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Difficulty: MediumNatural Radioactivity and Radiation Emissions

When a mixed beam containing α\alpha-particles, β\beta-particles, and γ\gamma-rays passes through a uniform electric field, the β\beta-particles undergo a significantly greater lateral deflection than the α\alpha-particles. Which of the following best explains this observation?

  1. The charge-to-mass ratio of a β\beta-particle is significantly greater than that of an α\alpha-particle.Answer
  2. B
    The magnitude of the electric force acting on a β\beta-particle is greater than that acting on an α\alpha-particle.
  3. C
    β\beta-particles travel at a much lower velocity than α\alpha-particles, allowing the electric field more time to act on them.
  4. D
    α\alpha-particles possess no net electric charge and are therefore unaffected by the electric field.

Answer

The charge-to-mass ratio of a β\beta-particle is significantly greater than that of an α\alpha-particle.
In a uniform electric field EE, a charged particle experiences an acceleration given by a=qEma = \frac{qE}{m}. The degree of deflection is directly proportional to this acceleration, and hence to the charge-to-mass ratio (qm\frac{q}{m}). Even though an α\alpha-particle carries twice the magnitude of charge of a β\beta-particle (2e2e compared to ee), an α\alpha-particle is about 73007300 times more massive than a β\beta-particle. Consequently, the charge-to-mass ratio of the β\beta-particle is thousands of times larger, causing it to undergo a significantly greater deflection in the field.

Step-by-Step Solution

1
Identify the relationship between electric field deflection and particle properties.
Deflection in a transverse electric field is proportional to acceleration a=qEma = \frac{qE}{m}, meaning deflection depends on the specific charge (charge-to-mass ratio, qm\frac{q}{m}).
Newton's second law (F=maF = ma) combined with Electrostatic force (F=qEF = qE) gives a=qEma = \frac{qE}{m}.
2
Compare the charges and masses of α\alpha-particles and β\beta-particles.
For an α\alpha-particle (24He2+^{4}_{2}\text{He}^{2+}), qα=+2eq_{\alpha} = +2e and mα≈7300mem_{\alpha} \approx 7300 m_e. For a β\beta-particle (−10e^{0}_{-1}\text{e}), qβ=−eq_{\beta} = -e and mβ=mem_{\beta} = m_e.
An alpha particle consists of 2 protons and 2 neutrons, making it much more massive than a single electron (beta particle).
3
Calculate and compare the charge-to-mass ratios.
∣qβmβ∣=eme\left|\frac{q_{\beta}}{m_{\beta}}\right| = \frac{e}{m_e}, while ∣qαmα∣≈2e7300me=e3650me\left|\frac{q_{\alpha}}{m_{\alpha}}\right| \approx \frac{2e}{7300 m_e} = \frac{e}{3650 m_e}. Thus, ∣qβmβ∣≈3650×∣qαmα∣\left|\frac{q_{\beta}}{m_{\beta}}\right| \approx 3650 \times \left|\frac{q_{\alpha}}{m_{\alpha}}\right|.
The vastly smaller mass of the beta particle dominates the ratio, resulting in a much larger charge-to-mass ratio and therefore a greater deflection.

Key Concept

Deflection of nuclear emissions in electric fields is governed by their charge-to-mass ratio (qm\frac{q}{m}).
Estimated Time:1m 15s
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Topics

Natural Radioactivity and Radiation Emissions

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