Exploring Binomial Expansion in Electric Dipole Fields

In summary, the authors of Resnick and Halliday's book use a binomial expansion to find the electric field of a dipole when x is larger than d. This is done by assuming that the term (d/2x)^2 is negligible compared to the rest of the equation. The simplified equation without using binomial expansion is actually off by a constant factor.
  • #1
manenbu
103
0
I'm learning the subject of electric fields from Resnick and Halliday's book, and they have an equation for the field of the dipole:
[tex]
E = \frac{1}{4\pi\epsilon_0}\frac{p}{x^3} \left[1+\left(\frac{d}{2x}\right)^2\right]^{-3/2}
[/tex]
Their next step is to find out what happens when x is larger than d, so they use a binomial expansion. Why to do that?
Why not just assume that [itex]\left(\frac{d}{2x}\right)^2[/itex] is equal to zero so the entire thing simplifies to:
[tex]
E = \frac{1}{4\pi\epsilon_0}\frac{p}{x^3}
[/tex]
Which is the same result as using binomial expansion?
 
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  • #2
Well, the result that you see afterwards (for a "pure" dipole) is simply the first order term of the binomial expansion. The other terms still exist, d/2x didn't simply go to zero, it's just assumed to be negligible compared to the term you've listed. It actually seems to me that the equation you listed is off by a constant factor, but if it's published I guess it is okay and I just don't know the context.
 

FAQ: Exploring Binomial Expansion in Electric Dipole Fields

What is the concept of binomial expansion in dipole?

Binomial expansion in dipole refers to the mathematical process of expanding a binomial expression into a series of terms, where each term has a coefficient and a variable raised to a power. In the context of dipoles, this expansion is used to calculate the electric potential and electric field at different points around a dipole.

How is binomial expansion used in dipole calculations?

In dipole calculations, binomial expansion is used to approximate the electric potential and electric field at different points around a dipole. By expanding the binomial expression, we can calculate the values of these parameters at different distances and angles from the dipole, which helps in understanding the behavior of dipoles in electric fields.

What are the key terms involved in binomial expansion in dipole?

The key terms involved in binomial expansion in dipole include the binomial expression, which consists of two terms, the coefficient, which is the number in front of the variable, and the variable, which is raised to a power. Other terms include the distance and angle between the dipole and the point of interest, and the electric potential and electric field values.

Can binomial expansion be used for any type of dipole?

Yes, binomial expansion can be used for all types of dipoles, including electric and magnetic dipoles. The formula for binomial expansion remains the same, but the values of the coefficients and variables may differ depending on the type of dipole and the specific scenario being analyzed.

Are there any limitations to using binomial expansion in dipole calculations?

One limitation of using binomial expansion in dipole calculations is that it is an approximation and may not give exact values. This is because it involves a finite number of terms, whereas the actual values may require an infinite number of terms for accurate calculation. Additionally, binomial expansion is most accurate for small distances and angles, and may become less accurate for larger values.

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