Textbook Question
A parallel-plate capacitor with a 1.0 mm plate separation is charged to 75 V. With what kinetic energy, in eV, must a proton be launched from the negative plate if it is just barely able to reach the positive plate?
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Ch 37: The Foundations of Modern Physics
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 37, Problem 23b
Consider the gold isotope 197Au. The gold nucleus has a diameter of 14.0 fm. What is the density of matter in a gold nucleus?
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Determine the volume of the gold nucleus, assuming it is spherical. The formula for the volume of a sphere is: , where is the radius of the nucleus. The radius is half the diameter, so .
Convert the radius from femtometers (fm) to meters, as the SI unit for density is kg/m³. Use the conversion factor: .
Calculate the mass of the gold nucleus. The mass of the nucleus is approximately equal to the mass of its nucleons (protons and neutrons). For gold-197, the mass is approximately , where is the atomic mass unit. Convert this to kilograms using the conversion factor: .
Calculate the density of the gold nucleus using the formula: , where is the mass of the nucleus and is its volume.
Substitute the values for mass and volume into the density formula, ensuring all units are consistent (mass in kilograms and volume in cubic meters). Simplify the expression to find the density of matter in the gold nucleus.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Nuclear Density
Nuclear density refers to the mass per unit volume of a nucleus. It is typically expressed in units of kilograms per cubic meter (kg/m³) and is remarkably high due to the compact arrangement of nucleons (protons and neutrons) within the nucleus. For most atomic nuclei, the density is approximately 2.3 x 10^17 kg/m³, which is significantly greater than the density of ordinary matter.
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Intro to Density
Volume of a Sphere
The volume of a sphere is calculated using the formula V = (4/3)πr³, where r is the radius of the sphere. In the context of a nucleus, this formula allows us to determine the volume based on its diameter. Understanding how to calculate the volume is essential for determining the density of the nucleus, as density is defined as mass divided by volume.
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05:21Volume Thermal Expansion
Isotopes and Atomic Mass
Isotopes are variants of a chemical element that have the same number of protons but different numbers of neutrons, resulting in different atomic masses. For example, the gold isotope 197Au has 79 protons and 118 neutrons. The atomic mass of an isotope is crucial for calculating the total mass of the nucleus, which is necessary for determining its density.
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Related Practice
Textbook Question
The factor γ appears in many relativistic expressions. A value γ = 1.01 implies that relativity changes the Newtonian values by approximately 1% and that relativistic effects can no longer be ignored. At what kinetic energy, in MeV, is γ = 1.01 for (a) an electron, (b) a proton, and (c) an alpha particle?
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Textbook Question
How many electrons, protons, and neutrons are contained in the following atoms or ions: (a) ¹⁰B, (b) ¹³N⁺, and (c) ¹⁷O⁺⁺⁺?
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Textbook Question
What is the velocity, as a fraction of c, of an electron with 2.0 GeV total energy? Hint: This problem uses relativity.
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Textbook Question
The fission process n + ²³⁵U → ²³⁶U → ¹⁴⁴Ba + ⁸⁹Kr + 3n converts 0.185 u of mass into the kinetic energy of the fission products. What is the total kinetic energy in MeV?
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Textbook Question
Identify the isotope that is 11 times as heavy as ¹²C and has 18 times as many protons as ⁶Li . Give your answer in the form ᴬS, where S is the symbol for the element. See Appendix C: Atomic and Nuclear Data.
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