The electrons that produce the picture in a TV set are accelerated by a very large electric force as they pass through a small region in the neck of the picture tube. This region is 1.6 cm in length, and the electrons enter with a speed of 1 × 105 m/s and leave with a speed of 2.5 × 106 m/s.
What is their acceleration over this 1.6 cm length?
Answer in units of m/s2.
How long is the electron in the accelerating region?
Answer in units of s.
d = V*t = 0.016.
1*10^5t = 0.016,
t = 1.6*10^-7 s.
V = Vo + a*t = 2.5*10^6.
1*10^5 + a*1.6*10^-7 = 2.5*10^6.
a =
would a be 1.5*10^13?
My ans. is 1.446*10^13.
average velocity is ... (1E5 + 2.5E6) / 2 = 1.3E6 m/s
transit time is ... t = d / v = 1.6E-2 m / 1.3E6 m/s = 1.2E-8 s
acceleration is ... (2.5E6 - 1E5) / 1.2E-8 = 2.0E14 m/s^2
To find the acceleration of the electrons over the given length, we can use the formula:
acceleration = (final velocity - initial velocity) / time
First, we need to convert the length of the region from centimeters to meters:
1.6 cm = 1.6 × 10^(-2) m
We are given:
Initial velocity (u) = 1 × 10^5 m/s
Final velocity (v) = 2.5 × 10^6 m/s
Length (s) = 1.6 × 10^(-2) m
We need to find the time (t) and acceleration (a).
To find the time, we can use the formula:
s = ut + (1/2)at^2
Since the acceleration is constant, the formula simplifies to:
s = ut + (1/2)at^2
1.6 × 10^(-2) m = (1 × 10^5 m/s)t + (1/2)a(t^2)
To find the time, we can rearrange the equation:
(1/2)a(t^2) + (1 × 10^5 m/s)t - 1.6 × 10^(-2) m = 0
This is a quadratic equation in terms of t. We can solve it using the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = 1/2a, b = 1 × 10^5 m/s, and c = -1.6 × 10^(-2) m.
Substituting these values into the equation and solving for t, we get two possible values for t.
Once we have the value of t, we can substitute it into the acceleration formula to find the acceleration (a):
acceleration = (final velocity - initial velocity) / time
So, to summarize:
1. Convert the length from centimeters to meters: 1.6 cm = 1.6 × 10^(-2) m.
2. Use the quadratic formula to find the time (t) by solving the equation: (1/2)a(t^2) + (1 × 10^5 m/s)t - 1.6 × 10^(-2) m = 0. This will give two possible values for t.
3. Substitute the value of t into the acceleration formula: acceleration = (final velocity - initial velocity) / time.
By following these steps, you can calculate the acceleration and the time for the given problem.