Question:
Suppose an electron is attracted towards the origin by a force \( K/r \), where K is a constant and r is the distance of the electron from the origin. By applying Bohr model to this system, the radius of the \( n^{th} \) orbit of the electron is found to be \( r_n \), and the kinetic energy of the electron to be \( T_n \), then which of the following is true?
Suppose an electron is attracted towards the origin by a force \( K/r \), where K is a constant and r is the distance of the electron from the origin. By applying Bohr model to this system, the radius of the \( n^{th} \) orbit of the electron is found to be \( r_n \), and the kinetic energy of the electron to be \( T_n \), then which of the following is true?
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In non-Coulombic central force fields, the behavior of orbital radii and energy levels can differ significantly from the standard hydrogen-like atom model.
Updated On: Jun 9, 2026
- \( T_n \) is independent of n, \( r_n \propto n \)
- \( T_n \propto 1/n, r_n \propto n \)
- \( T_n \propto 1/n, r_n \propto n^2 \)
- \( T_n \propto 1/n^2, r_n \propto n^2 \)
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The Correct Option is A
Solution and Explanation
Concept:
In the Bohr model, we have two primary conditions: the force provided by the field acting as centripetal force (\( \frac{mv^2}{r} = F(r) \)) and the quantization of angular momentum (\( mvr = \frac{nh}{2\pi} \)). Here, \( F(r) = \frac{K}{r} \).
Step 1: Solve the force equation for velocity \( v \).
$$ \frac{mv^2}{r} = \frac{K}{r} \implies mv^2 = K \implies v = \sqrt{\frac{K}{m}} $$ Since \( K \) and \( m \) are constants, the velocity \( v \) of the electron is constant in every orbit and is independent of the orbital radius \( r \) or the orbit number \( n \).
Step 2: Determine the Kinetic Energy (\( T_n \)).
$$ T_n = \frac{1}{2} mv^2 = \frac{1}{2} m \left( \sqrt{\frac{K}{m}} \right)^2 = \frac{K}{2} $$ Because \( T_n = K/2 \), the kinetic energy is constant and independent of the orbit number \( n \).
Step 3: Determine the orbit radius (\( r_n \)).
Apply Bohr's quantization condition \( mvr = \frac{nh}{2\pi} \): $$ r_n = \frac{nh}{2\pi mv} $$ Since \( n \), \( h \), \( m \), and \( v \) are constants (except for \( n \)), we have \( r_n \propto n \). $$\boxed{T_n \text{ is independent of n}, r_n \propto n}$$
Step 1: Solve the force equation for velocity \( v \).
$$ \frac{mv^2}{r} = \frac{K}{r} \implies mv^2 = K \implies v = \sqrt{\frac{K}{m}} $$ Since \( K \) and \( m \) are constants, the velocity \( v \) of the electron is constant in every orbit and is independent of the orbital radius \( r \) or the orbit number \( n \).
Step 2: Determine the Kinetic Energy (\( T_n \)).
$$ T_n = \frac{1}{2} mv^2 = \frac{1}{2} m \left( \sqrt{\frac{K}{m}} \right)^2 = \frac{K}{2} $$ Because \( T_n = K/2 \), the kinetic energy is constant and independent of the orbit number \( n \).
Step 3: Determine the orbit radius (\( r_n \)).
Apply Bohr's quantization condition \( mvr = \frac{nh}{2\pi} \): $$ r_n = \frac{nh}{2\pi mv} $$ Since \( n \), \( h \), \( m \), and \( v \) are constants (except for \( n \)), we have \( r_n \propto n \). $$\boxed{T_n \text{ is independent of n}, r_n \propto n}$$
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