91Ó°ÊÓ

A thin spherical shell of radius \({R_1}\)made of plastic carries a uniformly distributed negative charge \( - {Q_1}\). A thin spherical shell of radius \({R_2}\)made of glass carries a uniformly distributed positive charge \( + {Q_2}\). The distance between centers is \(L\), as shown in Figure 16.80. (a) Find the potential difference \({V_B} - {V_A}\). Location A is at the center of the glass sphere, and location \(B\) is just outside the glass sphere. (b) Find the potential difference \({V_C} - {V_B}\). Location \(B\) is just outside the glass sphere, and location \(C\) is a distance d to the right of \(B\). (c) Suppose the glass shell is replaced by a solid metal sphere with radius R2 carrying charge \( + {Q_2}\). Would the magnitude of the potential difference \({V_B} - {V_A}\) be greater than, less than, or the same as it was with the glass shell in place? Explain briefly, including an appropriate physics diagram.

A particle with charge\( + {q_1}\)and a particle with charge\( - {q_2}\)are located as shown in figure 16.91. What is the potential (relative to infinity) at location A.

What is the potential (relative to infinity) at location B, a distance h from a ring of radius a with charge –Q as shown in figure 16.94?

What is the maximum possible potential (relative to infinity) of the metal sphere of 10-cm radius? What is the maximum possible potential (relative to infinity) of the metal sphere of only 1-mm radius? These results hint at the reason why a highly charged piece of metal (with uniform potential throughout) tends to spark at places where the radius of curvature is small or at places where there are sharp points. Remember that breakdown electric strength for air is roughly\[{\bf{3 \times 1}}{{\bf{0}}^{\bf{6}}}\;\frac{{\bf{V}}}{{\bf{m}}}\].

Figure 16.60 shows a portion of a long, negatively charged rod. You need to calculate the potential difference${\mathbf{V}}_{\mathbf{A}}\mathbf{-}{\mathbf{V}}_{\mathbf{B}}$.

(a) What is the direction of the path (+y or −y)? (b) What is the sign of ${\mathbf{V}}_{\mathbf{A}}\mathbf{-}{\mathbf{V}}_{\mathbf{B}}$?

Locations$\mathbf{A}\mathbf{=}\mathbf{<}\mathbf{a}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{>}\mathbf{}\mathbf{and}\mathbf{}\mathbf{B}\mathbf{=}\mathbf{<}\mathbf{b}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{>}$are on the +x axis, as shown in Figure 16.61. Four possible expressions for the electric field along the x axis are given below. For each expression for the electric field, select the correct expression (1–8) for the potential difference${\mathbf{V}}_{\mathbf{A}}\mathbf{-}{\mathbf{V}}_{\mathbf{B}}$. In each case K is a numerical constant with appropriate units.

$$\begin{gathered} \mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{E}}\mathbf{=}<\frac{K}{x^2},0,0> \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{E}}\mathbf{=}<\frac{K}{x^3},0,0> \\ \mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{E}}\mathbf{=}<\frac{K}{x},0,0> \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{E}}\mathbf{=}<\mathrm{Kx},0,0> \\ \mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{}{\mathbf{V}}_{\mathbf{A}}\mathbf{-}{\mathbf{V}}_{\mathbf{B}}\mathbf{=}\mathbf{0} \\ \mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{}{\mathbf{V}}_{\mathbf{A}}\mathbf{-}{\mathbf{V}}_{\mathbf{B}}\mathbf{=}\mathbf{K}\left(a-b\right) \\ \mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{}{\mathbf{V}}_{\mathbf{A}}\mathbf{-}{\mathbf{V}}_{\mathbf{B}}\mathbf{=}\mathbf{K}\left(\frac{1}{a}-\frac{1}{b}\right) \\ \mathbf{\left(}\mathbf{4}\mathbf{\right)}\mathbf{}{\mathbf{V}}_{\mathbf{A}}\mathbf{-}{\mathbf{V}}_{\mathbf{B}}\mathbf{=}\mathbf{K}\left(\frac{1}{a^3}a-\frac{1}{b^3}b\right) \\ \mathbf{\left(}\mathbf{5}\mathbf{\right)}\mathbf{}{\mathbf{V}}_{\mathbf{A}}\mathbf{-}{\mathbf{V}}_{\mathbf{B}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{K}\left(b^2-a^2\right) \\ \mathbf{\left(}\mathbf{6}\mathbf{\right)}\mathbf{}{\mathbf{V}}_{\mathbf{A}}\mathbf{-}{\mathbf{V}}_{\mathbf{B}}\mathbf{=}\mathbf{K}\mathbf{}\mathbf{In}\left(\frac{b}{a}\right) \\ \mathbf{\left(}\mathbf{7}\mathbf{\right)}\mathbf{}{\mathbf{V}}_{\mathbf{A}}\mathbf{-}{\mathbf{V}}_{\mathbf{B}}\mathbf{=}\mathbf{K}\left(a^3-b^3\right) \\ \mathbf{\left(}\mathbf{8}\mathbf{\right)}\mathbf{}{\mathbf{V}}_{\mathbf{A}}\mathbf{-}{\mathbf{V}}_{\mathbf{B}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{K}\left(\frac{1}{a^2}-\frac{1}{b^2}\right) \end{gathered}$$

The graph in Figure 16.63 is a plot of electric potential versus distance from an object. Which of the following could be the object?

(1) A neutron, (2) A sodium ion (Na+), (3) A chloride ion (Cl−), (4) A proton, (5) An electron.

For each of the following statements, say whether it is true or false and explain why it is true or false. Be complete in your explanation, but be brief. Pay particular attention to the distinction between potential V and potential difference ∆V. (a) The electric potential inside a metal in equilibrium is always zero. (b) If there is a constant large positive potential throughout a region, the electric field in that region is large. (c) If you get close enough to a negative point charge, the potential is negative, no matter what other charges are around. (d) Near a point charge, the potential difference between two points a distance L apart is −E³¢. (e) In a region where the electric field is varying, the potential difference between two points a distance L apart is $\mathbf{-}\mathbf{(}{\mathbf{E}}_{\mathbf{f}}\mathbf{-}{\mathbf{E}}_{\mathbf{i}}\mathbf{)}\mathbf{L}\mathbf{.}$

What is the kinetic energy of a proton that is traveling at a speed of $\mathbf{3725}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$ ?

You move from location $\mathit{i}$ at $⟨\mathbf{2}\mathbf{,}\mathbf{7}\mathbf{,}\mathbf{5}âŸ\copyright \text{ }\mathbf{m}$to location$\mathit{f}$ at $⟨\mathbf{5}\mathbf{,}\mathbf{6}\mathbf{,}\mathbf{12}âŸ\copyright \text{ }\mathbf{m}$. All along this path there is a nearly uniform electric field of $⟨\mathbf{1000}\mathbf{,}\mathbf{200}\mathbf{,}−\mathbf{510}âŸ\copyright \text{ }\mathbf{N}\mathbf{/}\mathbf{C}$. Calculate ${\mathbf{V}}_{\mathbf{f}}\mathbf{−}{\mathbf{V}}_{\mathbf{i}}$, including sign and units.