The response of polar substances (e.g., HCl, H2 O, etc) to applied electric fields can be descri

The response of polar substances (e.g., HCl, H2 O, etc) to...

The response of polar substances (e.g., $\mathrm{HCl}, \mathrm{H}_2 \mathrm{O}$, etc) to applied electric fields can be described in terms of a classical model which attributes to each molecule a permanent electric dipole moment of magnitude $p$. (a) Write down a general expression for the average macroscopic polarization $\bar{p}$ (dipole moment per unit volume) for a dilute system of $n$ molecules per unit volume at temperature $T$ in a uniform electric field $E$. (b) Calculate explicitly an approximate result for the average macroscopic polarization $\bar{p}$ at high temperatures $(K T \gg p E)$.

The response of polar substances (e.g., $\mathrm{HCl}, \mathrm{H}_2 \mathrm{O}$, etc) to applied electric fields can be described in terms of a classical model which attributes to each molecule a permanent electric dipole moment of magnitude $p$.
(a) Write down a general expression for the average macroscopic polarization $\bar{p}$ (dipole moment per unit volume) for a dilute system of $n$ molecules per unit volume at temperature $T$ in a uniform electric field $E$.
(b) Calculate explicitly an approximate result for the average macroscopic polarization $\bar{p}$ at high temperatures $(K T \gg p E)$.

Key Concepts

Electric Dipole Moment

An electric dipole moment is a measure of the separation of positive and negative charges within a system. In the context of molecular systems, it describes how the charge distribution within a molecule gives rise to a polarity, which can interact with an external electric field. The magnitude and orientation of the dipole moment are crucial for determining how the molecule responds to external influences.

Macroscopic Polarization

Macroscopic polarization refers to the net dipole moment per unit volume in a material, arising from the collective alignment of microscopic dipoles. It is a key quantity in describing how materials respond to electric fields, influencing their dielectric properties. This polarization is calculated as an ensemble average over the dipole moments of all molecules present, weighted by their orientation distribution.

Statistical Mechanics and Boltzmann Distribution

Statistical mechanics provides the framework to compute average properties of systems with many particles by considering the distribution of states according to thermal energy. In this context, the Boltzmann distribution is used to weight the contribution of each molecular dipole based on its energy in the external electric field. This statistical averaging leads to expressions that describe how the macroscopic polarization emerges from individual molecular contributions.

High Temperature Approximation

At high temperatures (or equivalently when the thermal energy is much larger than the energy associated with aligning a dipole in the applied electric field), the response of the dipoles can be linearized. This approximation simplifies the expression for the ensemble-averaged polarization, often leading to a result where the macroscopic polarization is directly proportional to the applied electric field, reflecting a weak alignment of the dipoles.

Transcript

00:01 So here in this question, the series expansion of cot h of x is given by this 1 over x plus x over 3 minus s -q, o over 45 plus and continuous like that.

00:13 So here, alpha is given by p over e multiplied by a cut -h of x minus x to the power minus 1.

00:23 Then we have this alpha when break it down since we are substituting the cut -h into this particular equation.

00:30 We have x over 3 minus s cube over 45 and a continuous like that when we neglect the higher powers of x we are going to get this we are going to get alpha to be equal to p over e x all over 3 and we have here p over e p e p e all over 3 k t and that gives us p squared all over 3k t all over 3 k t all over 3 k t but that's not the end from here we can show that and this is not an accuracy of one part in 15 times x squared x to the power minus 2 so we are going to we can show or that this is to an accuracy of one part in let me change the color here in 15 x to the power minus 2 yes so so the series becomes the series, the series becomes x over 3, the series becomes x over 3 minus x cube all over 45 plus infinity.

01:52 So that's what we have for the end.

01:54 This actually is approximated to x.

01:58 This is approximated to x over 3, x over 3.

02:02 X over 3 okay so when we do this approximation the error we have committed is given by we can calculate our error i mean by doing our approximation there will be some errors in there and so that error is calculated by using this formula where we have the error to be equal to the approximated i'm going to use apk r o x to present the approximated so we have the approximated value you have the presumed value minus the actual value ac t represents actual the actual value okay so we can now do some substitutions in here so we have x over 3 minus x over 3 x cube all over 45 plus and the so from here we have the error to be equal to x cubed all over 45 minus and a continuous minus and a continues so that is the arrow so approximately approximately the arrow is giving by let me realize that's well.

03:47 Now, the error is given by xq approximately, we have the error to be s -quib over 45...

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