Solution (a)
(i) Features of Nuclear Forces: [1 Mark]
- Nuclear forces are short-range forces (effective only up to a few femtometers, $10^{-15} \text{ m}$).
- They are charge-independent; the force between two protons, two neutrons, or a proton and a neutron is approximately the same.
- (Note: Any two of these are sufficient for full marks).
(ii) Mass-Energy Equivalence: [2 Marks]
Step 1 (Governing Principle): [1 Mark]
Although the total number of nucleons (protons + neutrons) is conserved, the mass of the products is less than the mass of the reactants. This difference is known as the mass defect ($\Delta m$). The energy released is governed by Einstein's mass-energy equivalence relation:
$$E = \Delta m c^2$$Step 2 (Explanation): [1 Mark]
The "missing" mass ($\Delta m$) is converted into binding energy, which is released as kinetic energy of the products or gamma radiation. The conservation of nucleons does not imply the conservation of mass, as mass is a form of energy.
💡 EL10 Concept Insight: Think of it like a "packaging fee"—when nucleons bind together to form a nucleus, they release a tiny bit of their own mass as energy to stay tightly locked together, which is why the final nucleus weighs slightly less than its individual parts.
Solution (b)
(i) Alpha Particle Scattering: [2 Marks]
Step 1 (Graph Description): [1 Mark]
The graph of the number of scattered particles ($N$) versus scattering angle ($\theta$) shows an exponential decay: $N(\theta) \propto \frac{1}{\sin^4(\theta/2)}$.
Step 2 (Conclusions): [1 Mark]
- Most alpha particles pass through the foil undeflected, implying the atom is mostly empty space.
- A very small fraction of particles undergo large-angle scattering, implying the entire positive charge and mass of the atom are concentrated in a tiny central region called the nucleus.
(ii) Quantization in Planetary Motion: [1 Mark]
Step 1 (Reasoning): [1 Mark]
The quantum number $n$ for planetary motion is extremely large. For a planet like Earth, the angular momentum $L$ is of the order of $10^{40} \hbar$. The difference in angular momentum between two successive orbits ($n$ and $n+1$) is $\Delta L = \hbar$, which is negligible compared to the total angular momentum. Thus, the orbits appear continuous rather than discrete.
💡 EL10 Concept Insight: Imagine trying to count grains of sand on a beach; while they are technically individual pieces, there are so many that it just looks like one smooth, continuous pile. Planets are so massive compared to the tiny "quantum steps" that their orbits look like smooth circles instead of distinct, separate rings.