Electric Potential Energy Problems

The specific principles required are indicated in italics at the beginning of each problem.

1. Electric Potential Energy: While sitting in a restaurant with some friends, you notice that some "neon" signs are different in color than others. You know that these signs are essentially just gas sealed in a glass tube. The gas, when heated electrically, gives off light. One of your friends, who is an art major, and makes such signs as sculpture, tells you that the color of the light depends on which gas is in the tube. All "neon" signs are not made using neon gas. You know that the color of light tells you its energy. Red light is a lower energy than blue light. Since the light is given off by the atoms, which make up the gas, the different colors must depend on the structure of the different atoms of different gases. Suppose that atomic structure is as given by the Bohr theory which states that electrons are in uniform circular motion around a heavy, motionless nucleus in the center of the atom. This theory also states that the electrons are only allowed to have certain orbits. When an atom changes from one allowed orbit to another allowed orbit, it radiates light as required by the conservation of energy. Since only certain orbits are allowed, so the theory goes, only light of certain energies (colors) can be emitted. This seems to agree with the observations of your artist friend. You decide to test the theory by calculating the energy of light emitted by a simple atom when an electron makes a transition from one allowed orbit to another. You decide to consider hydrogen since you know it is the simplest atom with one electron and a nucleus consisting of one proton. You remember that the proton has a mass 2000 times that of an electron. When you get home you look in your textbook and find the electron mass is 9 x 10-31 kg and its charge is 1.6 x 10-19 C. The radius of the smallest allowed electron orbit for hydrogen is 0.5 x 10-10 meters, which determines the normal size of the atom. The next allowed orbit has a radius 4 times as large as the smallest orbit.
   
   
2. Electric Potential Energy: You have a great summer job working in a cancer research laboratory. Your team is trying to construct a gas laser that will give off light of an energy that will pass through the skin but be absorbed by cancer tissue. You know that an atom emits a photon (light) when an electron goes from a higher energy orbit to a lower energy orbit. Only certain orbits are allowed in a particular atom. To begin the process, you calculate the energy of photons emitted by a Helium ion in which the electron changes from an orbit with a radius of 0.30 nanometers to another orbit with a radius of 0.20 nanometers. A nanometer is 10-9 m. The helium nucleus consists of two protons and two neutrons.
   
   
3. Electric Potential Energy: Your job is to evaluate an electron gun designed to initiate an electron beam. The electrons have a 20 cm path from the heating element, which emits them to the end of the gun. This path is through a very good vacuum. For most applications, the electrons must reach the end of the gun with a speed of at least 107 m/s. After leaving the heating element, the electrons pass through a 5.0 mm diameter hole in the center of a 3.0 cm diameter charged circular disk. The disk's charge density is kept at 3.0 µC/m2. The heating element is a spherical electrode 0.10 mm in diameter that is kept at a very high charge of -0.10C. There is 1.0 cm between the heating element and the hole in the disk. Your first step is to determine if the electrons are going fast enough. Your boss has pointed that the hole in the disk is too large to ignore in your calculations. Using your physics text you find that the mass of the electron is 9.11 x 10-31 kg.
   
   
4. Electric Potential Energy - Gauss's Law: You have landed a summer job working with an Astrophysics group investigating the origin of high-energy particles in the galaxy. The group you are joining has just discovered a large spherical nebula with a radius 1.2 million km. The nebula consists of about 5 x 1010 hydrogen nuclei (protons) which appear to be uniformly distributed in the shape of a sphere. At the center of this sphere of positive charge is a very small neutron star. Your group had detected electrons emerging from the nebula. A friend of yours has a theory that the electrons are coming from the neutron star. To test that theory, she asks you to calculate the minimum speed that an electron would need to start from the neutron star and just make it to outside the nebula. From the inside cover of your trusty physics text you find that the charge of a proton (and an electron) is 1.6 x 10-19 C, the mass of the proton is 1.7 x 10-27 kg, and the mass of the electron is 9.1 x 10 -31 kg.
   
   
5. Electric Potential Energy, Gauss's Law: You are working in cooperation with the Public Health department to design an electrostatic trap for particles from auto emissions. The average particle enters the device and is exposed to ultraviolet radiation that knocks off electrons so that it has a charge of +3.0 x 10-8 C. This average particle is then moving at a speed of 900 m/s and is 15 cm from a very long negatively charged wire with a linear charge density of -8.0 x 10-6 C/m. The detector for the particle is located 7.0 cm from the wire. In order to design the proper kind of detector, your colleagues need to know the speed that an average emission particle will have if it hits the detector. They tell you that an average emission particle has a mass of 6.0 x 10-9 kg.
   
   
6. Electric Potential Energy, Heat Energy (Heat Capacity, Latent Heat): You are reading a newspaper report of a lightning strike in Jackson, Wyoming. Two men were sitting at a table outside a small cafe on a beautiful 30 oC day when a thunderstorm approached. Suddenly, a bolt of lightning struck a large aspen tree near their table. Needless to say, the men were very startled. One of the men remarked, "It just about scared the espresso out of me." They reported that when the bolt hit the tree and there was a loud hiss and a release of much steam from the tree. The lightning had boiled away some of the tree's sap. You are curious, and wonder how much water could be evaporated in this manner. So you study your physics book and make a few estimates and assumptions. You estimate that the electric potential difference between the tree and the thunderhead cloud was about 108 volts, and the amount of charge released by the bolt was about 50 Coulombs. You also assume that about 1% of the electrical energy was actually transferred into the sap, which is essentially water. The specific heat capacity of water is 4200 J/(kg oC) and its heat of vaporization is 2.3 x 106 J/kg.
   
   
7. Electric Potential Energy, Gravitational Force: NASA has asked your team of rocket scientists about the feasibility of a new satellite launcher that will save rocket fuel. NASA's idea is basically an electric slingshot that consists of 4 electrodes arranged in a horizontal square with sides of length d at a height h above the ground. The satellite is then placed on the ground aligned with the center of the square. A power supply will provide each of the four electrodes with a charge of +Q/4 and the satellite with a charge -Q. When the satellite is released from rest, it moves up and passes through the center of the square. At the instant it reaches the square's center, the power supply is turned off and the electrodes are grounded, giving them a zero electric charge. To test this idea, you decide to use energy considerations to calculate how big Q will have to be to get a 100 kg satellite to a sufficient orbit height. Assume that the satellite starts from 15 meters below the square of electrodes and that the sides of the square are each 5 meters. In your physics text you find the mass of the Earth to be 6.0 x 1024 kg.
   
   
8. Electric Potential Energy, Mechanical Energy: You have been able to get a part-time job in a University laboratory. The group is planning a set of experiments to study the forces between nuclei in order to understand the energy output of the Sun. To do this experiment, you shoot alpha particles from a Van de Graaf accelerator at a sheet of lead. The alpha particle is the nucleus of a helium atom and is made of 2 protons and 2 neutrons. The lead nucleus is made of 82 protons and 125 neutrons. The mass of the neutron is almost the same as the mass of a proton. To assure that you are actually studying the effects of the nuclear force, an alpha particle should come into contact with a lead nucleus. Assume that both the alpha particle and the lead nucleus have the shape of a sphere. The alpha particle has a radius of 1.0 x 10-13 cm and the lead nucleus has a radius 4 times larger. Your boss wants you to make two calculations:
   (a) What is the minimum speed of such an alpha particle if the lead nucleus is fixed at rest?
   (b) What is the potential difference between the two ends of the Van de Graaf accelerator if the alpha particle starts from rest at one end (from a bottle of helium gas)?