If we manage to find the maximum value of \(|f^{(n+1)}(c)|\) for \(c\) in the interval between \(a\) and \(x\text{,}\) then we can find the exact error in the Taylor polynomial approximation.
2.Approximating The Electric Field due to an Electric Dipole.
Consider an electric dipole consisting of two charges, \(+q\) and \(-q\text{,}\) separated by a distance \(d\text{.}\) The electric field at a point \(P\) located at a distance \(r\) from the positive charge along the axis of the dipole is given by:
\begin{equation*}
E = \frac{1}{4\pi\epsilon_0} \left( \frac{q}{r^2} - \frac{q}{(r+d)^2} \right)
\end{equation*}
Figure2.23.An electric dipole: charges \(+q\) and \(-q\) separated by a distance \(d\text{,}\) with the field point \(P\) a distance \(r\) from the positive charge along the axis.
Use the Taylor series to approximate the electric field \(E\) at point \(P\) for \(d \ll r\text{.}\) Show that the leading term in the approximation is proportional to \(d/r^3\text{.}\)
To approximate the electric field \(E\) at point \(P\) for \(d \ll r\text{,}\) we can use the Taylor series expansion for the function \(f(x) = \frac{1}{(r+x)^2}\) around \(x=0\text{.}\)
Figure2.24.The axial field of a dipole, \(E = \frac{1}{4\pi\epsilon_0}
\left(\frac{q}{r^2} - \frac{q}{(r+d)^2}\right)\text{,}\) compared with its leading Taylor term for \(d \ll r\text{.}\) The leading term \(\frac{2qd}{4\pi\epsilon_0 r^3} \propto \frac{d}{r^3}\) overshoots slightly at small \(r\) but converges to the exact field as \(r\) grows.
A dipole is often treated as a point dipole by replacing the exact axial field
\begin{equation*}
E = \frac{1}{4\pi\epsilon_0}\left( \frac{q}{r^2} - \frac{q}{(r+d)^2} \right)
\end{equation*}
with its leading-order Taylor term for \(d \ll r\text{,}\)
\begin{equation*}
E \approx \frac{1}{4\pi\epsilon_0}\cdot\frac{2qd}{r^3}.
\end{equation*}
Suppose this approximation must agree with the exact field to within a relative error of \(1\%\text{.}\) If the field point \(P\) is \(r = 3.0\ \text{cm}\) from the positive charge, how small must the charge separation \(d\) be for the point-dipole approximation to be valid?
The dominant error comes from the next term in the expansion. Using \(E = \dfrac{q}{4\pi\epsilon_0}\left( \dfrac{2d}{r^3} -
\dfrac{3d^2}{r^4} + \cdots \right)\text{,}\) form the relative error of the leading term and keep only the largest contribution.
the leading term is \(E_{\text{lead}} = \dfrac{q}{4\pi\epsilon_0}\cdot
\dfrac{2d}{r^3}\text{,}\) and the first neglected term is \(\dfrac{q}{4\pi\epsilon_0}\cdot\dfrac{3d^2}{r^4}\text{.}\) The relative error of the approximation is therefore
Notice that \(q\) and \(\epsilon_0\) cancel, so the relative error depends only on the ratio \(d/r\text{.}\) Requiring this to be at most \(1\% = 0.01\) gives
\begin{equation*}
\frac{3}{2}\cdot\frac{d}{r} \le 0.01
\qquad\Longrightarrow\qquad
d \le \frac{2}{3}(0.01)\,r.
\end{equation*}
So the point-dipole approximation is accurate to \(1\%\) only when the charges are separated by less than about \(0.2\ \text{mm}\) at this distance—consistent with Figure 2.24, where the two curves visibly merge as \(r\) grows relative to \(d\text{.}\)
