In an x-ray tube, electrons are accelerated in a uniform electric field and then

Question

In an x-ray tube, electrons are accelerated in a uniform electric field and then strike a metal target. Suppose an electron starting from rest is accelerated in a uniform electric field directed horizontally and having a magnitude of 2800 N/C . The electric field covers a region of space 12.5 cm wide.

What is the speed of the electron when it strikes the target?

Express your answer with the appropriate units.

Part B

How far does it fall under the influence of gravity during its flight?

Express your answer with the appropriate units.

Explanation / Answer

Hi,

Well, the problem is not telling us the initial position but it doesn't says the maximum heigh so it is irrelevant as we know the electron will strike the metal target. The forces exerted over the electron are the electric force (due to the electric field) and the weight.

The speed when the electron strikes the target can be calculated through velocity, but to do that we must first find the acceleration.

This question is very similar to the problems you find in projectile motion when you are working with kinematics, the difference is that now we have a force in the x axis so the acceleration has two components. Doing the balance of forces in each axis we have the following:

ay = g; as expected

ax = q*E/m = (1.6*10-9 C)*(2800 N/C)/(9.11*10-31 kg) = 4.92*1014 m/s2

So, in order to find the velocity in the x axis we have to use the distance and the acceleration we found.

vx = (2*a*x)1/2 = (2*4.92*1014*0.12)1/2 [m/s] = 1.09*107 [m/s]

For the y axis, we must find the time and then the velocity.

t = vx/ax = 1.09*107/4.92*1014 [s] = 2.22*10-8 [s]

vy = g*t = (9.8)*(2.22*10-8) [m/s] = 2.18*10-7 [m/s]

The magnitude of the velocity vector (the speed) would be:

v = (vx2 + vy2)1/2 [m/s] = ((2.18*10-7)2 + (1.09*107)2)1/2 [m/s] = 1.09*107 [m/s]

Finally, the distance in the y axis (which is the distance it falls due to gravity) can be calculated as follow:

y = vy2/2g = ((2.18*10-7)2/(2*9.8) [m] = 2.42*10-15 [m] = 2.42*10-13 [cm]

To solve this problem you use dynamics, kinematics and some concepts of electricity

Note: as you can see the gravity doesn't effect too much the electron because the electric force is much bigger. So, when you are calculating the velocity of the electron when strikes the metal, you can assume that is the same velocity that you have in the x axis.

I hope it helps

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