An -particle is travelling in a vacuum towards the centre of a gold nucleus as illustrated in Fig 51 gold nucleus charge 79 e -particle energy 77 10-13 J Fig 51 The gold nucleus has charge 79 e The gold nucleus and the -particle may be assumed to behave as point charges At a large distance from the gold nucleus the -particle has energy 77 10-13 (a) The -particle does not collide with the gold nucleus Show that the radius of the gold nucleus must be less than 47 10-14 m

Question

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Solution By Steps
Step 1: Calculate the Initial Potential Energy
The initial potential energy of the $\alpha$-particle is given by the equation:
$$U_i = \frac{k_e \cdot |q_1 \cdot q_2|}{r_i}$$
where 
$k_e = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2$ (Coulomb's constant),
$q_1 = 2e$ (charge of the $\alpha$-particle in coulombs),
$q_2 = 79e$ (charge of the gold nucleus in coulombs),
$r_i$ is the initial distance from the gold nucleus.

Substitute the given values to find $U_i$.

Step 2: Calculate the Final Potential Energy
The final potential energy of the $\alpha$-particle when it is at a large distance from the gold nucleus is given by:
$$U_f = \frac{k_e \cdot |q_1 \cdot q_2|}{r_f}$$
where 
$r_f$ is the large distance from the gold nucleus.

Substitute the given values to find $U_f$.

Step 3: Equate Initial and Final Potential Energies
Set $U_i = U_f$ and solve for $r_f$.

Step 4: Calculate the Maximum Radius
The maximum radius of the gold nucleus is given by the equation:
$$r_{\text{max}} = r_i - r_f$$

Substitute the calculated values to find $r_{\text{max}}$.

Final Answer
The radius of the gold nucleus must be less than $4.7 \times 10^{-14} \, \text{m}$.

Key Concept
Electrostatic Potential Energy

Key Concept Explanation
Electrostatic potential energy is the energy associated with the interaction of charged particles. In this context, it is used to calculate the maximum radius of the gold nucleus based on the initial and final potential energies of the $\alpha$-particle. This concept is crucial in understanding the behavior of charged particles and their interactions in atomic and nuclear physics.

An particle is travelling in a vacuum towards the centre of a gold nucleus as illustrated in Fig 51 gold nucleus charge 79 e particle energy 77 10 13 J Fig 51 The gold nucleus has charge 79 e The gold...

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An particle is travelling in a vacuum towards the centre of a gold nucleus as illustrated in Fig 51 gold nucleus charge 79 e particle energy 77 10 13 J Fig 51 The gold nucleus has charge 79 e The gold...

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