Chapter 25: The Electric Potential

A 2.0-mm-diameter glass bead is positively charged. The potential difference between a point 2.0 mm from the bead and a point 4.0 mm from the bead is 500 V. What is the charge on the bead?

Your lab assignment for the week is to measure the amount of charge on the 6.0-cm-diameter metal sphere of a Van de Graaff generator. To do so, you’re going to use a spring with a spring constant of 0.65 N/m to launch a small, 1.5 g bead horizontally toward the sphere. You can reliably charge the bead to 2.5 nC, and your plan is to use a video camera to measure the bead’s closest approach to the edge of the sphere as you change the compression of the spring. Your data is as follows:

Use an appropriate graph of the data to determine the sphere’s charge in nC. You can assume that the bead’s motion is entirely horizontal, that the spring is so far away that the bead has no interaction with the sphere as it’s launched, and that the approaching bead does not alter the charge distribution on the sphere.

A proton is fired from far away toward the nucleus of an iron atom. Iron is element number $26$, and the diameter of the nucleus is $9.0fm$. What initial speed does the proton need to just reach the surface of the nucleus? Assume the nucleus remains at rest.

A proton is fired from far away toward the nucleus of a mercury atom. Mercury is element number $80$, and the diameter of the nucleus is$14.0fm$ . If the proton is fired at a speed of$4.0\times {10}^{7}m/s$, what is its closest approach to the surface of the nucleus? Assume the nucleus remains at rest

In the form of radioactive decay known as alpha decay, an unstable nucleus emits a helium-atom nucleus, which is called an alpha particle. An alpha particle contains two protons and two neutrons, thus having mass$m=4u$and charge $q=2e$. Suppose a uranium nucleus with $92$protons decays into thorium, with $90$protons, and an alpha particle. The alpha particle is initially at rest at the surface of the thorium nucleus, which is $15fm$in diameter. What is the speed of the alpha particle when it is detected in the laboratory? Assume the thorium nucleus remains at rest

One form of nuclear radiation, beta decay, occurs when a neutron changes into a proton, an electron, and a neutral particle called a neutrino: $n\to p^{+}+e^{-}+v$where $n$is the symbol for a neutrino. When this change happens to a neutron within the nucleus of an atom, the proton remains behind in the nucleus while the electron and neutrino are ejected from the nucleus. The ejected electron is called a beta particle. One nucleus that exhibits beta decay is the isotope of hydrogen $3H$, called tritium, whose nucleus consists of one proton (making it hydrogen) and two neutrons (giving tritium an atomic mass $m=3u$). Tritium is radioactive, and it decays to helium: ${}^{3}H\to {}^{3}He+e^{-}+n$

a. Is charge conserved in the beta decay process? Explain.

b. Why is the final product a helium atom? Explain.

c. The nuclei of both ${}^{3}H\mathrm{and}{}^{\mathit{3}}He$ have radii of $1.5\times {10}^{-15}m$. With what minimum speed must the electron be ejected if it is to escape from the nucleus and not fall back?

What is the electric potential energy of the group of charges in FIGURE EX25.6?

FIGURE Q25.6 shows the electric potential along the x-axis.

a. Draw a graph of the potential energy of a $0.1\mathrm{C}$charged particle. Provide a numerical scale for both axes.

b. If the charged particle is shot toward the right from FIGURE Q25.6 $x=1\mathrm{m}$with$1.0\mathrm{J}$of kinetic energy, where is its turning point? Use your graph to explain.

Two$10-\mathrm{cm}$-diameter electrodes $0.50\mathrm{cm}$apart form a parallel plate capacitor. The electrodes are attached by metal wires to

the terminals of a $15\mathrm{V}$battery. After a long time, the capacitor is disconnected from the battery but is not discharged. What are the charge on each electrode, the electric field strength inside the capacitor, and the potential difference between the electrodes

a. Right after the battery is disconnected?

b. After insulating handles are used to pull the electrodes away from each other until they are $1.0\mathrm{cm}$apart?

c. After the original electrodes (not the modified electrodes of part b) are expanded until they are $20\mathrm{cm}$in diameter?

Electrodes of area $A$are spaced distance $d$apart to form a parallel-plate capacitor. The electrodes are charged to $\pm q$.

(a). What is the infinitesimal increase in electric potential energy $dU$if an infinitesimal amount of charge $dq$is moved from the negative electrode to the positive electrode?

(b). An uncharged capacitor can be charged to $\pm Q$by transferring charge $dq$over and over and over. Use your answer to part a to show that the potential energy of a capacitor charged to $\pm Q$is $U_{\text{cap}}=\frac{1}{2}Q\Delta V_{\mathrm{C}}$.