Question

In Bohr's atomic model, the electron is assumed to revolve in a circular orbit of radius 0.5 Å. If the speed of the electron is $2.2 \times 10^6$ m/s, then the current associated with the electron will be _________ $\times 10^{-2}$ mA. [Take $\pi$ as 22/7]

Hint:

The concept of an orbiting charge creating a current is fundamental. The equivalent current is always $I=q/T = qf$, where q is the charge, T is the period, and f is the frequency of revolution. This is also related to the magnetic dipole moment of the orbit.

Answer 1

Correct Answer: 112

To determine the current associated with the electron in Bohr's atomic model, we need to follow these steps:

The electron revolves in a circular orbit, creating a circular current loop. The formula for current (I) in such a loop due to one charge is given by:

I = \(\frac{q}{T}\),

where q is the charge of the electron, and T is the time period for one complete revolution. The charge of an electron q is approximately \(1.6 \times 10^{-19}\) C.

The time period T can be found using: 

T = \(\frac{2\pi r}{v}\),

where r is the radius of the orbit, and v is the speed of the electron. Here, r = 0.5 \times 10^{-10}\) m and v = 2.2 \times 10^6\) m/s.

Substituting the values:

T = \(\frac{2 \times \frac{22}{7} \times 0.5 \times 10^{-10}}{2.2 \times 10^6}\)

By calculating, T \approx 1.4286 \times 10^{-16}\) s.

Now, substituting T in the current formula:

I = \(\frac{1.6 \times 10^{-19}}{1.4286 \times 10^{-16}}\) C/s

Calculating this gives:

I \approx 1.12 \times 10^{-3}\) A

Converting the current to mA:

I \approx 1.12 \times 10^{-3} \times 10^3 = 1.12\) mA

Finally, expressing in terms of \( \times 10^{-2} \) mA:

The current associated with the electron is 112 \( \times 10^{-2}\) mA.

This value is validated to be within the expected range (112,112).