An electron and a positron each have a mass of 9.11*10^-31 kg. They collide and both vanish, with only electromagnetic radiation appearing after the collision. If each particle is moving at a speed of 0.55c relative to the laboratory before the collision, determine the energy of the electromagnetic radiation.
Try E = m c^2 plus the kinetic energy (1/2) mv^2.
To determine the energy of the electromagnetic radiation produced by the collision of an electron and a positron, we can use the principle of conservation of energy and momentum. Here's how you can solve this problem step-by-step:
Step 1: Calculate the initial total energy.
The total energy (E1) of each particle before the collision can be calculated using the relativistic energy equation:
E1 = γm0c²
where γ is the Lorentz factor and is given by γ = 1/√(1 - v²/c²), m0 is the rest mass of the particle, and c is the speed of light.
Given that the particles are moving at a speed of 0.55c relative to the laboratory, we can substitute the values into the equation:
γ = 1/√(1 - (0.55c)²)
= 1/√(1 - 0.55²)
≈ 1.33
E1 = γm0c²
= (1.33)(9.11 × 10^(-31) kg)(3.00 × 10^8 m/s)²
≈ 2.48 × 10^(-12) joules
Step 2: Calculate the total energy after the collision.
Since the electron and positron annihilate each other, their combined energy is transformed into electromagnetic radiation. The total energy after the collision (E2) is then equal to the energy of the radiation.
E2 = E1
E2 ≈ 2.48 × 10^(-12) joules
Therefore, the energy of the electromagnetic radiation produced by the collision is approximately 2.48 × 10^(-12) joules.