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 ||
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 ||
Classical Dynamics of Particles and Systems
5th Edition
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Stephen T. Thornton, Jerry B. Marion
Chapter12: Coupled Oscillations
Section: Chapter Questions
Problem 12.19P
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Question
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
||](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ef9f81c-8832-475e-81d0-37cf0d43f515%2Fd767c4d4-f135-4c23-b142-9d8a479150db%2Fbwxedvy.png&w=3840&q=75)
Transcribed Image Text: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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(a) Derivatives of the given function
VIEWExpand the given function as a Taylor series
VIEW(b) The magnitude of electric field at a point from the centre of the dipole
VIEWThe derivatives of the first term
VIEWExpand the above function as a Taylor series
VIEWUse Taylor series expansion to write the magnitude of electric field
VIEWRewrite the above equation in terms of dipole moment
VIEW
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