Problems

Consider a charge $Q_1(+5.0\mu \text{C})$ fixed at a site with another charge $Q_2$ (charge $+3.0\mu \text{C}$, mass $6.0\mu \text{g})$ moving in the neighboring space. (a) Evaluate the potential energy of $Q_2$ when it is 4.0 cm from $Q_1.$ (b) If $Q_2$ starts from rest from a point 4.0 cm from $Q_1,$ what will be its speed when it is 8.0 cm from $Q_1$? (Note: $Q_1$ is held fixed in its place.)

To form a hydrogen atom, a proton is fixed at a point and an electron is brought from far away to a distance of $0.529\times {10}^{\mathrm{-10}}\text{m,}$ the average distance between proton and electron in a hydrogen atom. How much work is done?

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 $1.67\times {10}^{\mathrm{-27}}\text{kg}\text{.}$

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

The electric field strength between two parallel conducting plates separated by 4.00 cm is $7.50\times {10}^4\text{V/m}$. (a) What is the potential difference between the plates? (b) The plate with the lowest potential is taken to be zero volts. What is the potential 1.00 cm from that plate and 3.00 cm from the other?

Two parallel conducting plates are separated by 10.0 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 cm from the zero volt plate (and 2.00 cm from the other) is 450 V? (b) What is the voltage between the plates?

An electron is to be accelerated in a uniform electric field having a strength of $2.00\times {10}^6\text{V/m}\text{.}$ (a) What energy in keV is given to the electron if it is accelerated through 0.400 m? (b) Over what distance would it have to be accelerated to increase its energy by 50.0 GeV?

The electric field in a region is pointed away from the z-axis and the magnitude depends upon the distance s from the axis. The magnitude of the electric field is given as $E=\frac{\alpha }{s}$ where $\alpha$ is a constant. Find the potential difference between points $P_1\text{and}{P}_2$, explicitly stating the path over which you conduct the integration for the line integral.

A 0.500-cm-diameter plastic sphere, used in a static electricity demonstration, has a uniformly distributed 40.0-pC charge on its surface. What is the potential near its surface?

If the potential due to a point charge is $5.00\times {10}^2\text{V}$ at a distance of 15.0 m, what are the sign and magnitude of the charge?

A research Van de Graaff generator has a 2.00-m-diameter metal sphere with a charge of 5.00 mC on it. Assume the potential energy is zero at a reference point infinitely far away from the Van de Graaff. (a) What is the potential near its surface? (b) At what distance from its center is the potential 1.00 MV? (c) An oxygen atom with three missing electrons is released near the Van de Graaff generator. What is its kinetic energy in MeV when the atom is at the distance found in part b?

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

Two charges $\mathrm{–2.0}\mu \text{C}\text{and}\text{+}2.0\mu \text{C}$ are separated by 4.0 cm on the z-axis symmetrically about origin, with the positive one uppermost. Two space points of interest $P_1\text{and}{P}_2$ are located 3.0 cm and 30 cm from origin at an angle $30\text{°}$ with respect to the z-axis. Evaluate electric potentials at $P_1\text{and}{P}_2$ in two ways: (a) Using the exact formula for point charges, and (b) using the approximate dipole potential formula.

Throughout a region, equipotential surfaces are given by $z=\text{constant}$. The surfaces are equally spaced with $V=100\text{V}$ for $z=0.00\text{m,}V=200\text{V}$ for $z=0.50\text{m,}V=300\text{V}$ for $z=1.00\text{m.}$ What is the electric field in this region?

Calculate the electric field of an infinite line charge, throughout space.

A very large sheet of insulating material has had an excess of electrons placed on it to a surface charge density of $\mathrm{–3.00}{\text{nC/m}}^2$. (a) As the distance from the sheet increases, does the potential increase or decrease? Can you explain why without any calculations? Does the location of your reference point matter? (b) What is the shape of the equipotential surfaces? (c) What is the spacing between surfaces that differ by 1.00 V?

Two large charged plates of charge density $\text{±}30\mu {\text{C/m}}^2$ face each other at a separation of 5.0 mm. (a) Find the electric potential everywhere. (b) An electron is released from rest at the negative plate; with what speed will it strike the positive plate?

Two parallel plates 10 cm on a side are given equal and opposite charges of magnitude $5.0\times {10}^{\mathrm{-9}}\text{C}\text{.}$ The plates are 1.5 mm apart. What is the potential difference between the plates?

Concentric conducting spherical shells carry charges Q and –Q, respectively. The inner shell has negligible thickness. What is the potential difference between the shells?

A solid cylindrical conductor of radius a is surrounded by a concentric cylindrical shell of inner radius b. The solid cylinder and the shell carry charges Q and –Q, respectively. Assuming that the length L of both conductors is much greater than a or b, what is the potential difference between the two conductors?

(a) What is the direction and magnitude of an electric field that supports the weight of a free electron near the surface of Earth? (b) Discuss what the small value for this field implies regarding the relative strength of the gravitational and electrostatic forces.

In a Geiger counter, a thin metallic wire at the center of a metallic tube is kept at a high voltage with respect to the metal tube. Ionizing radiation entering the tube knocks electrons off gas molecules or sides of the tube that then accelerate towards the center wire, knocking off even more electrons. This process eventually leads to an avalanche that is detectable as a current. A particular Geiger counter has a tube of radius R and the inner wire of radius a is at a potential of $V_0$ volts with respect to the outer metal tube. Consider a point P at a distance s from the center wire and far away from the ends. (a) Find a formula for the electric field at a point P inside using the infinite wire approximation. (b) Find a formula for the electric potential at a point P inside. (c) Use $V_0=900\text{V},a=3.00\text{mm,}R=2.00\text{cm,}$ and find the value of the electric field at a point 1.00 cm from the center.

To form a helium atom, an alpha particle that contains two protons and two neutrons is fixed at one location, and two electrons are brought in from far away, one at a time. The first electron is placed at $0.600\times {10}^{\mathrm{-10}}\text{m}$ from the alpha particle and held there while the second electron is brought to $0.600\times {10}^{\mathrm{-10}}\text{m}$ from the alpha particle on the other side from the first electron. See the final configuration below. (a) How much work is done in each step? (b) What is the electrostatic energy of the alpha particle and two electrons in the final configuration?

The probability of fusion occurring 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 $1.00\times {10}^{\mathrm{-12}}\text{m}\text{.}$ (b) At what temperature will atoms of a gas have an average kinetic energy equal to this needed electrical potential energy?

An electron enters a region between two large parallel plates made of aluminum separated by a distance of 2.0 cm and kept at a potential difference of 200 V. The electron enters through a small hole in the negative plate and moves toward the positive plate. At the time the electron is near the negative plate, its speed is $4.0\times {10}^5\text{m/s}\text{.}$ Assume the electric field between the plates to be uniform, and find the speed of electron at (a) 0.10 cm, (b) 0.50 cm, (c) 1.0 cm, and (d) 1.5 cm from the negative plate, and (e) immediately before it hits the positive plate.

(a) Will the electric field strength between two parallel conducting plates exceed the breakdown strength of dry air, which is $3.00\times {10}^6\text{V/m}$, if the plates are separated by 2.00 mm and a potential difference of $5.0\times {10}^3\text{V}$ is applied? (b) How close together can the plates be with this applied voltage?

A double charged ion is accelerated to an energy of 32.0 keV by the electric field between two parallel conducting plates separated by 2.00 cm. What is the electric field strength between the plates?

A lightning bolt strikes a tree, moving 20.0 C of charge through a potential difference of $1.00\times {10}^2\text{MV.}$ (a) What energy was dissipated? (b) What mass of water could be raised from $15\text{°}\text{C}$ 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.

(a) A sphere has a surface uniformly charged with 1.00 C. At what distance from its center is the potential 5.00 MV? (b) What does your answer imply about the practical aspect of isolating such a large charge?

In one of the classic nuclear physics experiments at the beginning of the twentieth 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 MeV, how close to the gold nucleus (79 protons) could it come before being deflected?