AQA · A-Level · Physics
Course version: 7408 · Version from Sept 2016
Students will need to be familiar with the Coulomb equation for the…
Full specification wording
Nuclear radius (A-level only): Students will need to be familiar with the Coulomb equation for the closest approach estimate.
When an alpha particle is fired directly at a nucleus, it slows as electrical potential energy replaces kinetic energy.
Checked against the AQA 7408 specification (opens in a new tab) on .
- Point 3 of 6
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
When an alpha particle is fired directly at a nucleus, it slows as electrical potential energy replaces kinetic energy. At the closest approach, all the initial kinetic energy has become electrostatic potential energy, so equating ½mv² with kQq/r lets you estimate the distance of closest approach and hence an upper limit for the nuclear radius. The Coulomb equation here is F = kQq/r², with k = 1/(4πε₀). You must be able to rearrange energy and force expressions, substitute charges in coulombs (alpha = 2e, nucleus = Ze) and solve for r. This is a method-based estimate: the alpha is treated as a point charge and only electrostatic repulsion is considered, so the value obtained is approximate.
What earns the marks
- State that at closest approach the alpha particle's kinetic energy is fully converted into electrostatic potential energy.
- Use the Coulomb potential energy expression E = kQq/r, with k = 1/(4πε₀), and equate it to ½mv².
- Substitute Q = Ze for the nucleus and q = 2e for the alpha particle, then rearrange to make r the subject.
- Recognise that the calculated distance is an upper limit for the nuclear radius because the alpha particle does not penetrate the nucleus.
- Carry out the arithmetic with consistent SI units, converting MeV to joules where necessary.
Common mistakes and how to put them right
Using the force equation F = kQq/r² instead of the potential energy equation when equating to kinetic energy.
Put it right by using E = kQq/r for the energy balance.
Forgetting that the alpha particle carries a charge of 2e rather than e.
Put it right by substituting q = 2e.
Leaving kinetic energy in MeV while r is required in metres.
Put it right by converting MeV to joules using 1 MeV = 1.6 × 10⁻¹³ J.
Exam tips
- Write the energy balance equation first, then substitute values, so the method is clear even if arithmetic slips.
- Check that every charge is expressed as a multiple of e and converted to coulombs before calculating.
- State explicitly that the result is an estimate of the upper limit of the nuclear radius, not an exact measurement.
Try the lesson
The start of the Nuclear radius (A-level only): Students will need to be familiar with the Coulomb equation for the closest approach estimate. lesson, made from checked teaching notes for this specification point.
Have a go first
An alpha particle of charge 2e is fired directly at a stationary gold nucleus of charge 79e. At the point of closest approach the separation is r. Which expression gives the electric potential energy of the alpha particle at this point?
Listen
When an alpha particle is fired directly at a nucleus, it slows as electrical potential energy replaces kinetic energy. At the closest approach, all the initial kinetic energy has become electrostatic potential energy, so equating ½mv² with kQq/r lets you estimate the distance of closest approach and hence an upper limit for the nuclear radius. The Coulomb equation here is F = kQq/r², with k = 1/(4πε₀). You must be able to rearrange energy and force expressions, substitute charges in coulombs (alpha = 2e, nucleus = Ze) and solve for r. This is a method-based estimate: the alpha is treated as a point charge and only electrostatic repulsion is considered, so the value obtained is approximate. A strong answer earns marks for these points. State that at closest approach the alpha particle's kinetic energy is fully converted into electrostatic potential energy. Use the Coulomb potential energy expression E = kQq/r, with k = 1/(4πε₀), and equate it to ½mv². Substitute Q = Ze for the nucleus and q = 2e for the alpha particle, then rearrange to make r the subject. Recognise that the calculated distance is an upper limit for the nuclear radius because the alpha particle does not penetrate the nucleus. A common slip: Using the force equation F = kQq/r² instead of the potential energy equation when equating to kinetic energy. Instead, using E = kQq/r for the energy balance. Exam tip: Write the energy balance equation first, then substitute values, so the method is clear even if arithmetic slips.
Key words
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Closest approach
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Coulomb potential energy expression
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Upper limit for the nuclear radius
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Alpha particle charge
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See every definition
- Closest approach
- The point at which all the alpha particle's initial kinetic energy has become electrostatic potential energy, allowing an estimate of the distance and hence an upper limit for the nuclear radius.
- Coulomb potential energy expression
- E = kQq/r, with k = 1/(4πε₀), equated to ½mv² to estimate the distance of closest approach.
- Upper limit for the nuclear radius
- The calculated distance of closest approach, which is an upper limit because the alpha particle does not penetrate the nucleus.
- Alpha particle charge
- A charge of 2e, substituted as q in the energy equation alongside Q = Ze for the nucleus.
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