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 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 cm length?
Answer in units of m/s^2
How long is the electron in the accelerating
region?
Answer in units of s

To find the acceleration of the electrons as they pass through the small region, we can use the equations of motion. The acceleration (a) of an object can be calculated using the following equation:

a = (vf - vi) / t

Where:
a = acceleration
vf = final velocity
vi = initial velocity
t = time taken

In this case, the initial velocity (vi) is 1 × 10^5 m/s, and the final velocity (vf) is 2.5 × 10^6 m/s. We need to determine the time taken (t) for the electrons to travel through the 1 cm length.

To find the time, we can use the equation:

distance = velocity × time

Since the distance is 1 cm, we need to convert it to meters by dividing by 100:

distance = 1 cm / 100 = 0.01 m

Rearranging the equation gives us:

time = distance / velocity

Plugging in the values:

time = 0.01 m / (1 × 10^5 m/s) = 1 × 10^-7 s

Now we have the time (t), we can substitute it back into the equation for acceleration:

a = (2.5 × 10^6 m/s - 1 × 10^5 m/s) / (1 × 10^-7 s)

Simplifying:

a = 2.4 × 10^7 m/s / 1 × 10^-7 s

a = 2.4 × 10^7 m/s × 10^7 s

a = 2.4 × 10^14 m/s^2

Thus, the acceleration of the electrons over this 1 cm length is 2.4 × 10^14 m/s^2.

To calculate the time it takes for the electron to travel through the accelerating region, we can use the equation we derived earlier:

time = distance / velocity

The distance is still 0.01 m, and the velocity of the electron as it leaves the region is 2.5 × 10^6 m/s.

Plugging in the values:

time = 0.01 m / (2.5 × 10^6 m/s)

Simplifying:

time = 4 × 10^-9 s

Thus, the electron spends 4 × 10^-9 seconds in the accelerating region.