An electron (m = 9 x 10-31 kg) leaves the cathode of a radio tube with zero initial velocity and travels in a straight line to the anode, which is 1 cm away. It reaches the anode with a velocity of 6 x 106 m s-1
i)calculate the acceleration and the force ?
ii) the time taken for the electron to reach the anode ?
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F = m a
a = change in v/ change in t
v = Vi + a t
6*10^6 = 0 + a t = a t
x = Xi + Vi t + (1/2) a t^2
10^-2 meters = 0 + 0 +(1/2) a t^2
so
a t^2 = 2*10^-2
and
a = 6*10^6/t
so
6*10^6/t * t^2 = 2*10^-2
6 t = 2*10^-10 seconds
t = (1/3)10^-10 second
then
a = 18 *10^4 m/s^2
now F = m a
by the way, you could have used the average speed
Vav = 3*10^6
so
t = 10^-2/3*10^6 = (1/3)*10^-8
(not 10^-10)
To solve this problem, we need to use the equations of motion and principles of classical physics.
i) Calculate the acceleration and the force:
We can use the kinematic equation v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement.
Given:
- Initial velocity (u) = 0 m/s (zero initial velocity)
- Final velocity (v) = 6 x 10^6 m/s
- Displacement (s) = 1 cm = 0.01 m
Rearranging the above equation to isolate the acceleration (a), we have:
a = (v^2 - u^2) / (2s)
Substituting the given values, we can calculate the acceleration:
a = (6 x 10^6 m/s)^2 - (0 m/s)^2 / (2 * 0.01 m)
a = 72 x 10^12 m^2/s^2 / 0.02 m
a = 3.6 x 10^15 m^2/s^2
Now, to calculate the force (F), we can use Newton's second law of motion, F = ma, where m is the mass of the electron.
Given:
- Mass of electron (m) = 9 x 10^-31 kg
- Acceleration (a) = 3.6 x 10^15 m^2/s^2
Substituting the values, we can calculate the force:
F = (9 x 10^-31 kg) * (3.6 x 10^15 m^2/s^2)
F = 32.4 x 10^-16 N
F = 3.24 x 10^-15 N
Therefore, the acceleration is 3.6 x 10^15 m^2/s^2 and the force is 3.24 x 10^-15 N.
ii) Calculate the time taken for the electron to reach the anode:
We can use the equation v = u + at, where t is the time taken.
Given:
- Initial velocity (u) = 0 m/s (zero initial velocity)
- Final velocity (v) = 6 x 10^6 m/s
- Acceleration (a) = 3.6 x 10^15 m^2/s^2
Rearranging the equation, we have:
t = (v - u) / a
Substituting the given values, we can calculate the time taken:
t = (6 x 10^6 m/s - 0 m/s) / 3.6 x 10^15 m^2/s^2
t = 1.67 x 10^-9 s
Therefore, the time taken for the electron to reach the anode is approximately 1.67 x 10^-9 seconds.