The Question is from neet-physics , and it was created by the author Mr. ANIL KUMAR.G

Question

The ratio of the speed of the electron in the first Bohr orbit of hydrogen and the speed of light is equal to (where $e,h$ and $c$ have their usual meanings)

Mr. ANIL KUMAR.G · Accepted Answer

## CORE INFORMATION
✓ **Answer**: D
## CONCEPT FRAMEWORK
	**Primary Concept**: Speed of electron in Bohr's orbit and its relation to speed of light.
	**Related Topics**: Bohr's atomic model, fine structure constant
	**Prerequisites**: Knowledge of Bohr's postulates, and the expression for the velocity of an electron in the nth Bohr orbit.
## SOLUTION PATHWAY
**Given Information**:
	* First Bohr orbit of hydrogen atom (n=1)
	* Speed of light = c
	* Electron charge = e
	* Planck's constant = h
**Method & Steps**:
	1. **Calculate the speed of the electron in the first Bohr orbit (v_1)**
		Reasoning: The speed of an electron in the nth Bohr orbit is given by $v_n = \frac{2\pi ke^2}{nh}$, where k is Coulomb's constant, which is equal to  1/(4πε₀) in SI units. For the first Bohr orbit (n=1), $v_1 = \frac{2\pi ke^2}{h}$.  Also, recall that the speed of electron in the first Bohr orbit can be expressed as $v_1 = \frac{e^2}{2\epsilon_0 h}$ 
		Formula/Rule: $v_n = \frac{2\pi ke^2}{nh}$, where k = 1/(4πε₀)
	2. **Express v_1 in terms of c, and the fine structure constant**
		Working: The fine structure constant is given by $\alpha = \frac{e^2}{2\epsilon_0 hc} = \frac{v_1}{c}$. Therefore, $v_1 = \alpha c$
		Key Point: The fine structure constant is a dimensionless constant approximately equal to 1/137
	3. **Find the ratio v_1/c**
		Result: The ratio is $\frac{v_1}{c} = \frac{\frac{e^2}{2\epsilon_0 h}}{c} =  \frac{e^2}{2\epsilon_0 hc} = \alpha$. The value of $\alpha$ is approximately $\frac{1}{137}$. Using k = 1/(4πε₀) and rearranging the equation from step 1, $\frac{v_1}{c} = \frac{2\pi e^2}{hc} \frac{1}{4\pi\epsilon_0} = \frac{2\pi e^2}{hc} \frac{k}{c} = \frac{e^2}{2\epsilon_0 hc} = \frac{2\pi ke^2}{hc}$.
		Verification: The ratio v1/c is equivalent to the fine structure constant $\alpha = \frac{e^2}{2\epsilon_0 hc}$ which is also equal to $\frac{2\pi e^2}{hc} \frac{1}{4\pi\epsilon_0} $.
 ## OPTION ANALYSIS
[A] :x:  Incorrect. This option has the Planck's constant in the numerator and the square of the electron charge in the denominator.
[B] :x: Incorrect. The Planck's constant is in the numerator and the speed of light is in the denominator.
[C] :x: Incorrect. The speed of light is in the numerator and the Planck's constant is in the denominator.
[D] ✓: Correct. This is the correct expression for the ratio of the electron's speed in the first Bohr orbit to the speed of light.
 ## LEARNING AIDS
	**Common Mistakes**:
		1. **Incorrect formula for electron speed**: Using the wrong formula for the velocity of an electron in a Bohr orbit.
		2. **Confusing constants**: Mixing up the constants or their positions in the formula.
	**Quick Tips**:
		1. **Remember the fine structure constant**: It provides a direct link between electron speed in the first Bohr orbit and the speed of light.
		2. **Start with the basic Bohr radius velocity formula**: Then rearrange to get the fine structure constant.

Mr. ANIL KUMAR.G · Answer

$2\pi \,hc/{e^2}$

Mr. ANIL KUMAR.G · Answer

${e^2}h/2\pi c$

Mr. ANIL KUMAR.G · Answer

${e^2}c/2\pi h$

Mr. ANIL KUMAR.G · Answer

The correct Option is "$2\pi {e^2}/hc$"

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