The 𝐇2^+ Ion. The 𝐇2^+ ion is composed of two protons, each...

The $\mathbf{H}_{2}^{+}$ Ion. The $\mathbf{H}_{2}^{+}$ ion is composed of two protons, each of charge $+e=1.60 \times 10^{-19} \mathrm{C},$ and an electron of charge $-e$ and mass $9.11 \times 10^{-31} \mathrm{kg} .$ The separation between the pro- tons is $1.07 \times 10^{-10} \mathrm{m}$ . The protons and the electron may be treated as point charges. (a) Suppose the electron is located at the point midway between the two protons. What is the potential energy of the interaction between the electron and the two protons? (Do not include the potential energy due to the interaction between the two protons.) (b) Suppose the electron in part (a) has a velocity of magnitude $1.50 \times 10^{6} \mathrm{m} / \mathrm{s}$ in a direction along the perpendicular bisector of the line connecting the two protons. How far from the point midway between the two protons can the electron move? Because the masses of the protons are much greater than the clectron mass, the motions of the protons are very slow and can be ignored. (Note: A realistic description of the electron motion requires the use of quantum mechanics, not Newtonian mechanics.)

Key Concepts

Conservation of Energy

Conservation of energy is a fundamental principle stating that the total energy in an isolated system remains constant over time. In the context of electrostatic and kinetic energy interactions, this principle allows one to equate kinetic energy and changes in potential energy to track the motion of charged particles, providing insights into how far a particle will travel under given conditions.

Superposition Principle

The superposition principle in electrostatics states that the total potential energy at a point due to several point charges is the algebraic sum of the potential energies from each individual charge. This principle simplifies the analysis of complex systems by allowing each interaction to be considered separately and then summed to find the overall potential energy.

Coulomb's Law

Coulomb's Law describes the force between two point charges, stating that the magnitude of the force is proportional to the product of the charges and inversely proportional to the square of the distance between them. This law is fundamental in evaluating both the forces and the potential energies in systems involving electrostatic interactions, such as in the calculation of the energy between charged particles.

Electrostatic Potential Energy

Electrostatic potential energy quantifies the work done in assembling a system of charges from infinity. It can be calculated between any pair of point charges using Coulomb's Law and is additive by applying the superposition principle. This concept is crucial in analyzing interactions in systems like ions or molecules where multiple charge interactions occur.

Transcript

00:01 Hello friends here it is given the hydrogen molecule ion h2 plus ion is composed of two protons each of charge this an electron of charge minus e and this the separation between the two protons is here it is a positive charge plus e plus e this separation r is given one 0 .07 into 10 to the power of minus 10 meter and electron mass of electron is 9 .11 10 to the power of minus 31 k .g.

01:20 And charge is 1 .6 into 10 to the power minus e.

01:25 So this is the electron here.

01:29 The proton is a point charge.

01:38 We have to calculate potential energy of interaction.

01:44 Between electrons and two protons.

01:53 In second part we have to calculate.

01:59 The electron has a velocity of magnitude in the direction perpendicular to vice sector.

02:06 The velocity of electron is given 1 .5 10 to the power 6 meter per second perpendicular to wide sector in this direction may not.

02:29 How far from the point between the the two proton can electron move then we have to find the distance it can move now let us see here first we will find the solution of a potential energy between the two so here you will get potential energy of interaction between the electron and proton is 1 upon 4 pi epsilon not minus 2 e square upon r r will be say this is r -d -s.

03:17 So r -d -s is this one and r -d -s is this one.

03:22 Midway.

03:24 So substitute the value 2 into 8 .98, 810 to the power minus 9, 1 .6 into 10 to the power minus 19.

03:40 And distance is 1 .07 -half 1 .07 by 2 into 10 to the power minus 10.

03:50 So on solving this potential energy you will get minus 8 .6 into 10 to the power minus 18 joules.

04:09 Just a moment...

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