Problem 25 - Essential Calculus Early Transcendentals

Question

An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are \(q\) and \(-q\) and are located at a distance \(d\) from each other, then the electric field \(E\) at the point \(P\) in the figure is $$E=\frac{q}{D^{2}}-\frac{q}{(D+d)^{2}}$$ By expanding this expression for \(E\) as a series in powers of \(d / D,\) show that \(E\) is approximately proportional to 1\(/ D^{3}\) when \(P\) is far away from the dipole.

Step-by-Step Solution

Answer

The electric field \( E \) is approximately \( \frac{2qd}{D^3} \) when \( P \) is far away from the dipole.

Step 1: Rewrite the Electric Field Expression

We start with the given expression for the electric field: \( E = \frac{q}{D^2} - \frac{q}{(D+d)^2} \). The goal is to expand this expression for \( E \) when \( P \) is far away from the dipole, meaning \( d \ll D \).

Step 2: Use Binomial Series Expansion

Consider the second term \( \frac{1}{(D+d)^2} \). We can use the binomial expansion for \( (1+\frac{d}{D})^{-2} \). The binomial series is: \( (1+x)^{-n} \approx 1 - nx + \frac{n(n+1)}{2}x^2 - \ldots \) for \(|x| < 1\). Here, \( x = \frac{d}{D} \) and \( n = 2 \).

Step 3: Apply the Binomial Expansion

Applying the binomial expansion to the expression \((D+d)^{-2}\), we get: \[\frac{1}{(D+d)^2} \approx \frac{1}{D^2} \left( 1 - 2\frac{d}{D} + 3\left(\frac{d}{D}\right)^2 - \ldots \right)\]

Step 4: Substitute the Expansion into the Electric Field Expression

Substitute the expansion from Step 3 back into the original expression for \(E\):\[E = \frac{q}{D^2} - q \left( \frac{1}{D^2} \left( 1 - 2\frac{d}{D} + 3\left(\frac{d}{D}\right)^2 \right) \right)\]Simplify the expression:\[E = \frac{q}{D^2} - \frac{q}{D^2} + \frac{2qd}{D^3} - \frac{3qd^2}{D^4} + \ldots\]

Step 5: Simplify and Find the Leading Term for Far Distance

The terms \( \frac{q}{D^2} \) cancel out, leaving us with:\[E \approx \frac{2qd}{D^3} - \frac{3qd^2}{D^4} + \ldots\]For large \( D \), the leading term in the expansion when \( d \ll D \) is \( \frac{2qd}{D^3} \). Thus, \( E \) is approximately proportional to \( \frac{1}{D^3} \) when \( P \) is far from the dipole.

Key Concepts

Electric Field

An electric field is a region around a charged particle where a force would be experienced by other charges. It is a fundamental concept in electromagnetism. In the case of an electric dipole, which is two equal but opposite charges separated by a distance, the electric field has unique properties. The electric field represents the way these charges interact with their surroundings.

For a point distant from the dipole, the field is calculated using the formula: \[ E = \frac{q}{D^2} - \frac{q}{(D+d)^2} \]where:

• \(q\) is the magnitude of the charge,
• \(D\) is the distance of the point \(P\) from the midpoint of the dipole,
• \(d\) is the separation between the charges.

Understanding this helps us analyze how the electric field behaves as you move either closer to or further away from the dipole.

Series Expansion

In mathematics and physics, series expansion is a method of expressing a function as the sum of terms of a series, usually infinite. This is particularly useful when a function is difficult to evaluate directly, but easier in parts or for approximations.

For this exercise, the series expansion is applied to the term \( \frac{1}{(D+d)^2} \) to simplify calculations when \( d \ll D \). When the point \( P \) is far from the dipole, this condition allows us to expand the expression in powers of \( \frac{d}{D} \),which simplifies the analysis.
Series expansions provide a powerful tool for approximating complex functions like this, making calculations more manageable.

Binomial Series

The binomial series is an expansion of the binomial theorem, which for positive integer exponents gives us polynomial expressions. When we deal with fractions and negative exponents, the binomial series comes in handy and can be expressed: \[(1+x)^{-n} \approx 1 - nx + \frac{n(n+1)}{2}x^2 - \ldots\]This is used when \(|x| < 1\).

In our exercise, \(x = \frac{d}{D} \) and \(n = 2\).Applying the formula helps transform \((1+\frac{d}{D})^{-2}\) into a form that is simpler to calculate. This method is extremely beneficial for engineers and physicists when dealing with small perturbations in their calculations, turning complex power functions into polynomials.

Proportionality

Proportionality is a mathematical relationship where one quantity varies as a constant multiple of another. In terms of electric fields, understanding how the field strength varies with distance is crucial.

For an electric dipole, the electric field strength at a point far from the dipole (\(d \ll D\)) can be approximated as proportional to \(\frac{1}{D^3}\). This tells us the rate at which the field decreases as the distance \(D\) increases.

• The simplification reveals that longer distances significantly weaken the field.
• This behavior aligns with the results of the binomial expansion.

The understanding of proportionality in electric fields is crucial as it helps in predicting how changes in one variable affect another, an essential aspect in the design and analysis of electric and electronic systems.