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Calculations of Electric Potential I01:15

Calculations of Electric Potential I

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Consider a ring of radius R with a uniform charge density λ. What will the electric potential be at point M, which is located on the axis of the ring at a distance x from the center of the ring?
The ring is divided into infinitesimal small arcs such that point M is equidistant from all the arcs. Here, the cylindrical coordinate system is used to calculate the electric potential at point M. A general element of the arc between angles θ and θ + dθ is of the length Rdθ and...
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Calculations of Electric Potential II01:27

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An electric dipole is a system of two equal but opposite charges, separated by a fixed distance. This system is used to model many real-world systems, including atomic and molecular interactions. One of these systems is the water molecule, but only under certain circumstances. These circumstances are met inside a microwave oven, where electric fields with alternating directions make the water molecules change orientation. This vibration is equivalent to heat at the molecular level.
Consider a...
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Potential-Energy Criterion for Equilibrium01:16

Potential-Energy Criterion for Equilibrium

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Potential energy or potential function plays an essential role in determining the stability of a mechanical system. If a system is subjected to both gravitational and elastic forces, the potential function of the system can be expressed as the algebraic sum of gravitational and elastic potential energy. If the system is in equilibrium and is displaced by a small amount, then the work done on the system equals the negative of the change in the system's potential energy from the initial to the...
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Thermodynamic Potentials01:26

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Thermodynamic potentials are state functions that are extremely useful in analyzing a thermodynamic system. They have dimensions of energy. The four important thermodynamic potentials are internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy. These thermodynamic potentials can be expressed using two of the following variables: pressure, volume, temperature, and entropy. These two variables are expressed as the rate of change of the thermodynamic potential with respect to other...
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Magnetic Vector Potential01:15

Magnetic Vector Potential

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In electrostatics, the electric field can be written as the negative gradient of the potential. In magnetostatics, the zero divergence of the magnetic field ensures that the magnetic field can be expressed as the curl of a vector potential. This potential is known as the magnetic vector potential.
Consider an ideal solenoid with n turns per unit length and radius R. If I is the current through the solenoid, the magnetic field inside the solenoid is expressed as the product of vacuum...
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Potential Energy00:52

Potential Energy

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The energy stored by a structure and location of matter in space is called potential energy. For instance, raising a kettlebell changes its spatial location and increases its potential energy. Similarly, a stretched rubber band contains potential energy which, under certain conditions, can be converted into other forms of energy, such as kinetic energy.
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ϕ(2) or not ϕ(2): testing the simplest inflationary potential.

Paolo Creminelli1, Diana López Nacir2, Marko Simonović3

  • 1Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy and Institute for Advanced Study, Princeton, New Jersey 08540, USA.

Physical Review Letters
|July 5, 2014
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Summary

The simplest inflationary model, a quadratic potential, serves as a benchmark. Future constraints can distinguish it from other models and limit deviations in the speed of sound and multi-field contributions.

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Area of Science:

  • Cosmology
  • Inflationary theory
  • Particle physics

Background:

  • The simplest inflationary model, V=1/2m(2)ϕ(2), is the benchmark for future cosmological constraints.
  • Deviations from this minimal scenario can be parametrized by a specific quantity that vanishes for a quadratic potential.

Purpose of the Study:

  • To explore the implications of future constraints on inflationary models.
  • To distinguish between a quadratic potential and a pseudo-Nambu-Goldstone boson model.
  • To set limits on the deviation from unity of the speed of sound and the contribution of a second field to perturbations.

Main Methods:

  • Analysis of a specific quantity related to the spectral index (n(s)) and tensor-to-scalar ratio (r).
  • Calculation of bounds on physical parameters based on future observational uncertainties.
  • Discussion of the impact of non-Gaussianity in light of recent BICEP2 results.

Main Results:

  • The quantity (n(s)-1)+r/4+11(n(s)-1)(2)/24 vanishes for a quadratic potential (up to cubic slow-roll corrections).
  • Future constraints can differentiate a quadratic potential from a pseudo-Nambu-Goldstone boson model (f≲30Mpl).
  • Limits are set on |c(s)-1|≲3×10(-2) (Λ≳2×10(16) GeV) and second field contributions ≲6×10(-2).

Conclusions:

  • The uncertainty in the spectral index is the limiting factor for these bounds.
  • The error on the number of e-folds (ΔN≃0.4) implies an error on reheating temperature (ΔT(rh)/T(rh)≃1.2).
  • The relevance of non-Gaussianity is discussed in the context of BICEP2 findings.