An electron revolving in nth Bohr orbit has magnetic moment mun. If mun propto nx, the value of x is:

Year	2026	2025	2024	Total
Questions	5	8	9	22

Year

2026

2025

2024

Total

Questions

Solution & Explanation

Related Formula

μₙ = i A = (e)/(T) π r²

Alternative relation using angular momentum:

μₙ = (e)/(2m) L

Core Logic

From Bohr's quantization condition, angular momentum $L$ $L$ is given by:

L = (nh)/(2π)

Therefore, the magnetic moment can be written directly as:

μₙ = (e)/(2m) ((nh)/(2π))

μₙ \propto n

Step 1: Direct Proportionality Analysis

We can also derive this via velocity and radius: $v \propto (1)/(n)$ $v \propto \frac{1}{n}$ and $r \propto n²$ $r \propto n^2$.

μₙ = (e v r)/(2) \propto ((1)/(n)) (n²) = n

Comparing with $μₙ \propto n^x$ $\mu_n \propto n^x$, we find $x = 1$ $x = 1$.

Pattern Recognition

Magnetic moment of an electron in a Bohr orbit is always directly proportional to the principal quantum number $n$ $n$. It is an integer multiple of the Bohr magneton ( $μB = (eh)/(4π m)$ $\mu_B = \frac{eh}{4\pi m}$).

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Class 12 Physics: Atoms

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Q53 jee_main_2024_30_jan_morning Hydrogen Energy Levels and Transitions

A electron of hydrogen atom on an excited state is having energy

Eₙ = -0.85 ~eV

. The maximum number of allowed transitions to lower energy level is ....

Numerical Answer. Answer: 6 to 6

Solution

Related Formula

Eₙ = -(13.6)/(n²) ~eV

Number of transitions = (n(n - 1))/(2)

Core Logic

First, identify the principal quantum number $n$ $n$ corresponding to the energy $-0.85 ~eV$ $-0.85 \mathrm{~eV}$. Then, use the combinatorics formula to find the total possible downward emission transitions.

Step 1: Find Quantum State

Eₙ = -(13.6)/(n²)

-0.85 = -(13.6)/(n²)

n² = (13.6)/(0.85) = 16

$n = 4$ $n = 4$

Step 2: Calculate Transitions

Maximum number of transitions from $n=4$ $n=4$ to lower levels ( $n=3, 2, 1$ $n=3, 2, 1$):

= (n(n - 1))/(2)

= (4(4 - 1))/(2) = (12)/(2) = 6

Pattern Recognition

Energy states in Hydrogen are heavily standardized: $n=1 arrow -13.6$ $n=1 \rightarrow -13.6$, $n=2 arrow -3.4$ $n=2 \rightarrow -3.4$, $n=3 arrow -1.51$ $n=3 \rightarrow -1.51$, $n=4 arrow -0.85$ $n=4 \rightarrow -0.85$. Recognize $-0.85 ~eV$ $-0.85 \mathrm{~eV}$ as state 4 immediately.

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Class 12 Physics: Atoms

Q36 jee_main_2024_31_jan_morning Hydrogen Spectrum

If the wavelength of the first member of Lyman series of hydrogen is

λ

. The wavelength of the second member will be

A. $(27)/(32) λ$ $\frac{27}{32} \lambda$
B. $(32)/(27)λ$ $\frac{32}{27}\lambda$
C. $(27)/(5) λ$ $\frac{27}{5} \lambda$
D. $(5)/(27)λ$ $\frac{5}{27}\lambda$

Solution

Related Formula

(1)/(λ) = R Z² [ (1)/(n₁²) - (1)/(n₂²) ]

Core Logic

For the first member of the Lyman series of hydrogen ( $n₁ = 1, n₂ = 2$ $n_1 = 1, n_2 = 2$):

(1)/(λ) = (13.6 Z²)/(hc) [ (1)/(1²) - (1)/(2²) ]

(1)/(λ) = (13.6 Z²)/(hc) [ (3)/(4) ] (i)

Step 2: Second Member Calculation

For the second member of the Lyman series ( $n₁ = 1, n₂ = 3$ $n_1 = 1, n_2 = 3$):

(1)/(λ') = (13.6 Z²)/(hc) [ (1)/(1²) - (1)/(3²) ]

(1)/(λ') = (13.6 Z²)/(hc) [ (8)/(9) ] (ii)

Step 3: Ratio

On dividing equation (i) by (ii):

(λ')/(λ) = (3/4)/(8/9) = (3)/(4) \times (9)/(8)

(λ')/(λ) = (27)/(32)

λ' = (27)/(32) λ

Pattern Recognition

Rydberg ratios between members of the same series are purely derived from the bracket terms $[1/n₁² - 1/n₂²]$ $[1/n_1^2 - 1/n_2^2]$. For Lyman 1st and 2nd, the ratio is $(3/4) / (8/9) = 27/32$ $(3/4) / (8/9) = 27/32$.

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Class 12 Physics: Atoms

An electron revolving in $nth$ $n^{\text{th}}$ Bohr orbit has magnetic moment $μₙ$ $\mu_{\mathrm{n}}$. If $μₙ \propto nx$ , the value of $x$ $\mathbf{x}$ is:

Solution & Explanation

Related Formula

Core Logic

Step 1: Direct Proportionality Analysis

Pattern Recognition

Chapter Mix

Reference Study Guides

More Atoms Previous-Year Questions — Page 5

Solution

Related Formula

Core Logic

Step 1: Find Quantum State

Step 2: Calculate Transitions

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 2: Second Member Calculation

Step 3: Ratio

Pattern Recognition

Chapter Mix

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An electron revolving in nth$n^{\text{th}}$ Bohr orbit has magnetic moment μₙ$\mu_{\mathrm{n}}$. If μₙ ∝ nx$\mu_{\mathrm{n}} \propto n^{\mathrm{x}}$, the value of x$\mathbf{x}$ is:

Solution & Explanation

Related Formula

Core Logic

Step 1: Direct Proportionality Analysis

Pattern Recognition

Chapter Mix

Reference Study Guides

More Atoms Previous-Year Questions — Page 5

Solution

Related Formula

Core Logic

Step 1: Find Quantum State

Step 2: Calculate Transitions

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 2: Second Member Calculation

Step 3: Ratio

Pattern Recognition

Chapter Mix

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An electron revolving in $nth$ $n^{\text{th}}$ Bohr orbit has magnetic moment $μₙ$ $\mu_{\mathrm{n}}$. If $μₙ \propto nx$ , the value of $x$ $\mathbf{x}$ is: