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.
Determine the time required for the electron to reach the end of the second stage.
To determine the time required for the electron to reach the end of the second stage, we first need to calculate the acceleration of the electron in the second section using the following equation:
v^2 = u^2 + 2a * s
where:
v = final velocity = 3.25 x 10^6 m/s
u = initial velocity = 4.56 x 10^5 m/s
a = acceleration
s = distance = 3.50 cm = 0.035 m
Rearranging the equation to solve for acceleration, we have:
a = (v^2 - u^2) / (2 * s)
a = ((3.25 x 10^6)^2 - (4.56 x 10^5)^2) / (2 * 0.035)
a = (10.5625 x 10^12 - 2.08336 x 10^11) / 0.07
a = 10.3541 x 10^12 / 0.07
a = 147773 x 10^12 m/s^2
a = 1.47773 x 10^15 m/s^2
Next, we can calculate the time required for the electron to reach the end of the second stage using the following equation:
v = u + at
where:
t = time
v = final velocity = 3.25 x 10^6 m/s
u = initial velocity = 4.56 x 10^5 m/s
a = acceleration = 1.47773 x 10^15 m/s^2
Rearranging the equation to solve for time, we have:
t = (v - u) / a
t = (3.25 x 10^6 - 4.56 x 10^5) / 1.47773 x 10^15
t = 2.794 x 10^6 / 1.47773 x 10^15
t = 1.887 x 10^-9 seconds
Therefore, the time required for the electron to reach the end of the second stage is approximately 1.887 x 10^-9 seconds (or 1.887 nanoseconds).