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Exercise:
A dipole consists of two po charges pm q at a distance d. abclist abc Derive a formal expression for the field strength on the line connecting the two charges as a function of the position x measured from the center between the charges. What is the asymptotic behaviour for x gg d? abc Derive a formal expression for the field strength on the perpicular bisector of the segment connecting the two charges as a function of the position y on the bisector. What is the asymptotic behaviour for ygg d? abclist

Solution:
abclist abc The partial field vectors vecE_+ and vecE_- are displayed in the figure. center includegraphicswidth.mm#image_path:dipolfield-along-x# center The net field strength E is given by E E_- - E_+ k_C qleftfracx-d/^-fracx+d/^right k_C q fracx+d/^-x-d/^leftx^-d/^right^ k_C q fracx^+x d/+d/^-x^+x d/-d/^leftx^-d/^right^ k_C fracq x dleftx^-d/^right^ For xgg d all mands but the leading term x^ are negligible in the denominator. The field strength is approximately equal to E &approx k_C fracq x dx^ k_C fracq dx^ abc The figure shows how the net electric field vector E deps on the partial field vectors E_+ and E_-. center includegraphicswidthmm#image_path:dipolfield-along-y# center Using similar triangles it follows fracEE_+ fracdr rightarrow E E_+fracdr k_C fracqr^fracdrk_C fracq dr^ k_C fracq dleftd/^+y^right^/ For ygg d the term in d can be neglected in the denominator. The field strength is approximately equal to E &approx k_Cfracq dy^^/ k_Cfracq dy^ abclist It turns out that the field strength of a dipole decreases like /r^ for rgg d. This is different from the behaviour of a single po charge where the field drops off like /r^ inverse square law.
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Exercise:
A dipole consists of two po charges pm q at a distance d. abclist abc Derive a formal expression for the field strength on the line connecting the two charges as a function of the position x measured from the center between the charges. What is the asymptotic behaviour for x gg d? abc Derive a formal expression for the field strength on the perpicular bisector of the segment connecting the two charges as a function of the position y on the bisector. What is the asymptotic behaviour for ygg d? abclist

Solution:
abclist abc The partial field vectors vecE_+ and vecE_- are displayed in the figure. center includegraphicswidth.mm#image_path:dipolfield-along-x# center The net field strength E is given by E E_- - E_+ k_C qleftfracx-d/^-fracx+d/^right k_C q fracx+d/^-x-d/^leftx^-d/^right^ k_C q fracx^+x d/+d/^-x^+x d/-d/^leftx^-d/^right^ k_C fracq x dleftx^-d/^right^ For xgg d all mands but the leading term x^ are negligible in the denominator. The field strength is approximately equal to E &approx k_C fracq x dx^ k_C fracq dx^ abc The figure shows how the net electric field vector E deps on the partial field vectors E_+ and E_-. center includegraphicswidthmm#image_path:dipolfield-along-y# center Using similar triangles it follows fracEE_+ fracdr rightarrow E E_+fracdr k_C fracqr^fracdrk_C fracq dr^ k_C fracq dleftd/^+y^right^/ For ygg d the term in d can be neglected in the denominator. The field strength is approximately equal to E &approx k_Cfracq dy^^/ k_Cfracq dy^ abclist It turns out that the field strength of a dipole decreases like /r^ for rgg d. This is different from the behaviour of a single po charge where the field drops off like /r^ inverse square law.
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Attributes & Decorations
Branches
Electrostatics
Tags
dipole, electric field, electrostatic
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Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
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Decoration
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