Exam 17: Electric Potential

A point charge of $+ 3.00 \mu \mathrm { C }$ and a second charge $Q$ are initially very far apart. If it takes $29.0 \mathrm {~J}$ of work to bring them to a final configuration in which the $+ 3.00 - \mu \mathrm { C }$ charge is at the point $x = 1.00$ $\mathrm { mm } , y = 1.00 \mathrm {~mm}$ , and the second charge $Q$ is at the point $x = 1.00 \mathrm {~mm} , y = 3.00 \mathrm {~mm}$ , find the magnitude of the charge $Q . \left( k = 1 / 4 \pi \varepsilon _ { 0 } = 8.99 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$

If a $\mathrm { Cu } ^ { 2 + }$ ion that is initially at rest accelerates through a potential difference of $12 \mathrm {~V}$ without friction, how much kinetic energy will it gain? $\left( e = 1.60 \times 10 ^ { - 19 } \mathrm { C } \right)$

An air-filled capacitor has a potential difference between the plates of $80 \mathrm {~V}$ . If the charge on each of the plates of the capacitor has magnitude $8.0 \mu \mathrm { C }$ , what is the electrical energy stored by this capacitor?

A 4.0-g bead carries a charge of $20 \mu \mathrm { C }$ . The bead is accelerated from rest through a potential difference $V$ , and afterward the bead is moving at $2.0 \mathrm {~m} / \mathrm { s }$ . What is the magnitude of the potential difference $V$ ?

Two $3.0 \mu \mathrm { C }$ charges lie on the $x$ -axis, one at the origin and the other at $2.0 \mathrm {~m}$ . What is the potential (relative to infinity) due to these charges at a point at $6.0 \mathrm {~m}$ on the $x$ -axis? $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times 10 ^ { 9 } \right.$ $\mathrm { N } \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 }$ )

Two very large parallel metal plates, separated by $0.20 \mathrm {~m}$ , are connected across a $12 - \mathrm { V }$ source of potential. An electron is released from rest at a location $0.10 \mathrm {~m}$ from the negative plate. When the electron arrives at a distance $0.050 \mathrm {~m}$ from the positive plate, how much kinetic energy has the electron gained? $\left( e = 1.60 \times 10 ^ { - 19 } \mathrm { C } \right)$

In the figure, $+ 4.0 - \mu \mathrm { C }$ and $- 4.0 - \mu \mathrm { C }$ point charges are located as shown. Now an additional $+ 2.00 - \mu \mathrm { C }$ point charge is placed at point $\mathrm { A }$ . What is the electric potential energy of this system of three charges, relative to infinity? $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 8.99 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$

A proton is accelerated from rest through a potential difference $V _ { 0 }$ and gains a speed $v _ { 0 }$ . If it were accelerated instead through a potential difference of $2 V _ { 0 }$ , what speed would it gain?

The three point charges shown in the figure form an equilateral triangle with sides $4.9 \mathrm {~cm}$ long. What is the electric potential (relative to infinity) at the point indicated with the dot, which is equidistant from all three charges? Assume that the numbers in the figure are all accurate to two significant figures. $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$

Two parallel circular plates, each with a radius of $7.0 \mathrm {~mm}$ and carrying equal-magnitude surface charge densities of $\pm 3.0 \mu \mathrm { C } / \mathrm { m } ^ { 2 }$ , are separated by a distance of $1.0 \mathrm {~mm}$ with only air between them. How much energy is stored in these plates? $\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \mathrm { C } ^ { 2 } / \mathrm { N } \cdot \mathrm { m } ^ { 2 } \right)$

Point charges $+ 4.00 \mu \mathrm { C }$ and $+ 2.00 \mu \mathrm { C }$ are placed at the opposite corners of a rectangle as shown in the figure. If these charges are released and are free to move with no friction, what is the maximum amount of kinetic energy they will gain? $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 8.99 \times 10 ^ { 9 } \mathrm {~N} \right.$ . $m ^ { 2 } / C ^ { 2 }$ )

A capacitor has a voltage of $391 \mathrm {~V}$ applied across its plates, and then the voltage source is removed. What is the potential difference across its plates if the space between them is then filled with mica, having a dielectric constant of $5.4$ ?

If it takes $50 \mathrm {~J}$ of energy to move $10 \mathrm { C }$ of charge from point $A$ to point $B$ , what is the magnitude of the potential difference between points $A$ and $B$ ?

Three point charges are placed at the following points in a horizontal $x - y$ plane: $+ 4.0 \mu \mathrm { C }$ is at $( 0.00 \mathrm {~m} , 0.50 \mathrm {~m} ) , + 1.0 \mu \mathrm { C }$ is at $( 0.20 \mathrm {~m} , 0.00 \mathrm {~m} )$ , and $- 5.0 \times \mu \mathrm { C }$ is at $( 0.20 \mathrm {~m} , 0.50 \mathrm {~m} )$ . Calculate the electrical potential (relative to infinity) at the origin due to these three point charges. $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$

The square plates of a $5000 - p F$ parallel-plate capacitor measure $50 \mathrm {~mm}$ by $50 \mathrm {~mm}$ and are separated by a dielectric that is $0.23 \mathrm {~mm}$ thickand totally fills the region between the plates. The voltage rating (the maximum safe voltage) of the capacitor is $400 \mathrm {~V}$ . What is the maximum energy that can be stored in this capacitor without damaging it? $\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \mathrm { C } ^ { 2 } / \mathrm { N } \cdot \mathrm { m } ^ { 2 } \right.$ )

$\mathrm { A } + 5.0 - \mathrm { nC }$ charge is at the point $( 0.00 \mathrm {~m} , 0.00 \mathrm {~m} )$ and a $- 2.0 - \mathrm { nC }$ charge is at $( 3.0 \mathrm {~m} , 0.00 \mathrm {~m} )$ . What work is required to bring a $1.0 - \mathrm { nC }$ charge from very far away to point $( 0.00 \mathrm {~m} , 4.0 \mathrm {~m} )$ ? $\left( k = 1 / 4 \pi \varepsilon _ { 0 } \right.$ $= 9.0 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 }$ )

What the electric energy density at a point $1.0 \mathrm {~cm}$ from a proton? $\left( e = 1.6 \times 10 - 19 \mathrm { C } , \varepsilon _ { 0 } = \right.$ $\left. 8.85 \times 10 ^ { - 12 } \mathrm { C } ^ { 2 } / \mathrm { N } \cdot \mathrm { m } ^ { 2 } , k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$

A $6.0 - \mu \mathrm { F }$ air-filled capacitor is connected across a 100-V potential source (a battery). After the battery fully charges the capacitor, it is left connected and the capacitor is immersed in transformer oil, which has a dielectric constant of 4.5. How much additional charge flows from the battery onto the capacitor during this process?

Measurements show that it takes $0.30 \mathrm {~mJ}$ of work to move $5.0 - \mu \mathrm { C }$ of charge from one plate to another of a certain air-filled capacitor while the potential difference between these plates is kep constant. What is the potential difference between the plates of this capacitor?

A uniform electric field has the strength of $7.0 \mathrm {~N} / \mathrm { C }$ . What is the electric energy density of this field? $\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \mathrm { C } ^ { 2 } / \mathrm { N } \cdot \mathrm { m } ^ { 2 } \right)$