- #1
Reprisal35
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Homework Statement
In a CRT tube, electrons are accelerated by a 20,000 V potential difference between the electron gun (the cathode) and the positive metal mesh 5.00 cm away.
a. What is the electron’s speed when it reaches the positive wire mesh?
b. How much time does it take the electron to reach the wire mesh?
Homework Equations
[itex]U = q∆V\\
K = \frac{1}{2}mv^2\\
F = ma\\
F = qE \\
E = \frac{V}{d}\\
d = d_0 + (v_0)(t) + \frac{1}{2}(a)(t^2)\\
[/itex]
The Attempt at a Solution
For part A, I got the following which is actually the same example we did in class.
[itex]
U = K\\
q∆V = \frac{1}{2}mv^2\\
(1.6*10^{-19}C)(20000\frac{J}{C}) = \frac{1}{2}(9.1*10^{-31}kg)(v^{2})\\
3.2*10^{-15}J = 4.55*10^{-31}kg(v^{2}) \\
(3.2*10^{-15}J)/(4.55*10^{-31}kg) = v^{2} \\
7.033*10^{15}\frac{m^{2}}{s^{2}} = v^{2} \\
v = √7.033*10^{15}\frac{m^{2}}{s^{2}} \\
v = 8.386*10^{7}\frac{m}{s}\\
[/itex]
Where I'm getting a bit paranoid is part B.
At first I thought, in order to get ##t##, all we needed to do was use the formula, ##t = \frac{d}{v}## and I got an answer of ##5.96*10^{-10}s##, but that seems to simple for this kind of problem.
So then I decided to use a combination of several formulas we learned in the past in order to determine the time it takes for an electron to travel a set distance.
Here's what I did:
[itex]
F = ma\\
F = qE \\
E = \frac{V}{d}\\
ma = \frac{qV}{d}\\
a= \frac{qV}{dm}\\
d = d_0 + (v_0)(t) + \frac{1}{2}(a)(t^2)\\
d = \frac{1}{2}(a)(t^2)\\
d = \frac{1}{2}[\frac{qV}{dm}](t^2)\\
2d = \frac{qV}{dm}(t^2)\\
\frac{2d^{2}m}{qV} = t^2\\
t = √\frac{2d^2m}{qV}\\
t = √\frac{(2)(.05m)^{2}(9.1*10^{-31}kg)}{(1.6*10^{-19}C)(20,000\frac{J}{C})}\\
t = √\frac{4.55*10^{-33}}{3.2*10^{-15}}\\
t = √1.422*10^{-18}s\\
t = 1.19*10^{-9}s\\
[/itex]
Now looking back at my second attempt, it seems a bit ridiculous to do all that to get ##t##. So my question is, is my second attempt for getting ##t## correct or did I mess up?