Question

Given below are two statements: 
Statement (I) : The dimensions of Planck’s constant and angular momentum are same. 
Statement (II) : In Bohr’s model, electron revolves around the nucleus in those orbits for which angular momentum is an integral multiple of Planck’s constant. 
In the light of the above statements, choose the most appropriate answer from the options given below:

• A. Both Statement I and Statement II are correct
• B. Statement I is incorrect but Statement II is correct
• C. Statement I is correct but Statement II is incorrect
• D. Both Statement I and Statement II are incorrect

Hint:

Planck’s constant \( h \) and angular momentum \( L \) have the same dimensional formula. Also, in Bohr’s model, angular momentum is quantized in integer multiples of \( h/2\pi \).

Answer 1

The Correct Option is C

To solve this question, let's analyze each statement separately:

Statement I: "The dimensions of Planck’s constant and angular momentum are same."

Planck's constant \( h \) has the dimensional formula \([M^1L^2T^{-1}]\). Angular momentum \( L \) (of a particle with mass \( m \), revolving with velocity \( v \), at a radius \( r \)) also has the dimensional formula given by \( mvr \), leading to \([M^1L^2T^{-1}]\).

Thus, Planck’s constant and angular momentum indeed have the same dimensions. So, Statement I is correct.

Statement II: "In Bohr’s model, electron revolves around the nucleus in those orbits for which angular momentum is an integral multiple of Planck’s constant."

According to Bohr’s model, the angular momentum \( L \) of an electron is quantized and given by:

\(L = n\frac{h}{2\pi}\) where \( n \) is a positive integer.

This expression shows that angular momentum is an integral multiple of \(\frac{h}{2\pi}\) (also known as reduced Planck's constant \( \hbar \)), not Planck's constant \( h \) itself. Hence, Statement II is incorrect because it states that angular momentum is an integral multiple of \( h \) without the \( \frac{1}{2\pi} \) factor.

Given the analysis:

• Statement I is correct.
• Statement II is incorrect.

Therefore, the correct answer is: Statement I is correct but Statement II is incorrect.

Answer 2

1. Statement I: The dimensions of Planck’s constant and angular momentum are the same. Planck’s constant \( h \) has the dimension of \( [ML^2 T^{-1}] \), where: 
- \( M \) is mass, 
- \( L \) is length, 
- \( T \) is time. 
Angular momentum \( L \) also has the dimension of \( [ML^2 T^{-1}] \), since it is given by the product of mass, length, and velocity. 
Hence, Statement I is correct. 
2. Statement II: In Bohr’s model, electron revolves around the nucleus in those orbits for which angular momentum is an integral multiple of Planck’s constant. According to Bohr’s model, the angular momentum \( L \) of an electron is quantized and is an integral multiple of Planck’s constant \( h \), i.e. \[ L = \frac{nh}{2\pi} \] where \( n \) is a positive integer. 
Hence, Statement II is also correct. 
Since both statements are correct, the correct answer is (3).