will stay in the excited state to observer O' is approximately 2.294 times longer than the time measured in the stationary frame of reference. Therefore, the excited state duration according to observer O' is 10⁻⁸ seconds × 2.294, which equals approximately 2.294 × 10⁻⁸ seconds. Clearly, this matches closest with option b) 2.22 × 10⁻⁸ seconds.