Question

What is the relationship between energy E and momentum p of a photon?
A. $E = pc$
B. $E = \dfrac{p}{c}$
C. $p = Ec$
D. $E = \dfrac{{{p^2}}}{c}$

Answer 1

Use the equations for Energy of a photon $E = h\nu$ and for the momentum of a photon as $p = \dfrac{{h\nu }}{c}$ and compute the relationship between them.

Complete step by step answer:

From the theories of the photoelectric effect and the particle nature of light, we know that
The energy of a photon, E, is given by the following equation,
$\Rightarrow E = h\nu$ .................Equation 1. Where $h$ is the Planck’s constant and $\nu$ is the frequency of the light, the photon of which is under consideration.
The momentum of the photon, p, is given by the following equation,
$\Rightarrow p = \dfrac{{h\nu }}{c}$ .............Equation 2. Where $h$ is the Planck’s constant, $\nu$ is the frequency of the light, the photon of which is under consideration and $c$ is the speed of light.
Substituting the value of E from Equation 1 to Equation 2 we get:
$\Rightarrow p = \dfrac{E}{c}$
Rearranging the terms, we obtain,
$\Rightarrow E = pc$
Which gives us the relationship between energy E and momentum p of a photon.
Therefore the correct option is A.

Note: Alternatively, notice that the question asks about the relationship between energy and momentum of a photon, which means that it is referring to the particle nature of light. Hence, Energy and Momentum should have their regular dimensions.
Therefore, comparing the dimensions of all four options we can determine that only $E = pc$ is correct –
Since, for energy, we have,
$\Rightarrow \dim (E) = M{L^2}{T^{ - 2}} = ML{T^{ - 1}}.L{T^{ - 1}}$ .... Equation 3
And for momentum, we have,
$\Rightarrow \dim (p) = ML{T^{ - 1}}$ ..............................Equation 4
Comparing equation 3 and 4 we get
$\Rightarrow \dim (E) = \dim (p).\dim (c)$ since $\dim (c) = \dim (v) = L{T^{ - 1}}$
Which is only true for option A. Options, B and C, are identical giving the relationship as $p = Ec$ which is incorrect dimensionally, and neither does the last option $E = \dfrac{{{p^2}}}{c}$ satisfy this requirement.