JEE Mains Physics · Formula sheet

Dual Nature of Radiation and Matter formulas

11 formulas, 3 reference tables and 39 common traps for JEE Mains Physics Dual Nature of Radiation and Matter, grouped by subtopic.

Full notes with worked examples

Photon Energy, Momentum and Threshold

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Photon energy and photons per second

Photon energy and photon count

E = h\nu = \frac{hc}{\lambda}, \qquad n = \frac{P}{E} = \frac{P\lambda}{hc}

$n$ photons emitted per second
$P$ power of the source

Photon momentum and the push of light

Photon momentum and force of light

p = \frac{h}{\lambda} = \frac{E}{c}, \qquad F_{\text{absorbed}} = \frac{P}{c}, \quad F_{\text{reflected}} = \frac{2P}{c}

Work function, threshold frequency and threshold wavelength

Threshold

\phi = h\nu_0 = \frac{hc}{\lambda_0}, \qquad \lambda_0\,(\text{nm}) = \frac{1240}{\phi\,(\text{eV})}

Common traps

At equal power, the longer wavelength sends more photons

Each red photon carries less energy than a blue one, so a red lamp must send out more of them to deliver the same power. Photons per second go as λ, not as 1/λ.

A power ratio is not a photon ratio

Power is the photon count times the energy of one photon, P = nhc/λ. When both the count and the wavelength change, write P₁/P₂ = (n₁/λ₁)/(n₂/λ₂) and solve for the unknown.

Use the hc the paper gives

Both 1240 eV nm and 1242 eV nm appear in JEE papers. The options are sometimes close enough that the wrong constant picks the wrong one.

Reflection doubles the push

Absorbed light delivers E/c; light reflected straight back delivers 2E/c, because its momentum reverses. Read the stem for absorbed or reflected before using either.

Energy and momentum rise together

Both E = hc/λ and p = h/λ grow as the wavelength falls. A statement that a shorter wavelength gives a photon less momentum or less energy is false.

Recoil takes energy too

A free body that emits a photon of energy hν loses more than hν of internal energy. The extra is the recoil kinetic energy p²/2M, with p = hν/c.

Brightness does not lower the threshold

A brighter lamp sends more photons of the same energy. If one photon cannot free an electron, a thousand of them cannot either, so a powerful lamp below threshold still ejects nothing.

Angular frequency is 2πν

When a stem gives ω in rad/s, the photon energy is hω/2π. Using hω makes the energy 6.28 times too large.

Longest wavelength, lowest frequency

The threshold is the LONGEST wavelength that ejects electrons and the LOWEST frequency that does. Shorter wavelengths and higher frequencies work; longer and lower do not.

Photoelectric Laws and Graphs

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What frequency and intensity each control in the photoelectric effect

Quantity	Raise the frequency (above threshold)	Raise the intensity (same frequency)
Maximum kinetic energy	Rises linearly: hν − φ	No change
Stopping potential	Rises linearly: (hν − φ)/e	No change Moving the lamp farther away dims it; the stopping potential stays the same.
Saturation current	Set by photons per second, not by their energy	Rises in proportion
Whether emission happens	Starts once ν passes ν₀	Never below ν₀, however bright
Delay before emission	None: emission is instant	None: emission is instant
Photons per second at fixed intensity	Falls, as n = IA/hν	Rises in proportion

Frequency sets the energy of each electron; intensity sets the number of electrons.

Reading photoelectric graphs

Graph	Shape	Slope	Intercepts, and what shifts the graph
Stopping potential against frequency	Straight line from ν₀ upward	h/e, the same for every metal	Meets the ν-axis at ν₀ and, extended, the V₀-axis at −φ/e; a larger φ shifts it right, parallel
Maximum kinetic energy against frequency	Straight line from ν₀ upward	h, the same for every metal	Meets the ν-axis at ν₀ and, extended, the K-axis at −φ
Photocurrent against collector voltage, two intensities, one frequency	Rises, then flattens at a saturation current	Flat once saturated	Both cut off at the same −V₀; the brighter light saturates higher Same cut-off voltage means same frequency.
Photocurrent against collector voltage, two frequencies, one intensity	Rises, then flattens at a saturation current	Flat once saturated	The higher frequency cuts off at the more negative voltage; the saturation level is the same
Photocurrent against intensity	Straight line through the origin	Constant for one metal and one frequency	Stays at zero below threshold at any intensity
Stopping potential against intensity	Horizontal line	Zero	Its height is set by the frequency

Every line here comes from eV₀ = hν − φ.

Common traps

Brighter light, same stopping potential

Raising the intensity sends more photons of the same energy. More electrons leave, so the current rises, but the fastest of them is no faster, so the stopping potential is unchanged.

Doubling the frequency more than doubles the kinetic energy

K = hν − φ. At 2ν it becomes 2hν − φ = 2K + φ, which is more than 2K. The stopping potential likewise more than doubles.

The collector is made negative, not the emitter

Electrons are stopped by making the collecting plate negative with respect to the emitting surface. Making the emitter itself negative pushes electrons away from it and helps the current.

The slope is the same for every metal

The slope of the stopping-potential line is h/e, built only from constants of nature. A larger work function moves the line to the right but does not tilt it.

The potential-axis intercept is negative

Extended back to ν = 0, the stopping-potential line meets the potential axis at −φ/e, below the origin. Its size gives the work function in eV.

Lowest threshold, fastest electrons

For the same light, the metal whose line starts at the lowest frequency has the smallest work function, so it gives out the most energetic electrons.

Einstein's Equation and Stopping Potential

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Einstein's photoelectric equation for one light

Einstein's photoelectric equation

h\nu = \frac{hc}{\lambda} = \phi + K_{\max}, \qquad K_{\max} = eV_0

Two wavelengths on the same metal

Eliminating the work function

e(V_1 - V_2) = hc\left(\frac{1}{\lambda_1} - \frac{1}{\lambda_2}\right)

Comparing the maximum speeds of photoelectrons

Speed of the fastest photoelectron

\tfrac{1}{2}mv_{\max}^{2} = h(\nu - \nu_0), \qquad \frac{v_1}{v_2} = \sqrt{\frac{\nu_1 - \nu_0}{\nu_2 - \nu_0}}

Common traps

Divide ω by 2π

A light wave written as sin(ωt) gives the angular frequency. The photon energy is hω/2π. Using hω makes it 6.28 times too large.

Two frequencies, use the higher

When the field holds two sine terms, the maximum kinetic energy comes from the higher frequency. The lower one gives slower electrons and is never the answer to a maximum.

Volts and electron-volts are the same number

A stopping potential of 2 V means the fastest electron had 2 eV. Working in eV needs no factor of e; converting to joules and back is where 1.6 × 10⁻¹⁹ errors come from.

Subtract, do not divide

Dividing the two Einstein equations leaves the work function in both numerator and denominator. Subtracting them removes it in one step.

A longer wavelength cannot give a larger stopping potential

Longer wavelength means less energy per photon, so less left over after the work function. A stem that gives the longer wavelength the larger stopping potential describes something impossible.

Doubling K is not halving λ

To double the kinetic energy the photon must carry φ + 2K. Halving the wavelength doubles the photon energy to 2φ + 2K, which gives more than 2K.

Subtract the work function before the square root

Speed goes as the square root of the leftover energy, k − 1 times φ, not of the photon energy kφ. Photons of 2φ and 8φ give speeds in the ratio 1 : √7, not 1 : 2.

A speed ratio is not an energy ratio

Kinetic energy goes as v². If the energies are in the ratio 1 : 4, the speeds are 1 : 2. Stopping at the energy ratio picks a wrong option.

de Broglie Wavelength of a Particle

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de Broglie wavelength and how it scales

de Broglie wavelength

\lambda = \frac{h}{p} = \frac{h}{\sqrt{2mK}} = \frac{h}{\sqrt{2mqV}}, \qquad \lambda_{e} = \frac{1.227}{\sqrt{V}}\ \text{nm}

de Broglie wavelength of an electron in an electric or magnetic field

Wavelength in a field

\lambda(t) = \frac{h}{m\,|\vec v(t)|}, \qquad \vec F = -e\vec E \ \text{(electron)}

Evidence for matter waves

Observation or device	What it shows	Key relation
Davisson–Germer experiment	Electrons scattered from a nickel crystal give a diffraction peak, so electrons behave as waves	At 54 V the peak is at 50°; the measured λ ≈ 0.165 nm matches h/p
Electron diffraction and interference	A beam of electrons spreads and makes fringes, like light	Fringe spacing grows with λ = h/p
Electron microscope	Resolves far finer detail than an optical microscope	Electron λ is a fraction of a nanometre, against 400–700 nm for light
Heavier particle at the same speed	Shorter wavelength, finer detail	λ = h/mv, so at equal speed λ ∝ 1/m
Photoelectric effect	Light arrives as particles, photons	E = hν per photon
Heisenberg uncertainty principle	Position and momentum cannot both be known exactly	Δx Δp ≥ h/4π
Everyday objects	No visible wave effects	For a large mass, h/mv is far smaller than any gap or slit

Wave behaviour is shown by diffraction and interference; particle behaviour by one-at-a-time energy exchange.

Common traps

λ goes as 1/√K, not 1/K

Momentum is √(2mK). Four times the kinetic energy halves the wavelength; it does not quarter it. The same holds for the accelerating voltage.

Use the particle's own charge

A charge q accelerated through V gains qV. An alpha particle has charge 2e, so through V it gains 2eV, not eV.

Extra energy is the change, not the new total

When a question asks how much energy must be ADDED, find the new kinetic energy and subtract the old one. The new total is a distractor.

The force on an electron is opposite to the field

For an electron, F = −eE. A field pointing against the motion pushes the electron forward and shortens its wavelength; a field along the motion slows it and lengthens the wavelength.

A sideways electric field still changes λ

A field at right angles to the motion adds a sideways velocity. The speed grows as √(v₀² + (eEt/m)²), so the wavelength falls, even though the original velocity component is unchanged.

A magnetic field never changes λ

The magnetic force is always at right angles to the velocity, so it does no work and the speed stays the same. The wavelength is unchanged at every instant.

Matter waves are not electromagnetic

An electron's wave is not light and does not need a charge. A neutron or a whole atom has a de Broglie wavelength too.

Diffraction means wave, photoelectric means particle

Electron diffraction is evidence for the wave nature of matter. The photoelectric effect is evidence for the particle nature of light. Swapping the two is a common wrong statement.

Comparing de Broglie Wavelengths

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Comparing particles at given kinetic energies or voltages

Ratio of wavelengths

\frac{\lambda_1}{\lambda_2} = \sqrt{\frac{m_2K_2}{m_1K_1}} = \sqrt{\frac{m_2q_2V_2}{m_1q_1V_1}}

Same de Broglie wavelength means same momentum

Momentum from wavelength

p = \frac{h}{\lambda}, \qquad K = \frac{p^{2}}{2m} = \frac{h^{2}}{2m\lambda^{2}}

Comparing a particle with a photon

Photon against particle

\lambda_{\text{photon}} = \frac{hc}{E}, \qquad \lambda_{\text{particle}} = \frac{h}{\sqrt{2mK}}

Common traps

The same voltage is not the same energy

Through the same V, a particle of charge q gains qV. An alpha particle, charge 2e, gains twice what a proton gains, so its mass AND its charge both enter the ratio.

Read the ratio the right way round

λ₁/λ₂ has the second particle's mass and energy on top. Inverting it gives the reciprocal, which is always one of the options.

At the same kinetic energy, heavier means shorter

λ ∝ 1/√m when K is fixed, so an alpha particle has the shortest wavelength and an electron the longest. A proton and a neutron come out nearly equal.

Equal wavelengths do not mean equal energies

The same λ means the same momentum. Kinetic energy is p²/2m, so the lighter particle carries more energy, in the inverse ratio of the masses.

Pieces from rest share one wavelength

A body at rest that splits sends its pieces off with equal and opposite momenta, so their wavelengths are equal. The masses do not enter.

At the same speed, λ goes as 1/m, not 1/√m

The square root belongs to the same-energy comparison. At equal speed, p = mv, so the wavelength ratio is the inverse mass ratio itself.

A photon's energy is pc, not p²/2m

Light has no rest mass, so p²/2m means nothing for it. Use E = pc, and λ = hc/E, for the photon; keep p²/2m for the particle.

Same wavelength, same momentum, different energies

When a particle and a photon share a wavelength, they share a momentum, but the slow particle's kinetic energy is much smaller than the photon's energy.

Dual Nature of Radiation and Matter formulas

Photon Energy, Momentum and Threshold

Photon energy and photons per second

Photon momentum and the push of light

Work function, threshold frequency and threshold wavelength

Common traps

Photoelectric Laws and Graphs

What frequency and intensity each control in the photoelectric effect

Reading photoelectric graphs

Common traps

Einstein's Equation and Stopping Potential

Einstein's photoelectric equation for one light

Two wavelengths on the same metal

Comparing the maximum speeds of photoelectrons

Common traps

de Broglie Wavelength of a Particle

de Broglie wavelength and how it scales

de Broglie wavelength of an electron in an electric or magnetic field

Evidence for matter waves

Common traps

Comparing de Broglie Wavelengths

Comparing particles at given kinetic energies or voltages

Same de Broglie wavelength means same momentum

Comparing a particle with a photon

Common traps

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