Electric Potential and Electric Field

A bare helium nucleus has two positive charges and a mass of \(6.64 \times {10^{ - 27}}{\rm{ }}kg\) . 

(a) Calculate its kinetic energy in joules at \(2.00\% \) of the speed of light. 

(b) What is this in electron volts? 

(c) What voltage would be needed to obtain this energy?

Integrated Concepts 

Singly charged gas ions are accelerated from rest through a voltage of \(13.0{\rm{ }}V\). At what temperature will the average kinetic energy of gas molecules be the same as that given these ions?

Integrated Concepts 

The temperature near the centre of the Sun is thought to be \(15{\rm{ }}million\) degrees Celsius \(\left( {1.5 \times {{10}^7}^o{\rm{ }}C} \right)\). Through what voltage must a singly charged ion be accelerated to have the same energy as the average kinetic energy of ions at this temperature?

The electric field strength between two parallel conducting plates separated by\(4.00\;cm\) is\(7.50 \times {10^4}\;V/m\). 

(a) What is the potential difference between the plates? 

(b) The plate with the lowest potential is taken to be at zero volts. What is the potential\(1.00\;cm\) from that plate (and\(3.00\;cm\) from the other)?

(a) Will the electric field strength between two parallel conducting plates exceed the breakdown strength for air \(\left( {3.0 \times {{10}^6}\;V/m} \right)\)/ if the plates are separated by \(2.00\;mm\) and a potential difference of \(5.0 \times {10^3}\;V\) is applied? 

(b) How close together can the plates be with this applied voltage?

Give the reason why a dielectric material increases capacitance compared with what it would be with air between the plates of a capacitor. What is the independent reason that a dielectric material also allows a greater voltage to be applied to a capacitor? (The dielectric thus increases \(C\) and permits a greater \(V\).

Voltage is the common word for potential difference. Which term is more descriptive, voltage or potential difference?

If the voltage between two points is zero, can a test charge be moved between them with zero net work being done? Can this necessarily be done without exerting a force? Explain.

What is the relationship between voltage and energy? More precisely, what is the relationship between potential difference and electric potential energy?

Voltages are always measured between two points. Why?

How are units of volts and electron volts related? How do they differ?

Which methods of radiation protection were used in the device shown in the first photo in Figure 32.35? Which were used in the situation shown in the second photo?

Figure 32.35 (a)

Figure 32.35 (b)

 (a) This x-ray fluorescence machine is one of the thousands used in shoe stores to produce images of feet as a check on the fit of shoes. They are unshielded and remain on as long as the feet are in them, producing doses much greater than medical images. Children were fascinated with them. These machines were used in shoe stores until laws preventing such unwarranted radiation exposure were enacted in the 1950s. (credit: Andrew Kuchling ) (b) Now that we know the effects of exposure to radioactive material, safety is a priority. (credit: U.S. Navy)

Discuss how potential difference and electric field strength are related. Give an example.

What is the strength of the electric field in a region where the electric potential is constant?

Will a negative charge, initially at rest, move toward higher or lower potential? Explain why.

In what region of space is the potential due to a uniformly charged sphere the same as that of a point charge? In what region does it differ from that of a point charge?

In what region of space is the potential due to a uniformly charged sphere the same as that of a point charge? In what region does it differ from that of a point charge?

Can the potential of a non-uniformly charged sphere be the same as that of a point charge? Explain.

What is an equipotential line? What is an equipotential surface?

Explain in your own words why equipotential lines and surfaces must be perpendicular to electric field lines.

Can different equipotential lines cross? Explain.

Does the capacitance of a device depend on the applied voltage? What about the charge stored in it?

Use the characteristics of the Coulomb force to explain why capacitance should be proportional to the plate area of a capacitor. Similarly, explain why capacitance should be inversely proportional to the separation between plates.

How does the polar character of water molecules help to explain water's relatively large dielectric constant? (Figure 19.19)

Sparks will occur between the plates of an air-filled capacitor at lower voltage when the air is humid than when dry. Explain why, considering the polar character of water molecules.

Water has a large dielectric constant, but it is rarely used in capacitors. Explain why.

Figure \({\rm{19}}{\rm{.28}}\) shows the electric field lines near two charges \(q1\) and \(q2\) , the first having a magnitude four times that of the second. Sketch the equipotential lines for these two charges, and indicate the direction of increasing potential.

Membranes in living cells, including those in humans, are characterized by a separation of charge across the membrane. Effectively, the membranes are thus charged capacitors with important functions related to the potential difference across the membrane. Is energy required to separate these charges in living membranes and, if so, is its source the metabolization of food energy or some other source?

If you wish to store a large amount of energy in a capacitor bank, would you connect capacitors in series or parallel? Explain

How does the energy contained in a charged capacitor change when a dielectric is inserted, assuming the capacitor is isolated and its charge is constant? Does this imply that work was done?

What happens to the energy stored in a capacitor connected to a battery when a dielectric is inserted? Was work done in the process?

Find the ratio of speeds of an electron and a negative hydrogen ion (one having an extra electron) accelerated through the same voltage, assuming non-relativistic final speeds. Take the mass of the hydrogen ion to be \(1.67 \times {10^{ - 27}}{\rm{ }}kg\).

An evacuated tube uses an accelerating voltage of \(40{\rm{ }}kV\) to accelerate electrons to hit a copper plate and produce \(x\) rays. Non-relativistically, what would be the maximum speed of these electrons?

Integrated Concepts 

(a) What is the average power output of a heart defibrillator that dissipates 400J of energy in 10.0 ms? 

(b) Considering the high-power output, why doesn’t the defibrillator produce serious burns?

Integrated Concepts 

A lightning bolt strikes a tree, moving 20.0 C of charge through a potential difference of 1.00 x 10² MV. 

(a) What energy was dissipated? 

(b) What mass of water could be raised from $\mathbf{15}\mathbf{°}\mathbf{ }\mathbf{C}$ to the boiling point and then boiled by this energy? 

(c) Discuss the damage that could be caused to the tree by the expansion of the boiling steam.

Integrated Concepts

A \(12.0\;V\)battery-operated bottle warmer heats \(50.0\;g\)of glass, \(2.50 \times {10^2}{\rm{ }}g\)of baby formula, and \(2.00 \times {10^2}\;g\)of aluminium from \({20.0^\circ }C\) to \({90.0^\circ }C\).

(a) How much charge is moved by the battery? 

(b) How many electrons per second flow if it takes \(5.00\)min to warm the formula? (Hint: Assume that the specific heat of baby formula is about the same as the specific heat of water.)

A battery-operated car utilizes a 12.0 V system. Find the charge the batteries must be able to move in order to accelerate the 750 kg car from rest to 25.0 m/s, make it climb a 2.00 x 10² m high hill, and then cause it to travel at a constant 25.0 m/s by exerting a 5.00 x 10² N force for an hour.

Membrane walls of living cells have surprisingly large electric fields across them due to separation of ions. (Membranes are discussed in some detail in Nerve Conduction—Electrocardiograms.) What is the voltage across an \(8.00{\rm{ }}nm\)–thick membrane if the electric field strength across it is \(5.50{\rm{ }}MV/m\)? You may assume a uniform electric field.

Fusion probability is greatly enhanced when appropriate nuclei are brought close together, but mutual Coulomb repulsion must be overcome. This can be done using the kinetic energy of high-temperature gas ions or by accelerating the nuclei toward one another. 

(a) Calculate the potential energy of two singly charged nuclei separated by 1.00 x 10^-12 m by finding the voltage of one at that distance and multiplying by the charge of the other. 

(b) At what temperature will atoms of a gas have an average kinetic energy equal to this needed electrical potential energy?

Unreasonable Results

(a) Find the voltage near a \(10.0\;cm\) diameter metal sphere that has \(8.00{\rm{ }}C\) of excess positive charge on it. 

(b) What is unreasonable about this result? 

(c) Which assumptions are responsible?

Consider a battery used to supply energy to a cellular phone. Construct a problem in which you determine the energy that must be supplied by the battery, and then calculate the amount of charge it must be able to move in order to supply this energy. Among the things to be considered are the energy needs and battery voltage. You may need to look ahead to interpret manufacturer's battery ratings in ampere-hours as energy in joules.

Show that units of V/m and N/m for electric field strength are indeed equivalent.

What is the strength of the electric field between two parallel conducting plates separated by 1.00 cm and having a potential difference (voltage) between them of $\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathbf{ }\mathbf{V}$?

How far apart are two conducting plates that have an electric field strength of \(4.50 \times {10^3}\;V/m\) between them, if their potential difference is \(15.0kV\)?

How far apart are two conducting plates that have an electric field strength of \(4.50 \times {10^3}\;V/m\) between them, if their potential difference is \(15.0kV\)?

The voltage across a membrane forming a cell wall is \(80.0{\rm{ }}mV\) and the membrane is \(9.00{\rm{ }}nm\) thick. What is the electric field strength? (The value is surprisingly large, but correct. Membranes are discussed in Capacitors and Dielectrics and Nerve Conduction—Electrocardiograms.) You may assume a uniform electric field.

Two parallel conducting plates are separated by \(10.0{\rm{ }}cm\), and one of them is taken to be at zero volts. (a) What is the electric field strength between them, if the potential \(8.00{\rm{ }}cm\) from the zero-volt plate (and \(2.00{\rm{ }}cm\) from the other) is \(450{\rm{ }}V\)? (b) What is the voltage between the plates?

Find the maximum potential difference between two parallel conducting plates separated by \(0.500{\rm{ }}cm\) of air, given the maximum sustainable electric field strength in air to be \(3.0 \times {10^6}{\rm{ }}V/m\).

A doubly charged ion is accelerated to an energy of \(32.0{\rm{ }}keV\) by the electric field between two parallel conducting plates separated by \(2.00{\rm{ }}cm\). What is the electric field strength between the plates?

An electron is to be accelerated in a uniform electric field having a strength of \({\bf{2}}.{\bf{00}} \times {\bf{1}}{{\bf{0}}^6}{\rm{ }}{\bf{V}}/{\bf{m}}\). (a) What energy in \(keV\) is given to the electron if it is accelerated through \({\bf{0}}.{\bf{400}}{\rm{ }}{\bf{m}}\)? (b) Over what distance would it have to be accelerated to increase its energy by \({\bf{50}}.{\bf{0}}{\rm{ }}{\bf{GeV}}\)?