Understanding the e/m Ratio for Subatomic Particles

The charge-to-mass ratio, denoted as ${e}/{m}$, is a fundamental property of charged particles. It represents the amount of electric charge a particle carries relative to its mass. This value is crucial in various physics experiments, such as determining the trajectory of particles in electric and magnetic fields. We need to find the increasing order of the ${e}/{m}$ ratio for neutron (n), proton (p), electron (e), and alpha particle (α).

Calculating e/m for Each Particle

Let's consider the approximate charge and mass of each particle:

• Electron (e): Charge $q_e = -e$ (where $e = 1.602 \times 10^{-19}$ C), Mass $m_e = 9.109 \times 10^{-31}$ kg. The magnitude of the ${e}/{m}$ ratio is $|{q_e}/{m_e}| = {e}/{m_e}$.
• Proton (p): Charge $q_p = +e$, Mass $m_p = 1.672 \times 10^{-27}$ kg. The ${e}/{m}$ ratio is ${q_p}/{m_p} = {e}/{m_p}$.
• Neutron (n): Charge $q_n = 0$, Mass $m_n = 1.675 \times 10^{-27}$ kg (approximately equal to proton mass). The ${e}/{m}$ ratio is ${q_n}/{m_n} = {0}/{m_n} = 0$.
• Alpha particle (α): An alpha particle is a helium nucleus (${^4_2He}$), consisting of 2 protons and 2 neutrons. Charge $q_\alpha = +2e$, Mass $m_\alpha \approx 2m_p + 2m_n \approx 4m_p$ (more precisely, $m_\alpha = 6.644 \times 10^{-27}$ kg). The ${e}/{m}$ ratio is ${q_\alpha}/{m_\alpha} = {2e}/{m_\alpha} \approx {2e}/{4m_p} = {e}/{2m_p}$.

Comparing the e/m Ratios

Now let's compare the ${e}/{m}$ values:

• For neutron (n), the ${e}/{m}$ ratio is 0.
• For proton (p), the ${e}/{m}$ ratio is ${e}/{m_p}$.
• For alpha particle (α), the ${e}/{m}$ ratio is approximately ${e}/{2m_p}$, which is half the value for a proton.
• For electron (e), the magnitude of the ${e}/{m}$ ratio is ${e}/{m_e}$. Since the electron mass ($m_e$) is much smaller than the proton mass ($m_p$) ($m_e \approx {m_p}/{1836}$), the ${e}/{m}$ ratio for the electron is much larger than for the proton or alpha particle: ${e}/{m_e} \approx 1836 \times ({e}/{m_p})$.

Increasing Order of e/m Ratio

Comparing the approximate ratios:

• Neutron (n): 0
• Alpha particle (α): $\approx 0.5 \times ({e}/{m_p})$
• Proton (p): $\approx 1 \times ({e}/{m_p})$
• Electron (e): $\approx 1836 \times ({e}/{m_p})$

Arranging these in increasing order of the ${e}/{m}$ ratio:

Neutron (n) < Alpha particle (α) < Proton (p) < Electron (e)

This gives the order: n, α, p, e. The neutron has the smallest ${e}/{m}$ ratio (zero) because it is neutral. The alpha particle has a lower ${e}/{m}$ ratio than the proton because it has a higher mass relative to its charge (2e charge, ~4m_p mass vs. e charge, m_p mass for proton). The electron has the largest ${e}/{m}$ ratio due to its extremely small mass compared to its charge.