The increasing order (lowest first) for the values of e/m (charge/mass) for

The correct option is "n, α, p, e".

To determine the increasing order of the charge-to-mass ratio ($\frac{e}{m}$) for the given subatomic particles (neutron, alpha particle, proton, and electron), we can evaluate the charge ($e$) and mass ($m$) for each in terms of fundamental constants: the elementary charge $q$ and the mass of a proton $m_p$.

1. Neutron (n):
A neutron is an electrically neutral particle.
Charge, $e_n=0$
Mass, $m_n\approx 1m_p$
Therefore, its charge-to-mass ratio is:
${\left(\frac{e}{m}\right)}_n=\frac{0}{m_p}=0$

2. Alpha particle (α):
An alpha particle is a helium nucleus (${\mathrm{He}}^{2+}$), containing 2 protons and 2 neutrons.
Charge, $e_{\alpha }=+2q$
Mass, $m_{\alpha }\approx 4m_p$
Therefore, its charge-to-mass ratio is:
${\left(\frac{e}{m}\right)}_{\alpha }=\frac{2q}{4m_p}=0.5\left(\frac{q}{m_p}\right)$

3. Proton (p):
A proton has a unit positive charge and unit mass.
Charge, $e_p=+1q$
Mass, $m_p=1m_p$
Therefore, its charge-to-mass ratio is:
${\left(\frac{e}{m}\right)}_p=\frac{q}{m_p}=1\left(\frac{q}{m_p}\right)$

4. Electron (e):
An electron carries a unit negative charge, but its mass is extremely small compared to a proton (approximately 1836 times lighter than a proton).
Charge magnitude, $e_e=1q$
Mass, $m_e\approx \frac{1}{1836}{m}_p$
Therefore, its charge-to-mass ratio (in magnitude) is:
${\left(\frac{e}{m}\right)}_e=\frac{q}{m_p/1836}$

Comparing the calculated values, we have:
$0<0.5\left(\frac{q}{m_p}\right)<1\left(\frac{q}{m_p}\right)<1836\left(\frac{q}{m_p}\right)$

Thus, the increasing order of $\frac{e}{m}$ (lowest first) is:
n < α < p < e