Question

An electron in hydrogen atom is moving round the nucleus with speed \(2.2 \times 10^6\) m/s in an orbit of radius \(5 \times 10^{-11}\) meter. Find the value of equivalent electric current.

Hint:

For the current problem, the concept is simple: current is charge flow per second. Find how many times the electron circles per second (the frequency \(f = v/2\pi r\)) and multiply by its charge (\(I = ef\)).

Answer 1

Step 1: Understanding the Concept:
An electron revolving in a circular orbit behaves like a tiny current loop. The equivalent electric current is defined as the total charge that passes through any point on the orbit per unit time.

Step 2: Key Formula or Approach:
The equivalent current \(I\) is given by: \[ I = \frac{\text{Charge}}{\text{Time Period}} = \frac{e}{T} \] The time period \(T\) for one revolution is the circumference of the orbit divided by the electron's speed \(v\): \[ T = \frac{2\pi r}{v} \] Combining these gives the formula for the current: \[ I = \frac{e}{ (2\pi r / v) } = \frac{ev}{2\pi r} \]

Step 3: Detailed Explanation (Calculation):
We are given the following values: \begin{itemize} \item Speed of electron, \(v = 2.2 \times 10^6\) m/s. \item Radius of orbit, \(r = 5 \times 10^{-11}\) m. \item Charge of an electron, \(e = 1.6 \times 10^{-19}\) C. \end{itemize} Substitute these values into the formula for current: \[ I = \frac{(1.6 \times 10^{-19} \text{ C}) \times (2.2 \times 10^6 \text{ m/s})}{2\pi \times (5 \times 10^{-11} \text{ m})} \] \[ I = \frac{3.52 \times 10^{-13}}{10\pi \times 10^{-11}} = \frac{3.52 \times 10^{-13}}{31.416 \times 10^{-11}} \] \[ I = \frac{3.52}{31.416} \times 10^{-2} \text{ A} \approx 0.112 \times 10^{-2} \text{ A} \] \[ I \approx 1.12 \times 10^{-3} \text{ A} = 1.12 \text{ mA} \]

Step 4: Final Answer:
The value of the equivalent electric current is approximately \(1.12 \times 10^{-3}\) A or 1.12 mA.