Answer the following questions:

$(a)$ guarks inside protons and neutrons are thought to carry fractional charges $[(+2 / 3) e ; (-1 / 3) e] .$ Why do they not show up in Millikan's oil-drop experiment?

$(b)$ What is so special about the combination $e / m ?$ Why do we not simply talk of $e$ and $m$ separately?

$(c)$ Why should gases be insulators at ordinary pressures and start conducting at very low pressures?

$(d)$ Every metal has a definite work function. Why do all photoelectrons not come out with the same energy if incident radiation is monochromatic? Why is there an energy distribution of photoelectrons?

$(e)$ The energy and momentum of an electron are related to the frequency and wavelength of the assoctated matter wave by the relations:

$E=h v, p=\frac{h}{\lambda}$

But while the value of $\lambda$ is physically significant, the value of $v$ (and therefore, the value of the phase speed $v \lambda$ ) has no physical significance. Why?

$(a)$ Quarks inside protons and neutrons carry fractional charges. This is because nuclear force increases extremely if they are pulled apart. Therefore, fractional charges may exist in nature;

observable charges are still the integral multiple of an electrical charge.

$(b)$ The basic relations for electric field and magnetic field are $\left(e V=\frac{1}{2} m v^{2}\right) \text { and }\left(e B v=\frac{m v^{2}}{r}\right)$ respectively

These relations include e (electric charge), v(velocity), $m$ (mass), V (potential), r(radius), and B (magnetic field). These relations give the value of velocity of an electron as $(v=\sqrt{2 V\left(\frac{e}{m}\right)})$ and $\left(v=B r\left(\frac{e}{m}\right)\right)$

It can be observed from these relations that the dynamics of an electron is determined not by $e$ and $m$ separately, but by the ratio $e/m.$

$(c)$ At atmospheric pressure, the ions of gases have no chance of reaching their respective electrons because of collision and recombination with other gas molecules. Hence, gases are insulators at atmospheric pressure. At low pressures, ions have a chance of reaching their respective electrodes and constitute a current. Hence, they conduct electricity at these pressures.

$(d)$ The work function of a metal is the minimum energy required for a conduction electron to get out of the metal surface. All the electrons in an atom do not have the same energy level. When a ray having some photon energy is incident on a metal surface, the electrons come out from different levels with different energies. Hence, these emitted electrons show different energy distributions.

$(e)$ The absolute value of energy of a particle is arbitrary within the additive constant. Hence, wavelength $(\lambda)$ is significant, but the frequency (v) associated with an electron has no direct physical significance.

Therefore, the product $v \lambda$ (phase speed) has no physical significance. Group speed is given as

$v_{G}=\frac{d v}{d k}$

$=\frac{d v}{d\left(\frac{1}{\lambda}\right)}=\frac{d E}{d P}=\frac{d\left(\frac{p^{2}}{2 m}\right)}{d P}=\frac{p}{m}$

This quantity has a physical meaning.