Understanding Specific Charge

The specific charge of a particle is defined as the ratio of its electric charge ($q$) to its mass ($m$). It is a fundamental property used to characterize charged particles.

The formula for specific charge is:

We need to find the ratio of the specific charge of a proton to the specific charge of an alpha (\(\alpha\))-particle.

Specific Charge of a Proton

A proton is a fundamental particle found in the nucleus of an atom.

• Charge of a proton: Let the elementary charge be \(e\). The charge of a proton is \(q_p = +e\).
• Mass of a proton: Let the mass of a proton be \(m_p\).

The specific charge of a proton is:

Specific Charge of an Alpha (\(\alpha\))-Particle

An alpha (\(\alpha\))-particle is the nucleus of a Helium atom (\(^{4}_{2}\text{He}\)). It consists of 2 protons and 2 neutrons.

• Charge of an alpha-particle: Since an alpha-particle has 2 protons, its total charge is \(q_\alpha = 2 \times (+e) = +2e\).
• Mass of an alpha-particle: A neutron has a mass very close to that of a proton. For simplicity in such calculations, the mass of a neutron is often taken as approximately equal to the mass of a proton (\(m_n \approx m_p\)). Therefore, the mass of an alpha-particle (2 protons + 2 neutrons) is approximately \(m_\alpha \approx 2m_p + 2m_n \approx 2m_p + 2m_p = 4m_p\).

The specific charge of an alpha-particle is:

Calculating the Specific Charge Ratio

We need to find the ratio of the specific charge of a proton to that of an alpha-particle, which is \(\frac{(q/m)_p}{(q/m)_\alpha}\).

Ratio = \(\frac{\frac{e}{m_p}}{\frac{e}{2m_p}}\)

To simplify the ratio, we can invert the denominator and multiply:

Ratio = \(\frac{e}{m_p} \times \frac{2m_p}{e}\)

Cancel out the common terms \(e\) and \(m_p\):

Ratio = \(\frac{\cancel{e}}{\cancel{m_p}} \times \frac{2\cancel{m_p}}{\cancel{e}}\)

Ratio = \(2\)

So the ratio of the specific charge of a proton and an alpha-particle is 2:1.

Summary of Specific Charges and Ratio

Ratio \((q/m)_p : (q/m)_\alpha\) = \(\frac{e}{m_p} : \frac{e}{2m_p}\)

Multiply both sides by \(2m_p\) to clear the denominators:

\(\left(\frac{e}{m_p} \times 2m_p\right) : \left(\frac{e}{2m_p} \times 2m_p\right)\)

\(2e : e\)

Divide both sides by \(e\):

\(2 : 1\)

Thus, the ratio is 2:1.

Revision Table: Key Particle Properties

Additional Information: Understanding Particles and Specific Charge

Specific charge is an important concept in physics, especially in fields like mass spectrometry and particle physics. It allows scientists to identify particles and study their behaviour in electric and magnetic fields.

• Charge (q): The electric charge is a fundamental property of matter. It can be positive or negative. The elementary charge \(e\) is the magnitude of the charge of a proton or electron.
• Mass (m): The mass of a particle determines its inertia. The masses of subatomic particles are typically measured in atomic mass units (amu) or MeV/c\(^2\).
• Alpha Particles: Alpha decay is a type of radioactive decay where an atomic nucleus emits an alpha particle. Alpha particles are relatively heavy and carry a significant positive charge, giving them a lower specific charge compared to protons or electrons.
• Applications: The specific charge is used in technologies like cathode ray tubes, mass spectrometers (which separate ions based on their specific charge), and particle accelerators.