# A proton in a well

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 Preview

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)

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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 ...

#### Solution Summary

The expert examines protons in a well with an electric potential.