Should the photon have mass?

Question

I know a few people have asked this question before and I'm trying not to duplicate it, but don't the laws of motion as well as relativity suggest that the photon should have mass? I was thinking that of you put in 0 as the value for mass into $F = ma$ or $E = mc^2$ the values for force and energy always come out as 0, can anyone help me to understand what's going on here?

Answer 1

The fact the photon mass is zero is discussed in some detail in the answers to Why can't photons have a mass? However since you are specifically asking why the equations for the energy and force don't imply the photon has a mass let me address this point.

The equation $E=mc^2$ is a special case of the more general equation:

$$ E^2 = p^2c^2 + m^2c^4 $$

where $p$ is the momentum and $m$ is the rest mass. A photon has zero rest mass and its energy is therefore given by setting $m=0$ in the equation above to get:

$$ E = pc $$

And the momentum of a photon is given by:

$$ p = \frac{h}{\lambda} $$

So we get:

$$ E = \frac{hc}{\lambda} = h\nu $$

which should be familiar as the equation for the energy of a photon. The fact a photon has a non-zero energy does not mean it has to have a mass.

The argument for the force is related. The equation $F=ma$ can also be written as:

$$ F = \frac{dp}{dt} $$

And as before the momentum is given by $p=h/\lambda$. The fact that light can exert a force is because photons possess momentum not because photons have a mass.