An electron with an initial speed of 4.56 x 10^5m/s enters the second section of a particle accelerator that is 3.50 cm long. In this section, the electron is accelerated to a speed of 3.25 x 10^6 m/s.
Particle accelerators are known to increase the velocity of a particle to 99.9% of the speed of light (2.997 x10^8m/s). What magnitude of acceleration and time is required for the electron to reach this velocity in the second stage?
First, we need to calculate the acceleration of the electron in the second stage of the particle accelerator:
We can use the equation v^2 = u^2 + 2as, where:
- v is the final velocity (3.25 x 10^6 m/s),
- u is the initial velocity (4.56 x 10^5 m/s),
- a is the acceleration,
- s is the distance (3.50 cm = 0.035 m).
Rearranging the equation to solve for acceleration:
a = (v^2 - u^2) / 2s
a = ((3.25 x 10^6)^2 - (4.56 x 10^5)^2) / (2 * 0.035)
a = (1.05625 x 10^13 - 2.0736 x 10^11) / 0.07
a ≈ 1.486 x 10^13 m/s^2
Next, we can calculate the time required for the electron to reach 99.9% of the speed of light:
Using the equation v = u + at, where:
- v is the final velocity (2.997 x 10^8 m/s),
- u is the initial velocity (3.25 x 10^6 m/s),
- a is the acceleration (1.486 x 10^13 m/s^2),
- t is the time.
Rearranging the equation to solve for time:
t = (v - u) / a
t = (2.997 x 10^8 - 3.25 x 10^6) / 1.486 x 10^13
t ≈ 19.94 × 10^-5 seconds
Therefore, the magnitude of acceleration required for the electron to reach 99.9% of the speed of light in the second stage is approximately 1.486 x 10^13 m/s^2, and the time required is approximately 19.94 × 10^-5 seconds.