Dual Nature of Radiation and Matter: Quick Revision

A brief note on Dual Nature of Radiation and Matter. Read it before a test. Try each question at the end before you open its answer.

The photoelectric effect

• Photocurrent: at a large positive collector voltage it is in proportion to the intensity (the saturation current).
• Stopping potential $V_0$: the collector voltage $-V_0$ at which the current is zero. $K_{\max} = eV_0$.
• Frequency: $V_0$ rises in a straight line with the frequency, and does not depend on the intensity.
• Threshold frequency $\nu_0$: below it no electron leaves, whatever the intensity.
• No delay: emission starts in less than $10^{-9}$ s.
• The wave theory of light fails on all three: the energy, the threshold and the delay.

Einstein's equation

• Photon energy: $E = h\nu = hc/\lambda$, with $hc = 1240$ eV nm.
• Equation: $K_{\max} = h\nu - \phi$, and at the threshold $h\nu_0 = \phi$.
• Graph of $V_0$ against $\nu$: a straight line of slope $h/e = 4.14 \times 10^{-15}$ V s, cutting the $\nu$ axis at $\nu_0$ and the $V_0$ axis at $-\phi/e$.
• Photons each second: $N = P/E$. Photocurrent: $I = \eta Ne$.

Matter waves

• Photon momentum: $p = h/\lambda$. A photon has no rest mass.
• De Broglie wavelength: $\lambda = h/p = h/mv$. At a given energy, $\lambda = h/\sqrt{2mK}$.
• Electrons speeded up by $V$ volt: $\lambda = 1.227/\sqrt{V}$ nm.
• Davisson and Germer: electrons diffract from a nickel crystal. Peaks at $d\sin\varphi = n\lambda$.
• Waves show only when a gap is about as wide as the wavelength.

A worked example

Light of wavelength $413$ nm falls on a metal with a work function of $1.9$ eV. Find the stopping potential.

1. Photon energy. $E = \dfrac{1240\ \text{eV nm}}{413\ \text{nm}} = 3.0$ eV.
2. Largest kinetic energy. $K_{\max} = 3.0\ \text{eV} - 1.9\ \text{eV} = 1.1$ eV.
3. Stopping potential. $V_0 = 1.1$ V.

Practise with new numbers

Check yourself

1. A photocell is lit by light of the same colour but twice the intensity. What changes: the saturation current, the stopping potential, or both?

2. Two photons have wavelengths of $450$ nm and $600$ nm. Which has the larger momentum?

3. Why does a moving cricket ball not show diffraction?