calculate the energy released when an electron annihilates a positron?

To calculate the energy released when an electron annihilates a positron, we can use Einstein's famous equation, E=mc². This equation tells us that energy (E) is equal to mass (m) times the speed of light (c) squared. In the case of electron-positron annihilation, the mass of both particles is converted into energy, typically in the form of gamma-ray photons.

Understanding the Mass of Electron and Positron

Both the electron and the positron have the same mass, which is approximately 9.11 x 10^-31 kilograms. When they annihilate each other, their combined mass is converted into energy.

Calculating the Total Mass

The total mass involved in the annihilation is the sum of the masses of the electron and the positron:

• Mass of electron (m_e) = 9.11 x 10^-31 kg
• Mass of positron (m_p) = 9.11 x 10^-31 kg
• Total mass (m_total) = m_e + m_p = 2 x 9.11 x 10^-31 kg = 1.82 x 10^-30 kg

Applying Einstein's Equation

Now that we have the total mass, we can substitute it into Einstein's equation to find the energy released:

E = m_total x c²

Here, the speed of light (c) is approximately 3.00 x 10⁸ meters per second. Plugging in the values:

• E = (1.82 x 10^-30 kg) x (3.00 x 10⁸ m/s)²
• E = (1.82 x 10^-30 kg) x (9.00 x 10¹⁶ m²/s²)
• E ≈ 1.64 x 10^-13 joules

Converting Joules to Electronvolts

In particle physics, it's often more convenient to express energy in electronvolts (eV). To convert joules to electronvolts, we use the conversion factor: 1 eV = 1.6 x 10^-19 joules.

So, to convert our result:

• E ≈ (1.64 x 10^-13 joules) / (1.6 x 10^-19 joules/eV)
• E ≈ 1.025 x 10⁶ eV

Final Energy Released

The energy released during the annihilation of an electron and a positron is approximately 1.02 MeV (mega-electronvolts), which is a significant amount of energy for such a small mass. This energy is typically emitted in the form of two gamma-ray photons, each carrying about 0.511 MeV of energy.

This process not only illustrates the principles of mass-energy equivalence but also plays a crucial role in various applications, including positron emission tomography (PET) in medical imaging. Understanding these concepts helps us appreciate the fundamental interactions in particle physics and their implications in technology and medicine.