Read the following passage and answer the accompanying questions - Leaving Cert Physics - Question 10 - 2007

The kinetic energy gained by the proton when accelerated through a potential difference is given by the equation:

$K.E. = qV$

Where:

• $q$ = charge of the proton = $1.602 imes 10^{-19} ext{ C}$
• $V$ = potential difference = $700,000 ext{ V}$

Plugging in the values:

$K.E. = 1.602 imes 10^{-19} imes 700,000 = 1.1214 imes 10^{-10} ext{ J}$

The kinetic energy can also be expressed as: $K.E. = \frac{1}{2}mv^2$

Thus,

$v = \sqrt{\frac{2 \cdot K.E.}{m}}$

Where $m$ is the mass of the proton = $1.6726 imes 10^{-27} ext{ kg}$. Therefore:

$v = \sqrt{\frac{2 \cdot 1.1214 imes 10^{-10}}{1.6726 imes 10^{-27}}} = 1.16 imes 10^{7} ext{ m/s}$

The nuclear equation for the disintegration of a lithium-6 nucleus when bombarded with a proton can be written as follows:

$^6_3Li + ^1_1p \rightarrow ^4_2He + ^2_1H$

In this reaction:

• $^6_3Li$ is the lithium nucleus.
• $^1_1p$ is the incoming proton.
• $^4_2He$ is the helium nucleus produced.
• $^2_1H$ is the hydrogen-2 (deuterium) produced.

To calculate the energy released, we need to find the mass defect and convert it to energy using Einstein's equation:

$E = mc^2$

The masses involved are:

• Mass of lithium nucleus: $1.1646 imes 10^{-26} ext{ kg}$
• Mass of proton: $1.6726 imes 10^{-27} ext{ kg}$
• Mass of helium nucleus: $6.6434 imes 10^{-27} ext{ kg}$
• Mass of deuterium: $3.000 imes 10^{-27} ext{ kg}$

The mass before the reaction (reactants): $m_{reactants} = 1.1646 imes 10^{-26} + 1.6726 imes 10^{-27} = 1.33186 imes 10^{-26} ext{ kg}$

The mass after the reaction (products): $m_{products} = 6.6434 imes 10^{-27} + 3.000 imes 10^{-27} = 9.6434 imes 10^{-27} ext{ kg}$

Then, the mass defect is: $\Delta m = m_{reactants} - m_{products} = 1.33186 imes 10^{-26} - 9.6434 imes 10^{-27} ext{ kg} = 3.6752 imes 10^{-27} ext{ kg}$

Now we convert this mass defect to energy: $E = \Delta m imes c^2 = 3.6752 imes 10^{-27} imes (2.9979 imes 10^{8})^2$

Calculating: $E ≈ 3.30 imes 10^{-10} ext{ J}$

Electrons and positrons share several properties but also have key differences:

• Charge: Electrons are negatively charged (-1e), while positrons are positively charged (+1e).
• Mass: Both have the same mass, approximately $9.11 imes 10^{-31} ext{ kg}$.
• Spin: Both particles have a spin of $1/2$, making them fermions.
• Interaction with matter: Electrons interact with negatively charged matter, while positrons can annihilate upon interaction with electrons, producing photons.

In summary, while they possess equal mass and spin, the critical difference lies in their electric charge.

When an electron meets a positron, they can annihilate each other, resulting in the production of gamma-ray photons. This annihilation process typically results in the release of two photons, each carrying energy corresponding to the mass of the electron and positron, and can be described by the equation: $E = 2mc^2$ where $m$ is the mass of the electron (or positron). This process conserves energy and momentum.

Fermi's theory of radioactive decay proposed that during beta decay, a neutron decays into a proton, an electron, and an antineutrino. The inclusion of the antineutrino provided a mechanism for conserving both energy and momentum in the decay process. The antineutrino carries away any excess momentum, ensuring that the overall conservation laws are upheld. This was a significant advancement in understanding weak interactions and decay processes in particle physics.