Class 12 Physics NCERT Solutions

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Dual Nature of Radiation and Matter Class 12 Physics NCERT Solutions

The complete NCERT exercise solutions for Chapter 11, Dual Nature of Radiation and Matter — 11 questions from 11.1 to 11.11, each worked through step by step in the CBSE marking pattern. The photoelectric effect, Einstein's equation, the threshold frequency, particle nature of light and de Broglie's matter waves.

Class:12Subject:PhysicsChapter:11

4 Key Formulas24 Practice MCQs

DWritten byDeep Narayan

UpdatedSeptember 2026

Key Concept Summary

How many questions are in NCERT Class 12 Physics Chapter 11?

Chapter 11 carries 11 exercise questions, numbered 11.1 to 11.11. All of them are solved step by step on this page, along with the chapter's key formulas and exam pointers.

Table of Contents12 sections

Revision & Practice

Chapter Overview

The photoelectric effect fixes light as a stream of photons of energy E = hν, and Planck's constant falls out of the slope of the cut-off voltage versus frequency plot. X-rays give the high-frequency end, where the accelerating voltage sets the shortest wavelength, and de Broglie's matter waves make every particle a wave with λ = h/p. Every question below is from the NCERT Class 12 textbook (rationalised edition), solved line by line in the board pattern.

Board pattern

Photons: E = hc/λ, p = h/λ. Photoelectric equation: K_max = hν − φ₀ = eV₀, with slope of V₀-vs-ν plot = h/e (so h = e × slope). Threshold condition: emission needs hν ≥ φ₀. X-rays: the maximum frequency and minimum wavelength come from ν_max = eV/h and λ_min = c/ν_max = hc/eV. de Broglie: λ = h/(mv). Use h = 6.63 × 10⁻³⁴ J s, e = 1.6 × 10⁻¹⁹ C unless told otherwise.

$Ex 11.1 — Maximum frequency and minimum wavelength of X-rays$
$Ex 11.2 — Caesium photoemission: kinetic energy, stopping potential, speed$
$Ex 11.3 — Maximum kinetic energy from cut-off voltage$
$Ex 11.4 — He-Ne laser: photon energy, momentum, photon rate, atom speed$
$Ex 11.5 — Planck’s constant from the cut-off voltage slope$
$Ex 11.6 — Cut-off voltage for given threshold and incident frequency$
$Ex 11.7 — Photoelectric emission check for 330 nm light$
$Ex 11.8 — Threshold frequency from emitted electron speed$
$Ex 11.9 — Work function of an argon-laser emitter$
$Ex 11.10 — de Broglie wavelength of bullet, ball and dust particle$
$Ex 11.11 — Photon wavelength equals its de Broglie wavelength$

NCERT Exercise 11.1 — Maximum Frequency and Minimum Wavelength of X-rays

1Exercise question

Step-by-step solution

1The maximum photon energy equals the electron's full kinetic energy: hν_max = eV.
2(a) ν_max = eV/h = (1.6 × 10⁻¹⁹ × 30 × 10³)/(6.63 × 10⁻³⁴) = 7.24 × 10¹⁸ Hz.
3(b) The shortest wavelength corresponds to ν_max: λ_min = c/ν_max = (3 × 10⁸)/(7.24 × 10¹⁸) = 4.14 × 10⁻¹¹ m.

Final answer

(a) ν_max = 7.24 × 10¹⁸ Hz. (b) λ_min = 4.14 × 10⁻¹¹ m.

NCERT Exercise 11.2 — Caesium Photoemission

1Exercise question

Step-by-step solution

1Photon energy: hν = (6.63 × 10⁻³⁴ × 6 × 10¹⁴) J = 3.978 × 10⁻¹⁹ J = 2.49 eV.
2(a) K_max = hν − φ₀ = 2.49 − 2.14 = 0.346 eV ≈ 0.35 eV.
3(b) Stopping potential: eV₀ = K_max ⇒ V₀ = 0.346 V ≈ 0.35 V.
4(c) K_max = ½mv²_max ⇒ v_max = √(2K_max/m) = √(2 × 5.54 × 10⁻²⁰)/(9.1 × 10⁻³¹) = 3.49 × 10⁵ m/s.

Final answer

(a) K_max = 0.346 eV (0.35 eV). (b) V₀ = 0.35 V. (c) v_max = 3.49 × 10⁵ m/s.

NCERT Exercise 11.3 — Kinetic Energy from Cut-off Voltage

1Exercise question

Step-by-step solution

1K_max = eV₀, where V₀ = 1.5 V is the stopping (cut-off) potential.
2K_max = 1.6 × 10⁻¹⁹ × 1.5 = 2.4 × 10⁻¹⁹ J = 1.5 eV.

Final answer

K_max = 1.5 eV = 2.4 × 10⁻¹⁹ J.

NCERT Exercise 11.4 — He-Ne Laser Photon Number and Atom Speed

1Exercise question

Step-by-step solution

1(a) Energy of one photon: E = hc/λ = (6.63 × 10⁻³⁴ × 3 × 10⁸)/(632.8 × 10⁻⁹) = 3.14 × 10⁻¹⁹ J = 1.96 eV.
2Momentum: p = E/c = h/λ = (6.63 × 10⁻³⁴)/(632.8 × 10⁻⁹) = 1.05 × 10⁻²⁷ kg m s⁻¹.
3(b) Photons per second: N = P/E = (9.42 × 10⁻³)/(3.14 × 10⁻¹⁹) = 3.0 × 10¹⁶ s⁻¹.
4(c) For equal momentum: v = p/m_H = (1.05 × 10⁻²⁷)/(1.67 × 10⁻²⁷) = 0.63 m/s.

Final answer

(a) E = 3.14 × 10⁻¹⁹ J (1.96 eV), p = 1.05 × 10⁻²⁷ kg m s⁻¹. (b) 3.0 × 10¹⁶ photons/s. (c) v = 0.63 m/s.

NCERT Exercise 11.5 — Planck's Constant from the Slope

1Exercise question

Step-by-step solution

1From the photoelectric equation eV₀ = hν − φ₀, so V₀ = (h/e)ν − φ₀/e.
2The slope of V₀ versus ν is h/e.
3h = e × slope = (1.6 × 10⁻¹⁹)(4.12 × 10⁻¹⁵) = 6.6 × 10⁻³⁴ J s.

Final answer

h = 6.6 × 10⁻³⁴ J s.

NCERT Exercise 11.6 — Cut-off Voltage from Threshold Frequency

1Exercise question

Step-by-step solution

1Work function φ₀ = hν₀; photon energy hν, so K_max = h(ν − ν₀).
2K_max = (6.63 × 10⁻³⁴)(8.2 − 3.3) × 10¹⁴ = 3.25 × 10⁻¹⁹ J.
3V₀ = K_max/e = (3.25 × 10⁻¹⁹)/(1.6 × 10⁻¹⁹) = 2.03 V.

Final answer

V₀ = 2.03 V.

NCERT Exercise 11.7 — Emission Check for 330 nm Light

1Exercise question

Step-by-step solution

1Photon energy: E = hc/λ = (6.63 × 10⁻³⁴ × 3 × 10⁸)/(330 × 10⁻⁹) J = 6.03 × 10⁻¹⁹ J = 3.77 eV.
2Compare with the work function: 3.77 eV < 4.2 eV.
3Since hν < φ₀, emission cannot occur.

Final answer

No — the photon energy (3.77 eV) is less than the work function (4.2 eV).

NCERT Exercise 11.8 — Threshold Frequency from Emitted Speed

1Exercise question

Step-by-step solution

1Maximum kinetic energy: K_max = ½mv² = ½(9.1 × 10⁻³¹)(6.0 × 10⁵)² = 1.64 × 10⁻¹⁹ J = 1.02 eV.
2Photon energy: hν = (6.63 × 10⁻³⁴)(7.21 × 10¹⁴) = 4.78 × 10⁻¹⁹ J = 2.99 eV.
3Work function: φ₀ = hν − K_max = 2.99 − 1.02 = 1.97 eV.
4Threshold frequency: ν₀ = φ₀/h = (1.97 × 1.6 × 10⁻¹⁹)/(6.63 × 10⁻³⁴) = 4.74 × 10¹⁴ Hz.

Final answer

ν₀ = 4.74 × 10¹⁴ Hz.

NCERT Exercise 11.9 — Work Function of an Argon-Laser Emitter

1Exercise question

Step-by-step solution

1Photon energy: hν = hc/λ = (6.63 × 10⁻³⁴ × 3 × 10⁸)/(488 × 10⁻⁹) J = 4.08 × 10⁻¹⁹ J = 2.55 eV.
2Photoelectric equation: φ₀ = hν − eV₀ = 2.55 − 0.38 = 2.17 eV.

Final answer

φ₀ = 2.17 eV.

NCERT Exercise 11.10 — de Broglie Wavelengths of Macroscopic Objects

1Exercise question

Step-by-step solution

1de Broglie wavelength: λ = h/(mv) = (6.63 × 10⁻³⁴)/(mv).
2(a) λ = (6.63 × 10⁻³⁴)/(0.040 × 1000) = 1.66 × 10⁻³⁵ m.
3(b) λ = (6.63 × 10⁻³⁴)/(0.060 × 1.0) = 1.11 × 10⁻³² m.
4(c) λ = (6.63 × 10⁻³⁴)/(1.0 × 10⁻⁹ × 2.2) = 3.01 × 10⁻²⁵ m.

Final answer

(a) 1.66 × 10⁻³⁵ m. (b) 1.11 × 10⁻³² m. (c) 3.01 × 10⁻²⁵ m.

NCERT Exercise 11.11 — Photon and de Broglie Wavelengths

1Exercise question

Step-by-step solution

1A photon of energy E = hν has momentum p = E/c = hν/c.
2Since c = νλ, we get p = h/λ, i.e. λ = h/p.
3The de Broglie wavelength of the same photon is λ_dB = h/p = h/(h/λ) = λ.
4Hence the electromagnetic wavelength and the photon's de Broglie wavelength are identical.

Final answer

λ_EM = h/p = λ_dB, since the photon's momentum is p = h/λ.

Quick Revision

Key formulas at a glance

Memorise these equations — direct application numericals and derivations in CBSE & JEE frequently hinge on these.

Einstein's photoelectric equation

K_{ma x} = h ν - ϕ

Threshold frequency

ν_{0} = \frac{ϕ}{h}

de Broglie wavelength

λ = \frac{h}{m v} = \frac{h}{p}

Energy of a photon

E = h ν = \frac{h c}{λ}

Exam Strategy

How this chapter is asked

High-yield question patterns observed across CBSE boards, JEE Main & Advanced, and NEET.

The stopping potential measures the maximum kinetic energy of the emitted electrons, so the graph of Vs against frequency is a straight line whose slope is h/e and whose intercept gives the work function.
Doubling the intensity of light doubles the photocurrent but leaves the stopping potential unchanged, because intensity sets how many photons arrive, not how energetic each one is.

FAQ

Frequently asked questions

How many questions are in NCERT Class 12 Physics Chapter 11 (Dual Nature of Radiation and Matter)?

There are 11 exercise questions in this chapter, numbered 11.1 to 11.11. Every one is solved step by step on this page in the official NCERT numbering.

Which formulas come up in Dual Nature of Radiation and Matter Class 12 Physics?

The formulas this chapter's questions actually turn on are: Einstein's photoelectric equation, Threshold frequency, de Broglie wavelength, Energy of a photon. They are listed with their expressions in the key formulas section below, and the solved questions show where each one is used.

Is Dual Nature of Radiation and Matter important for JEE Main and NEET?

Important — the photoelectric graph questions and de Broglie wavelength calculations are classic one-mark and two-mark items in JEE Main and NEET.

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