Exam 17: Electric Potential

Each plate of a parallel-plate air-filled capacitor has an area of $0.0040 \mathrm {~m} ^ { 2 }$ , and the separation of the plates is $0.020 \mathrm {~mm}$ . An electric field of $8.2 \times 10 ^ { 6 } \mathrm {~N} / \mathrm { C }$ is present between the plates. What is the energy density in the region between 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 with horizontal plates has a plate separation of $5.0 \mathrm {~cm}$ . If the potential difference between the plates is $2000 \mathrm {~V}$ , with the top plate at the higher potential, what are the magnitude and direction of the electric field between the plates?

A 12.0-V battery (potential source) is connected across a $6.00 - \mu \mathrm { F }$ air-filled capacitor. (a) How much energy can be stored this way? (b) How much excess charge is on each plate of the capacitor?

Two ideal parallel-plate capacitors are identical in every respect except that one has twice the plate area of the other. If the smaller capacitor has capacitance $C$ , the larger one has capacitance

Four point charges of magnitude $6.00 \mu \mathrm { C }$ and are at the corners of a square $2.00 \mathrm {~m}$ on each side. Two of the charges are positive, and two are negative. What is the electric potential at the center of this square, relative to infinity, due to these charges? $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 8.99 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$

A parallel-plate capacitor consists of two parallel, square plates having dimensions $1.0 \mathrm {~cm}$ by $1.0$ $\mathrm { cm }$ . The plates are separated by $1.0 \mathrm {~mm}$ , and the space between them is filled with Teflon, which has a dielectric constant of $2.1$ . What is the capacitance of this capacitor? $\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \mathrm { C } ^ { 2 } / \mathrm { N } \right.$ . $\mathrm { m } ^ { 2 }$ )

Each plate of an ideal air-filled parallel-plate capacitor has an area of $0.0020 \mathrm {~m} ^ { 2 }$ , and the separation of the plates is $0.090 \mathrm {~mm}$ . An electric field of $2.1 \times 10 ^ { 6 } \mathrm {~N} / \mathrm { C }$ is present between the plates. What is the surface charge density on the plates? $\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \mathrm { C } ^ { 2 } / \mathrm { N } \cdot \mathrm { m } ^ { 2 } \right)$

Four charged particles (two having a charge $+ Q$ and two having a charge $- Q$ ) are arranged in the $x y$ -plane, as shown in the figure. These particles are all equidistant from the origin. The electric potential (relative to infinity) at point $\mathrm { P }$ on the $z$ -axis due to these particles, is

The figure shows an arrangement of two particles each having charge $Q = - 6.8 \mathrm { nC }$ and each separated by $5.0 \mathrm {~mm}$ from a proton. If the two particles are held fixed at their locations and the proton is set into motion as shown, what is the minimum speed the proton needs to totally escape from these particles? $\left( m _ { \text {proton } } = 1.67 \times 10 ^ { - 27 } \mathrm {~kg} , e = 1.60 \times 10 ^ { - 19 } \mathrm { C } , k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times 10 ^ { 9 } \mathrm {~N} \right.$ . $\mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 }$ )

An ideal air-filled parallel plate capacitor with plate a separation of $4.0 \mathrm {~cm}$ has a plate area of $0.040$ $\mathrm { m } ^ { 2 }$ . What is the capacitance of this capacitor with air between these plates? $\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \right.$ $\mathrm { C } ^ { 2 } / \mathrm { N } \cdot \mathrm { m } ^ { 2 }$ )

A very small $4.8 - \mathrm { g }$ particle carrying a charge of $+ 9.9 \mu \mathrm { C }$ is fired with an initial speed of $8.0 \mathrm {~m} / \mathrm { s }$ directly toward a second small $7.8 - \mathrm { g }$ particle carrying a charge of $+ 5.2 \mu \mathrm { C }$ . The second particle is held fixed throughout this process. If these particles are initially very far apart, what is the closest they get to each other? $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$

$\mathrm { A } + 5.0 - \mu \mathrm { C }$ point charge is $12 \mathrm {~cm}$ from a $- 5.0 - \mu \mathrm { C }$ point charge. What is the magnitude of the electric field they produce at the point on the line connecting them where their electric potential (relative to infinity) is zero? $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$

A parallel-plate air-filled capacitor is made from two plates that are $0.070 \mathrm {~m}$ on each side and spaced $9.2 \mathrm {~mm}$ apart. What must the potential difference between the plates be to produce an energy density of $0.062 \mathrm {~J} / \mathrm { m } 3$ in the region between them? $\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \right.$ $\left. \mathrm { C } ^ { 2 } / \mathrm { N } \cdot \mathrm { m } ^ { 2 } \right)$

Two $+ 6.0 - \mu \mathrm { C }$ charges are placed at two of the vertices of an equilateral triangle having sides $2.0 \mathrm {~m}$ Iong. What is the electric potential at the third vertex, relative to infinity, due to these charges? $( k =$ $1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 }$ )

After a proton with an initial speed of $1.50 \times 10 ^ { 5 } \mathrm {~m} / \mathrm { s }$ has increased its speed by accelerating through a potential difference of $0.100 \mathrm { kV }$ , what is its final speed? $\left( e = 1.60 \times 10 ^ { - 19 } \mathrm { C } , m _ { \text {proton } } = \right.$ $1.67 \times 10 ^ { - 27 } \mathrm {~kg}$ )

An air-filled 20- $\mu \mathrm { F }$ capacitor has a charge of $60 \mu \mathrm { C }$ on its plates. How much energy is stored in this capacitor?

The figure shows a group of three particles, all of which have charge $Q = 8.8 \mathrm { nC }$ . How much work did it take to assemble this group of charges if they all started out extremely far from each other? $( k$ $\left. = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times 10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 } \right)$ $3.0 \mathrm {~cm}$

At a distance $d$ from a point charge $Q$ , the energy density in its electric field is $u$ . If we now go to a distance $d / 2$ from the charge, what is the energy density at the new location?

An electric dipole with $\pm 5.0 \mu \mathrm { C }$ point charges is positioned so that the positive charge is $1.0 \mathrm {~mm}$ to the right of the origin and the negative charge is at the origin. How much work does it take to bring a $3.0 - \mu \mathrm { C }$ point charge from very far away to the point $x = 3.0 \mathrm {~mm} , y = 0.0 \mathrm {~mm}$ ? $\left( k = 1 / 4 \pi \varepsilon _ { 0 } = 9.0 \times \right.$ $10 ^ { 9 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { C } ^ { 2 }$ )

Two isolated copper plates, each of area $0.40 \mathrm {~m} ^ { 2 }$ , carry opposite charges of magnitude $7.08 \times 10 ^ { - 10 }$ C. They are placed opposite each other in parallel alignment, with a spacing of $4.0 \mathrm {~cm}$ between them. What is the potential difference between the plates? $\left( \varepsilon _ { 0 } = 8.85 \times 10 ^ { - 12 } \mathrm { C } ^ { 2 } / \mathrm { N } \cdot \mathrm { m } ^ { 2 } \right)$