A proton in a well
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The figure [attached] shows electric potential V along an x axis. A proton is to be released at x = 3.5 cm with initial kinetic energy 3.5 eV. The scale of the vertical axis is set by Vs = 10.0 V.
(a) If it is initially moving in the negative direction, it either reaches a turning point (if so, what is the x coordinate of that point) or it escapes from the plotted region (if so, what is its speed at x = 0)?
(b) If it is initially moving in the positive direction, it either reaches a turning point (if so, what is the x coordinate of that point) or it escapes from the plotted region (if so, what is its speed at x = 6.0 cm)?
(c) What is the electric force (including sign) on the proton if the proton moves just to the left of x = 3.0 cm?
(d) What is the electric force (including sign) on the proton if the proton moves just to the right of x = 5.0 cm?
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Solution Summary
The expert examines protons in a well with an electric potential.
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The particle is a proton.
Thus, the potential energy of such a particle under potential V(x) is simply
(1.1)
Where e is the charge of an electron.
First, let's express the potential energy in mathematical terms:
(1.2)
Where is given in electron-volts while x is given in centimeters.
The proton has initial kinetic energy at , so the total mechanical energy is:
(1.3)
The total mechanical energy is constant, so for any x:
(1.4)
In our case and , thus so the total mechanical energy is:
(1.5)
If we plot the potential and total mechanical energy on the same graph it looks like:
A turning point occurs when the total energy equals the potential energy.
If initially the ...
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