Exam 17: Electric Potential

Two $5.0 - \mu \mathrm { C }$ point charges are $12 \mathrm {~cm}$ apart. What is the electric potential (relative to infinity) of this combination at the point where the electric field due to these charges is zero? $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times \right.$ $10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 }$ )

An air-filled capacitor carries enough charge to store $6.00 \mathrm {~mJ}$ of potential energy. It is then accidentally filled with water in such a way as not to discharge its plates. How much energy does it continue to store after it is filled? The dielectric constant for water is 78 and for air it is $1.0006$ .

A 3.0-pFcapacitor consists of two large closely-spaced parallel plates that have surface charge densities of $\pm 1.0 \mathrm { nC } / \mathrm { mm } ^ { 2 }$ . If the potential across the plates is $23.0 \mathrm { kV }$ with only air between them, find the surface area of each of the plates. $\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \mathrm { C } ^ { 2 } / \mathrm { N } \cdot \mathrm { m } ^ { 2 } \right.$ )

An ideal air-filled parallel-plate capacitor consists of plates that are $1.0 \mathrm {~mm}$ apart and have an area of $1.5 \times 10 ^ { - 4 } \mathrm {~m} ^ { 2 }$ . The capacitor is connected to a $12 - \mathrm { V }$ potential source (battery). $\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \mathrm { C } ^ { 2 } / \mathrm { N } \cdot \mathrm { m } ^ { 2 } \right.$ ) (a) What is the capacitance of this capacitor? (b) How much charge is on each of its plates? (c) What is the strength of the electric field between the plates?

The equipotential surfaces for two spherical conductors are shown in the figure, with the value of potential marked on the line for each surface. (a) If the distance between points $A$ and $B$ is $2.5 \mathrm {~cm}$ what is the approximate intensity of the electric field between these two points? (b) If the distance between points $C$ and $D$ is $2.5 \mathrm {~cm}$ what is the approximate intensity of the electric field between these two points?

An electron is released from rest at a distance of $9.00 \mathrm {~cm}$ from a fixed proton. How fast will the electron be moving when it is $3.00 \mathrm {~cm}$ from the proton? $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 8.99 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } , e = 1.60 \right.$ $\left. \times 10 ^ { - 19 } \mathrm { C } , m _ { \text {electron } } = 9.11 \times 10 ^ { - 31 } \mathrm {~kg} , m _ { \text {proton } } = 1.67 \times 10 ^ { - 27 } \mathrm {~kg} \right)$

A $15 - \mu \mathrm { F }$ capacitor is connected to a 50 - $\mathrm { V }$ battery and becomes fully charged. The battery is removed and a slab of dielectric, having a dielectric constant of $5.0$ , is inserted between the plates and completely fills the space between them. (a) What is the capacitance of the capacitor after the slab is inserted? (b) What is the potential difference across the capacitor with the dielectric inserted.

A $- 6.5 - \mu C$ point charge is $8.0 \mathrm {~cm}$ from a $- 17 - \mu \mathrm { C }$ charge. What is the electric energy density at the point midway between them? $\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \mathrm { C } ^ { 2 } / \mathrm { N } \cdot \mathrm { m } ^ { 2 } , k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \right.$ $\left. \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$

Two very small $+ 3.00 - \mu \mathrm { C }$ charges are at the ends of a meter stick. Find the electric potential (relative to infinity) at the center of the meter stick. $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 8.99 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$

A $6.9 \mu \mathrm { C }$ negative point charge has a positively charged particle in an elliptical orbit about it. If the mass of the positively charged particle is $1.0 \mu \mathrm { g }$ and its distance from the point charge varies from $4.0 \mathrm {~mm}$ to $16.0 \mathrm {~mm}$ , what is the maximum potential difference through which the positive object moves? $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$

Two tiny particles having charges $q _ { 1 } = + 56.0 \mathrm { nC }$ and $q _ { 2 } = - 46.0 \mathrm { nC }$ are separated by $0.500 \mathrm {~m}$ and held in place, as shown in the figure. A third particle, having a charge of $54.0 \mathrm { nC }$ is placed at the point $A$ , which is $0.18 \mathrm {~m}$ to the left of $q 2$ . How much work is needed to move the third particle from point $A$ to point $B$ , which is $0.40 \mathrm {~m}$ to the left of $q 1$ . All the points in the figure lie on the same line. $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$

A space probe approaches a planet, taking measurements as it goes. If it detects a potential difference of $6000 \mathrm { MV }$ between the altitudes of $253,000 \mathrm {~km}$ and $276,000 \mathrm {~km}$ above the planet's surface, what is the approximate electric field strength produced by the planet at $264,500 \mathrm {~km}$ above the surface? Assume the electric field strength is approximately constant at these altitudes.

A proton that is initially at rest is accelerated through an electric potential difference of magnitude $500 \mathrm {~V}$ . What speed does the proton gain? $\left( e = 1.60 \times 10 ^ { - 19 } \mathrm { C } , m \right.$ proton $\left. = 1.67 \times 10 ^ { - 27 } \mathrm {~kg} \right)$

A square is $1.0 \mathrm {~m}$ on a side. Point charges of $+ 4.0 \mu \mathrm { C }$ are placed in two diagonally opposite corners. In the other two corners are placed charges of $+ 3.0 \mu \mathrm { C }$ and $- 3.0 \mu \mathrm { C }$ . What is the potential (relative to infinity) at the midpoint of the square? $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$

What charge accumulates on the plates of a $2.0 - \mu \mathrm { F }$ air-filled capacitor when it is charged until the potential difference across its plates is $100 \mathrm {~V}$ ?

A proton moves $0.10 \mathrm {~m}$ along the direction of an electric field of magnitude $3.0 \mathrm {~V} / \mathrm { m }$ . What is the change in kinetic energy of the proton? $\left( e = 1.60 \times 10 ^ { - 19 } \mathrm { C } \right)$

Three point charges, $- 2.00 \mu \mathrm { C } , + 4.00 \mu \mathrm { C }$ , and $+ 6.00 \mu \mathrm { C }$ , are located along the $x$ -axis as shown in the figure. What is the electric potential (relative to infinity) at point $\mathrm { P }$ due to these charges? $( k =$ $1 / 4 \pi \varepsilon _ { 0 } = 8.99 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 }$ )

A 25.0-V potential source (a battery) is connected across the plates of a $6.66 - \mu \mathrm { F }$ air-filled parallel-plate capacitor having plates that are $1.22 \mathrm {~mm}$ apart. What energy density does it produce between the plates? $\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \mathrm { C } ^ { 2 } / \mathrm { N } \cdot \mathrm { m } ^ { 2 } \right)$

How much energy is necessary to place three $+ 2.0 - \mu \mathrm { C }$ point charges at the vertices of an equilateral triangle of side $2.0 \mathrm {~cm}$ if they started out extremely far away? $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times 10 ^ { 9 } \mathrm {~N} \right.$ $\cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 }$ )

An ideal parallel-plate capacitor having circular plates of diameter $D$ that are a distance $d$ apart stores energy $U$ when it is connected across a fixed potential difference. If you want to triple the amount of energy stored in this capacitor by changing only the size of its plates, the diameter should be changed to