|| CALC The electric field in a region of space is $E_x=5000x\text{V}\text{/}\text{m}$, where x is in meters.

1. Graph $E_x$ versus x over the region $-1\text{m}\le x\le 1\text{m}$.

2. Find an expression for the potential V at position x. As a reference, let $V=0\text{V}$ at the origin.

3. Graph V versus x over the region $-1\text{m}\le x\le 1\text{m}$.

|| CALC The electric field in a region of space is $E_x=-1000x^2\text{V}\text{/}\text{m}$, where x is in meters.

1. Graph $E_x$ versus x over the region $-1\text{m}\le x\le 1\text{m}$.

2. What is the potential difference between $x_{\text{i}}=-20\text{cm}$ and $x_{\text{f}}=30\text{cm}?$

|| INT CALC An infinitely long cylinder of radius R has linear charge density $\text{λ}$. The potential on the surface of the cylinder is $V_0$, and the electric field outside the cylinder is $E_r=\text{λ}\text{/}2\pi {ϵ}_{0}r$. Find the potential relative to the surface at a point that is distance r from the axis, assuming $r>R$.

|| FIGURE P26.41 is an edge view of three charged metal electrodes. Let the left electrode be the zero point of the electric potential. What are V and $\vec{E}$ at (a) $x=0.5\text{cm,}$ (b) $x=1.5\text{cm,}$ and (c) $x=2.5\text{cm}?$

FIGURE P26.41

||

1. CALC Use the methods of Chapter 25 to find the potential at distance x on the axis of the charged rod shown in FIGURE P26.42.

   FIGURE P26.42

   

   FIGURE P26.42 Full Alternative Text

2. Use the result of part a to find the electric field at distance x on the axis of a rod.

|| CALC Use the on-axis potential of a charged disk from Chapter 25 to find the on-axis electric field of a charged disk.

|| CALC Engineers discover that the electric potential between two electrodes can be modeled as $V\left(x\right)=V_0\text{ln}(1+x/d)$, where $V_0$ is a constant, x is the distance from the first electrode in the direction of the second, and d is the distance between the electrodes. What is the electric field strength midway between the electrodes?

|| CALC The electric potential in a region of space is $V=(150x^2-200y^2)\text{V,}$ where x and y are in meters. What are the strength and direction of the electric field at $(x\text{,}y)=(2.0\text{m, }2.0\text{m})?$ Give the direction as an angle cw or ccw (specify which) from the positive x-axis.

|| CALC The electric potential near the origin is $V=200\text{/}\sqrt{x^2+y^2}$, where x and y are in meters. What are the strength and direction of the electric field at $\left(x\text{,}y\right)=(2.0\text{m, }1.0\text{m})?$ Give the direction as an angle cw or ccw (specify which) from the positive x-axis.

|| It is postulated that the radial electric field of a group of charges falls off as $E_r=C\text{/}{r}^n$, where C is a constant, r is the distance from the center of the group, and n is an unknown exponent. To test this hypothesis, you make a field probe consisting of two needle tips spaced 1.00 mm apart. You orient the needles so that a line between the tips points to the center of the charges, then use a voltmeter to read the potential difference between the tips. After you take measurements at several distances from the center of the group, your data are as follows:

Use an appropriate graph of the data to determine the constants $C$ and $n$.

|| CALC The electric field in a region of space is $\vec{E}=(800xî-600yĵ)\text{V}\text{/}\text{m}$, where x and y are in m. The zero of electric potential is at the origin. What are (a) the electric field and (b) the electric potential at the point $(x\text{,}y)=(2.0\text{m,}1.0\text{m})?$

Hint: The potential difference is the same along any path connecting two points.

|| INT CALC Two positive point charges q are located on the y-axis at $y=\pm a$.

1. Write an expression for the electric potential at position x on the x-axis.

2. The binomial approximation is $(1+z{)}^n\approx 1+nz$ if $z\ll 1$. Use this to find an expression for the potential along the x-axis if $x\ll a$—that is, very near the origin.

3. Symmetry dictates that the electric field along the x-axis has only an x-component: $E_y=E_z=0$. Find an expression for $E_x$ if $x\ll a$.

4. What is the force on an electron at position x on the x-axis, with $x\ll a?$

5. Your answer to part d shows that an electron experiences a linear restoring force, so it will undergo simple harmonic motion. What is the oscillation frequency in GHz for an electron moving between two 1.0 nC charges separated by 2.0 mm?

|| The metal spheres in FIGURE P26.50 are charged to $\pm 300\text{V}$. Draw this figure on your paper, then draw a plausible contour map of the potential, showing and labeling the $-300\text{V,}-200\text{V}$, $-100\text{V}$, . . . , 300 V equipotential surfaces.

FIGURE P26.50

| Metal sphere 1 has a positive charge of 6.0 nC. Metal sphere 2, which is twice the diameter of sphere 1, is initially uncharged. The spheres are then connected together by a long, thin metal wire. What are the final charges on each sphere?

||Two $2.0\mathrm{cm}\times 2.0\mathrm{cm}$ metal electrodes are spaced 1.0 mm apart and connected by wires to the terminals of a 9.0 V battery.

1. What are the charge on each electrode and the potential difference between them?

   The wires are disconnected, and insulated handles are used to pull the plates apart to a new spacing of 2.0 mm.

2. What are the charge on each electrode and the potential difference between them?

||| The electric potential is 40 V at point A near a uniformly charged sphere. At point B, $2.0\mu \text{m}$ farther away from the sphere, the potential has decreased by 0.16 mV. How far is point A from the center of the sphere?

| Find expressions for the equivalent capacitance of (a) N identical capacitors C in parallel and (b) N identical capacitors C in series.

| Two $2.0\mathrm{cm}\times 2.0\mathrm{cm}$ metal electrodes are spaced 1.0 mm apart and connected by wires to the terminals of a 9.0 V battery.

1. What are the charge on each electrode and the potential difference between them?

   While the plates are still connected to the battery, insulated handles are used to pull them apart to a new spacing of $2.0\text{mm}$.

2. What are the charge on each electrode and the potential difference between them?

|| What are the charge on and the potential difference across each capacitor in FIGURE P26.56?

FIGURE P26.56

|| What are the charge on and the potential difference across each capacitor in FIGURE P26.57?

FIGURE P26.57

|| What are the charge on and the potential difference across each capacitor in FIGURE P26.58?

FIGURE P26.58

| Six identical capacitors with capacitance C are connected as shown in FIGURE P26.59.

1. What is the equivalent capacitance of these six capacitors?

2. What is the potential difference between points a and b?

FIGURE P26.59

|| Initially, the switch in FIGURE P26.60 is in position A and capacitors $C_2$ and $C_3$ are uncharged. Then the switch is flipped to position B. Afterward, what are the charge on and the potential difference across each capacitor?

FIGURE P26.60

|| Initially, the switch in FIGURE P26.61 is in position A and capacitors $C_2$ and $C_3$ are uncharged. Then the switch is flipped to position B. Afterward, the voltage across $C_1$ is 4.0 V. What is the emf of the battery?

FIGURE P26.61

|| A battery with an emf of 60 V is connected to the two capacitors shown in FIGURE P26.62. Afterward, the charge on capacitor 2 is $450\mu \text{C}$. What is the capacitance of capacitor 2?

FIGURE P26.62

|| Capacitors $C_1=10\mu \text{F}$ and $C_2=20\mu \text{F}$ are each charged to 10 V, then disconnected from the battery without changing the charge on the capacitor plates. The two capacitors are then connected in parallel, with the positive plate of $C_1$ connected to the negative plate of $C_2$ and vice versa. Afterward, what are the charge on and the potential difference across each capacitor?

|| INT An isolated $5.0\mu \text{F}$ parallel-plate capacitor has 4.0 mC of charge. An external force changes the distance between the electrodes until the capacitance is $2.0\mu \text{F}$. How much work is done by the external force?

|| INT You’ve built a device that uses the energy from a rapidly discharged capacitor to launch the capacitor straight up. One capacitor, with a mass of 3.5 g, is launched to a height of 1.6 m after having been charged to 100 V. What is its capacitance in $\mu \text{F}?$

|| INT The flash unit in a camera uses a 3.0 V battery to charge a capacitor. The capacitor is then discharged through a flashlamp. The discharge takes $10\mu \text{s,}$ and the average power dissipated in the flashlamp is 10 W. What is the capacitance of the capacitor?

|| The label rubbed off one of the capacitors you are using to build a circuit. To find out its capacitance, you place it in series with a $10\mu \text{F}$ capacitor and connect them to a 9.0 V battery. Using your voltmeter, you measure 6.0 V across the unknown capacitor. What is the unknown capacitor’s capacitance?

||| CALC High-frequency signals are often transmitted along a coaxial cable, such as the one shown in FIGURE P26.68. For example, the cable TV hookup coming into your home is a coaxial cable. The signal is carried on a wire of radius $R_1$ while the outer conductor of radius $R_2$ is grounded (i.e., at $V=0\text{V}$). An insulating material fills the space between them, and an insulating plastic coating goes around the outside.

1. Find an expression for the capacitance per meter of a coaxial cable. Assume that the insulating material between the cylinders is air.

2. Evaluate the capacitance per meter of a cable having $R_1=0.50\text{mm}$ and $R_2=3.0\text{mm}$.

FIGURE P26.68

|| A capacitor that uses 0.50-mm-thick paper as a dielectric is charged to the maximum sustainable voltage. What are (a) the capacitor voltage and (b) the strength of the induced electric field?

|| CALC The current that charges a capacitor transfers energy that is stored in the capacitor’s electric field. Consider a $2.0\mu \text{F}$ capacitor, initially uncharged, that is storing energy at a constant 200 W rate. What is the capacitor voltage $2.0\mu \text{s}$ after charging begins?

|| A vacuum-insulated parallel-plate capacitor with plate separation d has capacitance $C_0$. What is the capacitance if an insulator with dielectric constant $\kappa$ and thickness $d\text{/}2$ is slipped between the electrodes without changing the plate separation?

|| BIO A typical cell has a membrane potential of $-70\text{mV,}$ meaning that the potential inside the cell is 70 mV less than the potential outside due to a layer of negative charge on the inner surface of the cell wall and a layer of positive charge on the outer surface. This effectively makes the cell wall a charged capacitor. Because a cell’s diameter is much larger than the wall thickness, it is reasonable to ignore the curvature of the cell and think of it as a parallel-plate capacitor. How much energy is stored in the electric field of a 50-$\mu$m-diameter cell with a 7.0-nm-thick cell wall whose dielectric constant is 9.0?

||| BIO A nerve cell in its resting state has a membrane potential of $-70\text{mV,}$ meaning that the potential inside the cell is 70 mV less than the potential outside due to a layer of negative charge on the inner surface of the cell wall and a layer of positive charge on the outer surface. This effectively makes the cell wall a charged capacitor. When the nerve cell fires, sodium ions, ${\text{Na}}^{+}$, flood through the cell wall to briefly switch the membrane potential to $+40\text{mV}$. Model the central body of a nerve cell—the soma—as a $50\text{-}\mu \text{m}$-diameter sphere with a 7.0-nm-thick cell wall whose dielectric constant is 9.0. Because a cell’s diameter is much larger than the wall thickness, it is reasonable to ignore the curvature of the cell and think of it as a parallel-plate capacitor. How many sodium ions enter the cell as it fires?

|| Derive Equation 26.33 for the induced surface charge density on the dielectric in a capacitor.

$2az\text{V/m}=-{\displaystyle \frac{dV}{dz}}$, where a is a constant with units of $\text{V}\text{/}{\text{m}}^2$

$V(z=0)=10\text{V}$

$400\text{ nC}=(100\text{ V})C$

$C=\frac{(8.85\times {10}^{-12}{\text{ C}}^2{\text{/N m}}^2)(0.10\text{ m}\times 0.10\text{ m})}{d}$

$({\displaystyle \frac{1}{3\mu \text{F}}}+{\displaystyle \frac{1}{6\mu \text{F}}}{)}^{-1}+C=4\mu \text{F}$

||| CALC An electric dipole at the origin consists of two charges $\pm q$ spaced distance s apart along the y-axis.

1. Find an expression for the potential $V\left(x\text{,}y\right)$ at an arbitrary point in the xy-plane. Your answer will be in terms of q, s, x, and y.

2. Use the binomial approximation to simplify your result of part a when $s\ll x$ and $s\ll y$.

3. Assuming $s\ll x$ and y, find expressions for $E_x$ and $E_y$, the components of $\vec{E}$ for a dipole.

4. What is the on-axis field $\vec{E}?$ Does your result agree with Equation 23.10?

5. What is the field $\vec{E}$ on the bisecting axis? Does your result agree with Equation 23.11?

||| CALC Charge is uniformly distributed with charge density $\rho$ inside a very long cylinder of radius R. Find the potential difference between the surface and the axis of the cylinder.

||| CALC Consider a uniformly charged sphere of radius R and total charge Q. The electric field $E_{\text{out}}$ outside the sphere $(r\ge R)$ is simply that of a point charge Q. In Chapter 24, we used Gauss’s law to find that the electric field $E_{\text{in}}$ inside the sphere $(r\le R)$ is radially outward with field strength

1. The electric potential $V_{\text{out}}$ outside the sphere is that of a point charge Q. Find an expression for the electric potential $V_{\text{in}}$ at position r inside the sphere. As a reference, let $V_{\text{in}}=V_{\text{out}}$ at the surface of the sphere.

2. What is the ratio $V_{\text{center}}\text{/}{V}_{\text{surface}}?$

3. Graph V versus r for $0\le r\le 3R$.

|||

1. CALC Find an expression for the capacitance of a spherical capacitor, consisting of concentric spherical shells of radii $R_1$ (inner shell) and $R_2$ (outer shell).

2. A spherical capacitor with a 1.0 mm gap between the shells has a capacitance of 100 pF. What are the diameters of the two spheres?

||| Two 5.0-cm-diameter metal disks separated by a 0.50-mm-thick piece of Pyrex glass are charged to a potential difference of 1000 V. What are (a) the surface charge density on the disks and (b) the surface charge density on the glass?

||| Each capacitor in FIGURE CP26.83 has capacitance C. What is the equivalent capacitance between points a and b?

FIGURE CP26.83