Q.40 Physics ⭐ 360° Option Elimination Active

NEET 2026 Physics Question 40 Solution

Topic: Atoms & Nuclei · Official NTA Booklet Code 50

Question

Consider that an electron is revolving in an excited state of Hydrogen atom with velocity √25.6 × 10⁵ m s⁻¹. The radius of the orbit is x × 10⁻⁹ m. The value of x is: [Take the mass of electron to be 9×10⁻³¹ kg, charge of electron = −1.6×10⁻¹⁹ C and 1/(4πε₀) = 9×10⁹ N m² C⁻²]

✓

Verified Against NTA Final Answer Key

Option (4) : x = 1 (r = 1 * 10^-9 m)

📚 Atoms & Nuclei Medium

📋 Given Parameters & Problem Setup

Orbital Velocity v = sqrt(25.6) * 10^5 m/s ==> v^2 = 25.6 * 10^10 m^2/s^2

Electron Mass m = 9 * 10^-31 kg

Electron Charge e = 1.6 * 10^-19 C ==> e^2 = 2.56 * 10^-38 C^2

Coulomb Constant 1 / (4*pi*epsilon_0) = 9 * 10^9 N m^2 C^-2

Given Form of Radius r = x * 10^-9 m

Target Variable Value of x

⚡

The 30-Second Exam Hall Shortcut: ⚡ Dynamic Balance Cancellation Trick

⏱️ Bypasses calculating the quantum principal number n.

Centripetal force m*v^2 / r = k*e^2 / r^2 ==> r = k*e^2 / (m*v^2)!

1 Notice the numbers chosen by NTA: k = 9*10^9 and m = 9*10^-31 (The 9s cancel!).

2 e^2 = (1.6*10^-19)^2 = 2.56*10^-38.

3 v^2 = 25.6*10^10.

4 Ratio e^2 / v^2 = (2.56 / 25.6) * (10^-38 / 10^10) = 0.1 * 10^-48 = 10^-49.

5 Combine with (k / m) = (9*10^9) / (9*10^-31) = 10^40.

6 r = (k / m) * (e^2 / v^2) = 10^40 * 10^-49 = 10^-9 m = 1 * 10^-9 m.

7 Therefore, x = 1. Solved in 20 seconds!

🛡️

Why You Can 100% Trust This Shortcut: Newtonian centripetal balance m*v^2/r = k*e^2/r^2 holds exactly for any Bohr orbit. NTA specifically calibrated sqrt(25.6) and m=9 to yield an exact integer 1.

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Formulate Centripetal Force Balance for Electron Orbit

For an electron of mass m moving at speed v in a circular orbit of radius r around a proton (+e):

\frac{m v^2}{r} = \frac{1}{4\pi \epsilon_0} \frac{e^2}{r^2}

• One factor of r cancels from both sides.
• m v^2 r = \frac{1}{4\pi \epsilon_0} e^2.

Step 2

Isolate Orbital Radius r

Rearrange to solve explicitly for orbital radius r:

r = \left( \frac{1}{4\pi \epsilon_0} \right) \frac{e^2}{m v^2}

• Notice that this formula does not even require knowing the principal quantum number n!

Step 3

Substitute Constants and Evaluate x

Substitute the numerical values provided in the question header:

r = \frac{(9 \times 10^9) \times (1.6 \times 10^{-19})^2}{(9 \times 10^{-31}) \times (\sqrt{25.6} \times 10^5)^2} = \frac{(9 \times 10^9) \times (2.56 \times 10^{-38})}{(9 \times 10^{-31}) \times (25.6 \times 10^{10})}

• Cancel 9 from numerator and denominator.
• 2.56 / 25.6 = 0.1 = 10^-1.
• Powers of 10: 10^9 * 10^-38 / (10^-31 * 10^10) = 10^-29 / 10^-21 = 10^-8.
• r = 10^-1 * 10^-8 = 10^-9 m = 1 * 10^-9 m.
• Comparing with r = x * 10^-9 m: x = 1.
• Matches Option (4) exactly.

⭐

360° Option Elimination Engine Key Feature

Diagnose every choice: Know exactly why the right option works and why distractors fail.

4-Option Deep Scan

1 Option (1)

✗ Incorrect

Verdict & Scientific Reason:

Used principal quantum number n.

⚠️

Student Fallacy (Why candidates pick this trap): Identifies the state as n = 4 and assumes x = n.

Note: Radius scales as r = n^2 * a_0; radius is 1 nm, not 4 nm.

2 Option (2)

✗ Incorrect

Verdict & Scientific Reason:

Arithmetic error.

⚠️

Student Fallacy (Why candidates pick this trap): Divides 25.6 by 9.

Note: The factor 9 cancels out completely.

3 Option (3)

✗ Incorrect

Verdict & Scientific Reason:

Forgot square in velocity.

⚠️

Student Fallacy (Why candidates pick this trap): Uses v instead of v^2 in the denominator.

Note: Centripetal acceleration is v^2 / r, requiring v^2.

4 Option (4)

✓ 100% Correct

Verdict & Scientific Reason:

100% mathematically exact; confirmed by Official NTA Final Answer Key.

💡

Why this is airtight: Flawless cancellation of coefficients and powers of 10.

Note: Exact physics.

⚠️

NTA Trap Alert: Attempting to Find n First (The Wasted 3-Minute Loop)

Many candidates calculate v = (2.18 * 10^6) / n, square both sides to find n, and then use r = n^2 * 0.529 A, introducing decimal rounding errors.

🛡️ How to avoid the -1 mark: Whenever v and constants are given directly, use r = k*e^2 / (m*v^2) in a single direct step!

📖

Core NEET Syllabus Standard

NEET UG Physics Core Standards · Atoms Standard Benchmark

Core Scientific Principle Tested:

"Dynamic equilibrium of electron in orbit: m*v^2/r = e^2 / (4pi*epsilon_0*r^2). Orbital radius is directly linked to orbital speed by r = e^2 / (4pi*epsilon_0*m*v^2)."

Syllabus Focus: Bohr Model of Hydrogen Atom: Orbital Velocity and Radius

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

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NEET 2026 Physics Question 40 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Dynamic Balance Cancellation Trick

Step-by-Step Rigorous Workflow & Mathematical Proof

Formulate Centripetal Force Balance for Electron Orbit

Isolate Orbital Radius r

Substitute Constants and Evaluate x

360° Option Elimination Engine Key Feature

NTA Trap Alert: Attempting to Find n First (The Wasted 3-Minute Loop)

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

🔗 Related Questions — Atoms & Nuclei

Geiger-Marsden Experiment: Scattered Alpha Particles vs Scattering Angle Plot

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