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A helium ion (He++) that comes within about 10 fm of the center of the nucleus of an atom in the sample may induce a nuclear reaction instead of simply scattering. Imagine a helium ion with a kinetic energy of 3.0 MeV heading straight toward an atom at rest in the sample. Assume that the atom stays fixed. What minimum charge can the nucleus of the atom have such that the helium ion gets no closer than 10 fm from the center of the atomic nucleus? (1 fm = 1 × 1015 m, and e is the magnitude of the charge of an electron or a proton.) (a) 2e; (b) 11e; (c) 20e; (d) 22e.

Short Answer

Expert verified
The minimum charge is 22e. Option (d).

Step by step solution

01

Understanding the Problem

You need to find the minimum charge (in terms of the elementary charge e) of an atomic nucleus such that a helium ion with a kinetic energy of 3.0 MeV does not come closer than 10 fm. A helium ion He++ has a charge of 2e.
02

Calculate the Coulomb Potential Energy

The potential energy U between the helium ion and the atomic nucleus, given their charges and separation distance, is given by the formula: U=k(Ze)(2e)r where k=8.99×109 N m2/C2 is the Coulomb constant, Z is the charge of the nucleus, e is the elementary charge 1.6×1019 C, and r=10 fm=10×1015 m.
03

Equate Potential Energy to Kinetic Energy

To find the minimum charge Z, equate the calculated potential energy U to the given kinetic energy. Convert the kinetic energy from MeV to joules: 3.0 MeV=3.0×1.6×1013 J=4.8×1013 J. Set U=4.8×1013 J.
04

Solve for the Charge Z

Substitute the known values and solve for Z:(8.99×109)(Z1.6×1019)(21.6×1019)10×1015=4.8×1013 Simplifying gives Z22.
05

Select the Minimum Charge Option

From the given options: (a) 2e, (b) 11e, (c) 20e, (d) 22e, the calculated Z=22 matches the final option (d).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coulomb potential energy
Coulomb potential energy is a crucial concept in nuclear physics and relates to the energy between two charged particles due to their electric fields. It is calculated using the formula:
  • U=k(q1)(q2)r
Here, U represents the potential energy, k is Coulomb's constant (8.99×109 N m2/C2), q1 and q2 are the charges of the particles, and r is the distance separating them.
This energy is important when studying nuclear reactions, especially when charged particles such as ions approach each other. The interaction energy increases as the particles get closer, making it essential for determining the conditions under which nuclear reactions occur. Understanding Coulomb potential energy helps predict whether a reaction will be elastic (bouncing apart) or inelastic (leading to nuclear reaction or fusion).
In our exercise, we need to calculate the Coulomb potential energy between a helium ion and an atomic nucleus to ensure that the ion does not come closer than a specific distance. This calculation will allow us to understand how much energy is needed to overcome the repulsive force and result in a potential nuclear reaction.
Helium ions
Helium ions, also known as alpha particles, are composed of two protons and two neutrons, making them quite massive and positively charged. They are often denoted as He++ due to their double positive charge, which means their charge is equal to 2e, where e is the elementary charge 1.6×1019 C.
These ions are utilized in various nuclear physics applications, particularly in experiments investigating nuclear reactions and radiation interactions. When helium ions are introduced to other atoms, the interactions can lead to different scenarios, depending on the energy levels and conditions, such as scattering or triggering a reaction.
In this specific exercise, a helium ion approaches an atomic nucleus. The understanding of the interactions is pivotal to knowing how close the ion can approach without initiating a nuclear reaction. This distance is heavily influenced by the ion's kinetic energy and the surrounding potential energy field, determined by the atomic charges.
Nuclear reactions
Nuclear reactions are processes where two nuclei or nuclear particles collide, leading to a new formation of particles or nuclei. These reactions release or absorb considerable amounts of energy due to changes in the nuclei's composition.
There are various types of nuclear reactions, with the most prominent being:
  • Fission - where a heavy nucleus splits into smaller nuclei.
  • Fusion - where two light nuclei combine to form a heavier nucleus.
  • Collision - can be elastic, leading to scattering, or inelastic, causing changes within the nuclei.
Helium ions are often used to study such reactions due to their straightforward structure, and significant positive charge, making them excellent candidates for initiating nuclear events when interacting with a target nucleus.
In our exercise, the focus is on preventing the helium ion from coming closer than a certain distance (10 fm) to another nucleus, to avoid a nuclear reaction, which could potentially alter the nucleus's structure or release energy.
Elementary charge
The elementary charge is a fundamental constant in physics, symbolized as e, and its value is 1.6×1019 C. It represents the smallest unit of electric charge, found in electrons and protons.
In calculations concerning electric forces, fields, and potential energies, the elementary charge serves as the standard unit for expressing charge equally for both positive and negative particles. This means that understanding and correctly using e is crucial for dealing with any electron or proton-related interactions in physics.
Our exercise involves
  • Helium ion (He++), which has a charge of 2e
  • Nucleus charge assumed as Ze,
where Z represents an integer indicating the number of elemental charges in the nucleus.
In context, applying the elementary charge helps calculate the force interactions and potential energy of ions approaching each other, providing a clear understanding of whether conditions will permit a nuclear reaction or lead to scattering. The charge interactions outlined in the exercise are fundamental to predicting how close particles may approach each other in nuclear physics settings.

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Most popular questions from this chapter

A small metal sphere, carrying a net charge of q1=2.80 μC, is held in a stationary position by insulating supports. A second small metal sphere, with a net charge of q2=7.80 μC and mass 1.50 g, is projected toward q1. When the two spheres are 0.800 m apart, q2, is moving toward q1 with speed 22.0 m/s (Fig. E23.5). Assume that the two spheres can be treated as point charges. You can ignore the force of gravity. (a) What is the speed of q2 when the spheres are 0.400 m apart? (b) How close does q2 get to q1?

An infinitely long line of charge has linear charge density 5.00 ×1012 C/m. A proton (mass 1.67 ×1027 kg, charge +1.60 ×1019 C) is 18.0 cm from the line and moving directly toward the line at 3.50 ×103 m/s. (a) Calculate the proton's initial kinetic energy. (b) How close does the proton get to the line of charge?

A point charge q1=+2.40 μC is held stationary at the origin. A second point charge q2=4.30 μC moves from the point x= 0.150 m, y= 0 to the point x= 0.250 m, y= 0.250 m. How much work is done by the electric force on q2?

When radium-226 decays radioactively, it emits an alpha particle (the nucleus of helium), and the end product is radon-222. We can model this decay by thinking of the radium-226 as consisting of an alpha particle emitted from the surface of the spherically symmetric radon-222 nucleus, and we can treat the alpha particle as a point charge. The energy of the alpha particle has been measured in the laboratory and has been found to be 4.79 MeV when the alpha particle is essentially infinitely far from the nucleus. Since radon is much heavier than the alpha particle, we can assume that there is no appreciable recoil of the radon after the decay. The radon nucleus contains 86 protons, while the alpha particle has 2 protons and the radium nucleus has 88 protons. (a) What was the electric potential energy of the alpharadon combination just before the decay, in MeV and in joules? (b) Use your result from part (a) to calculate the radius of the radon nucleus.

An insulating spherical shell with inner radius 25.0 cm and outer radius 60.0 cm carries a charge of +150.0 μC uniformly distributed over its outer surface. Point a is at the center of the shell, point b is on the inner surface, and point c is on the outer surface. (a) What will a voltmeter read if it is connected between the following points: (i) a and b; (ii) b and c; (iii) c and infinity; (iv) a and c? (b) Which is at higher potential: (i) a or b; (ii) b or c; (iii) a or c? (c) Which, if any, of the answers would change sign if the charge were 150 μC?

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