2. Neutron Star - Magnetic Field Maxwell's classical theory of electromagnetism is linear, meaning light does...

 

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2. Neutron Star - Magnetic Field Maxwell's classical theory of electromagnetism is linear, meaning light does not interact with light. However, in Quantum Electrodynamics (QED), photons can interact with each other, for example, via the Feynman diagram shown below. This diagram describes two photons interacting via the intermediate creation and annihilation of virtual electron-positron pairs. (The solid lines with arrows indicate electrons; reversing an arrow turns it into a positron.) (a) In order for such photon-photon interactions to occur, the photons must have an energy at least sufficient to create an electron-positron pair at rest. What is this mini- mum required photon energy, Ey, and the corresponding photon frequency, f? (Symbolic answers, not numerical.) (b) Suppose we have a uniform magnetic field of strength B in the 2-direction. If an electron is given an initial speed in the ry-plane, calculate the frequency, f, of its subsequent circular motion. Use non-relativistic physics and express your answer in terms of e (the magnitude of the charge on the electron), m (the mass of the electron), and B, and any numerical factors that arise. Note that an electron in circular motion is accelerating, and so will emit photons (at the same frequency as the circular motion). (c) Combine the previous two parts to determine the minimum strength of magnetic field, B, required for electrons (or positrons) moving in this magnetic field to emit virtual photons with sufficient energy to interact with each other. Let's call this critical value of the strength of the magnetic field BQED. Work out BQED both symbolically and numerically (in Tesla). (d) Convert the numerical value of the magnetic field strength, BQED, that you calculated in the previous part, into an equivalent mass density (a numerical answer in kg m-3) and compare this with the mass density of lead (11,400 kg m-³). (e) Suppose we start with an ordinary star with radius Rstar and surface magnetic field strength 1 Gauss (convert this to Tesla). Using the argument discussed in the notes and lectures, what would Rstar need to be in order that, if it collapsed to a neutron star of radius 13 km, its surface magnetic field strength would be equal to BQED? Express Rstar in terms of the radius of the Sun, Ro. The largest stars in the universe have a radius of about 1800R. Is it plausible that there exist neutron stars with a surface magnetic field strength on the order of BQED or larger? 2. Neutron Star - Magnetic Field Maxwell's classical theory of electromagnetism is linear, meaning light does not interact with light. However, in Quantum Electrodynamics (QED), photons can interact with each other, for example, via the Feynman diagram shown below. This diagram describes two photons interacting via the intermediate creation and annihilation of virtual electron-positron pairs. (The solid lines with arrows indicate electrons; reversing an arrow turns it into a positron.) (a) In order for such photon-photon interactions to occur, the photons must have an energy at least sufficient to create an electron-positron pair at rest. What is this mini- mum required photon energy, Ey, and the corresponding photon frequency, f? (Symbolic answers, not numerical.) (b) Suppose we have a uniform magnetic field of strength B in the 2-direction. If an electron is given an initial speed in the ry-plane, calculate the frequency, f, of its subsequent circular motion. Use non-relativistic physics and express your answer in terms of e (the magnitude of the charge on the electron), m (the mass of the electron), and B, and any numerical factors that arise. Note that an electron in circular motion is accelerating, and so will emit photons (at the same frequency as the circular motion). (c) Combine the previous two parts to determine the minimum strength of magnetic field, B, required for electrons (or positrons) moving in this magnetic field to emit virtual photons with sufficient energy to interact with each other. Let's call this critical value of the strength of the magnetic field BQED. Work out BQED both symbolically and numerically (in Tesla). (d) Convert the numerical value of the magnetic field strength, BQED, that you calculated in the previous part, into an equivalent mass density (a numerical answer in kg m-3) and compare this with the mass density of lead (11,400 kg m-³). (e) Suppose we start with an ordinary star with radius Rstar and surface magnetic field strength 1 Gauss (convert this to Tesla). Using the argument discussed in the notes and lectures, what would Rstar need to be in order that, if it collapsed to a neutron star of radius 13 km, its surface magnetic field strength would be equal to BQED? Express Rstar in terms of the radius of the Sun, Ro. The largest stars in the universe have a radius of about 1800R. Is it plausible that there exist neutron stars with a surface magnetic field strength on the order of BQED or larger?

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a In order for two photons to interact the photons must have an energy at least sufficient to create ... View the full answer