Electric Potential and Electric Field

A \({\bf{0}}.{\bf{500}}{\rm{ }}{\bf{cm}}\) diameter plastic sphere, used in a static electricity demonstration, has a uniformly distributed \({\bf{40}}.{\bf{0}}{\rm{ }}{\bf{pC}}\) charge on its surface. What is the potential near its surface?

How far from a \(1.00{\rm{ }}\mu C\) point charge will the potential be \(100{\rm{ }}V\)? At what distance will it be \(2.00 \times {10^2}{\rm{ }}V\)?

(a) A sphere has a surface uniformly charged with \(1.00{\rm{ }}C\). At what distance from its centre is the potential \(5.00{\rm{ }}MV\)? 

(b) What does your answer imply about the practical aspect of isolating such a large charge?

What are the sign and magnitude of a point charge that produces a potential of \(-2.00{\rm{ }}V\) at a distance of \(1.00{\rm{ mm}}\)?

In nuclear fission, a nucleus splits roughly in half. (a) What is the potential \(2.00 \times {10^{ - 14}}{\rm{ }}m\) from a fragment that has \(46\) protons in it? (b) What is the potential energy in \(MeV\) of a similarly charged fragment at this distance?

A research Van de Graaff generator has a \(2.00 - m\)-diameter metal sphere with a charge of \(5.00{\rm{ }}mC\) on it. (a) What is the potential near its surface? (b) At what distance from its centre is the potential \(1.00{\rm{ }}MV\)? (c) An oxygen atom with three missing electrons is released near the Van de Graaff generator. What is its energy in \(MeV\) at this distance?

An electrostatic paint sprayer has a \(0.200{\rm{ }}m\) diameter metal sphere at a potential of \(25.0{\rm{ }}kV\) that repels paint droplets onto a grounded object. (a) What charge is on the sphere? (b) What charge must a \(0.100{\rm{ }}mg\) drop of paint have to arrive at the object with a speed of \(10.0{\rm{ }}m/s\)?

In one of the classic nuclear physics experiments at the beginning of the \(20{\rm{ }}th\) century, an alpha particle was accelerated toward a gold nucleus, and its path was substantially deflected by the Coulomb interaction. If the energy of the doubly charged alpha nucleus was \(5.00{\rm{ }}MeV\), how close to the gold nucleus (\(79\) protons) could it come before being deflected?

(a) What is the potential between two points situated \(10{\rm{ }}cm\) and \(20{\rm{ }}cm\) from a \(3.0{\rm{ }}\mu C\) point charge? (b) To what location should the point at \(20{\rm{ }}cm\) be moved to increase this potential difference by a factor of two?

(a) Sketch the equipotential lines near a point charge \( + q\) . Indicate the direction of increasing potential. (b) Do the same for a point charge \( - 3q\) .

Sketch the equipotential lines for the two equal positive charges shown in Figure . Indicate the direction of increasing potential.

Sketch the equipotential lines a long distance from the charges shown in Figure \({\rm{19}}{\rm{.28}}\). Indicate the direction of increasing potential.

Sketch the equipotential lines in the vicinity of two opposite charges, where the negative charge is three times as great in magnitude as the positive. See Figure \({\rm{19}}{\rm{.28}}\) for a similar situation. Indicate the direction of increasing potential.

Sketch the equipotential lines surrounding the two conducting plates shown in Figure \({\rm{19}}{\rm{.30}}\), given the top plate is positive and the bottom plate has an equal amount of negative charge. Be certain to indicate the distribution of charge on the plates. Is the field strongest where the plates are closest? Why should it be?

What is the potential $\mathbf{0}\mathbf{.}\mathbf{530}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{10}}\mathbf{ }\mathbf{m}$ from a proton (the average distance between the proton and electron in a hydrogen atom)?

If the potential due to a point charge is $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{2}}\mathbf{ }\mathbf{V}$ at a distance of 15.0 m, what are the sign and magnitude of the charge?

Sketch the equipotential lines in the vicinity of the negatively charged conductor in Figure 19.29. How will these equipotentials look a long distance from the object?

(a) Sketch the electric field lines in the vicinity of the charged insulator in Figure 19.31. Note its non-uniform charge distribution. 

(b) Sketch equipotential lines surrounding the insulator. Indicate the direction of increasing potential.

The naturally occurring charge on the ground on a fine day out in the open country is \( - 1.00n{\rm{ }}C/{m^2}\). 

(a) What is the electric field relative to ground at a height of \(3.00{\rm{ }}m\)? 

(b) Calculate the electric potential at this height. 

(c) Sketch electric field and equipotential lines for this scenario.

Find the charge stored when \(5.50{\rm{ }}V\) is applied to an \(8.00{\rm{ }}\mu F\) capacitor.

Calculate the voltage applied to a  \({\bf{2}}{\bf{.00}}\;{\bf{\mu F}}\) capacitor when it holds \({\bf{3}}{\bf{.10}}\;{\bf{\mu C}}\) of charge.

What voltage must be applied to an \({\bf{8}}{\bf{.00 nF}}\) capacitor to store \({\bf{0}}{\bf{.160}}\;{\bf{mC}}\) of charge?

What capacitance is needed to store \(3.00{\bf{ }}\mu C\) of charge at a voltage of \(120\;V\)?

What is the capacitance of a large Van de Graaff generator's terminal, given that it stores \(8.00{\rm{ }}mC\)of charge at a voltage of \(12.0{\rm{ }}MV\)?

Find the capacitance of a parallel plate capacitor having plates of area \(5.00\;{m^2}\) that are separated by \(0.100\;mm\)of Teflon.

(a) A certain parallel plate capacitor has plates of area \(4.00\;{m^2}\), separated by \(0.01\;mm\) of nylon, and stores \(0.170\;C\) of charge. What is the applied voltage? (b) What is unreasonable about this result? (c) Which assumptions are responsible or inconsistent?

In open heart surgery, a much smaller amount of energy will defibrillate the heart. 

(a) What voltage is applied to the \(8.00{\rm{ }}\mu F\)capacitor of a heart defibrillator that stores \(40.0{\rm{ }}J\) of energy? 

(b) Find the amount of stored charge.

The lesser electric ray (Narcine bancroftii) maintains an incredible charge on its head and a charge equal in magnitude but opposite in sign on its tail (Figure 19.32). 

(a) Sketch the equipotential lines surrounding the ray. 

(b) Sketch the equipotential when the ray is near a ship with a conducting surface. 

(c) How could this charge distribution be of use to the ray?

What charge is stored in a $\mathbf{180}\mathbf{\mu F}$ capacitor when 120V is applied to it?

What charge is stored in the capacitor in Example 19.8?

(a) What is the capacitance of a parallel plate capacitor having plates of area 1.50 m²  that are separated by 0.0200 mm of neoprene rubber? (b) What charge does it hold when 9.00 V is applied to it?

A prankster applies 450V  to an $\mathbf{80}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\mu F}$  capacitor and then tosses it to an unsuspecting victim. The victim's finger is burned by the discharge of the capacitor through 0.200 g of flesh. What is the temperature increase of the flesh? Is it reasonable to assume no phase change?

Find the total capacitance of the combination of capacitors in Figure 19.33.

Suppose you want a capacitor bank with a total capacitance of 0.750F and you possess numerous 1.50mF capacitors. What is the smallest number you could hook together to achieve your goal, and how would you connect them?

What total capacitances can you make by connecting a $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\mu F}$ and an $\mathbf{8}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\mu F}$capacitor together?

Find the total capacitance of the combination of capacitors shown in Figure 19.34 .

Find the total capacitance of the combination of capacitors shown below

(a) An $\mathbf{8}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\mu F}$capacitor is connected in parallel to another capacitor, producing a total capacitance of $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\mu F}$. What is the capacitance of the second capacitor? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

(a) What is the energy stored in the $\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\mu F}$ capacitor of a heart defibrillator charged to $\mathbf{9}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{ }\mathbf{V}$? (b) Find the amount of stored charge.

A $\mathbf{165}\mathbf{ }\mathbf{\mu F}$capacitor is used in conjunction with a motor. How much energy is stored in it when 119 V is applied?

Suppose you have a \(9.00\;V\) battery, a \(2.00{\rm{ }}\mu F\) capacitor, and a  \(7.40{\rm{ }}\mu F\)capacitor. 

(a) Find the charge and energy stored if the capacitors are connected to the battery in series.

 (b) Do the same for a parallel connection.

A nervous physicist worries that the two metal shelves of his wood frame bookcase might obtain a high voltage if charged by static electricity, perhaps produced by friction. 

(a) What is the capacitance of the empty shelves if they have area \(1.00 \times {10^2}\;{m^2}\) and are \(0.200\;m\) apart? 

(b) What is the voltage between them if opposite charges of magnitude \(2.00nc\) are placed on them? 

(c) To show that this voltage poses a small hazard, calculate the energy stored.

Show that for a given dielectric material the maximum energy a parallel plate capacitor can store is directly proportional to the volume of dielectric (Volume=A.d. Note that the applied voltage is limited by the dielectric strength.

Construct Your Own Problem 

Consider a heart defibrillator similar to that discussed in Example 19.11 . Construct a problem in which you examine the charge stored in the capacitor of a defibrillator as a function of stored energy. Among the things to be considered are the applied voltage and whether it should vary with energy to be delivered, the range of energies involved, and the capacitance of the defibrillator. You may also wish to consider the much smaller energy needed for defibrillation during open-heart surgery as a variation on this problem.

Question: Unreasonable Results

(a) On a particular day, it takes \(9.60 \times {10^3}\;J\) of electric energy to start a truck's engine. Calculate the capacitance of a capacitor that could store that amount of energy at \(12.0\;V\).

(b) What is unreasonable about this result?

(c) Which assumptions are responsible?