Hello, Though I can derive the Taylor series of 1/(1-x), I cannot prove that the x needs to be x << 1. Could you please help me with that? In addtion, how do I derive the formula for the electric field as indicated in (b)? I guess that the x is d/z... but how I can connect this info with the (a) to apply 1/(1-x) Taylor series? f(m)(a)(x - a)",(a) The Taylor series representation of a function (x) is given byƐn=0n!where f("(a) is the nth derivative of f at the point a. By choosing a=0, show that theTaylor expansion of the function 1/(1-x) is 1 +x + x² + x³ +(b) Hence, show that the field magnitude at a point of distance z from the centre of anfor x<<1.qdelectric dipole is given by E =2n5023as the one shown below:E = E+) - E-)%3D4πεο r-)4πεο r-)4 TE(z – 3d)?4 TE,(z + ¿d)?-For very small d/z, we Taylor expand the above and ignorehigher order terms:For very small4περΖ24TEOZ2 (1 +2)4περζ2(6-1)[ignore higher terms of Taylor expansions]qd2περΖ32περ23where p = qd = electric dipole moment||

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Answered: f(m)(a) (x - a)", (a) The Taylor series…

College Physics

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ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Hello,

Though I can derive the Taylor series of 1/(1-x), I cannot prove that the x needs to be x << 1. Could you please help me with that?

In addtion, how do I derive the formula for the electric field as indicated in (b)? I guess that the x is d/z... but how I can connect this info with the (a) to apply 1/(1-x) Taylor series?

f(m)(a)
(x - a)",
(a) The Taylor series representation of a function (x) is given byƐn=0
n!
where f("(a) is the nth derivative of f at the point a. By choosing a=0, show that the
Taylor expansion of the function 1/(1-x) is 1 +x + x² + x³ +
(b) Hence, show that the field magnitude at a point of distance z from the centre of an
for x<<1.
qd
electric dipole is given by E =
2n5023
as the one shown below:
E = E+) - E-)
%3D
4πεο r-)
4πεο r-)
4 TE(z – 3d)?
4 TE,(z + ¿d)?
-
For very small d/z, we Taylor expand the above and ignore
higher order terms:
For very small
4περΖ2
4TEOZ2 (1 +2)
4περζ2
(6-1)
[ignore higher terms of Taylor expansions]
qd
2περΖ3
2περ23
where p = qd = electric dipole moment
||

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