Problem 80 - University Physics with Modern Physics

Question

Three point charges are placed on the \(y\) -axis: a charge \(q\) at \(y=a,\) a charge \(-2 q\) at the origin, and a charge \(q\) at \(y=-a\) . Such an arrangement is called an electric quadrupole. (a) Find the magnitude and direction of the electric field at points on the positive \(x\) -axis. (b) Use the binomial expansion to find an approximate expression for the electric field valid for \(x \gg a\) . Contrast this behavior to that of the electric field of a point charge and that of the electric field of a dipole.

Step-by-Step Solution

Answer

The electric field magnitude at positive x-axis is approx. \( -\frac{3kqa^2}{x^4} \) for \(x\gg a\), showing quartic decrease.

Step 1: Identify Charge Positions and Configuration

We have three point charges: a charge \( q \) at position \( (0, a) \), a charge \( -2q \) at the origin \( (0, 0) \), and a charge \( q \) at position \( (0, -a) \). This setup forms an electric quadrupole along the \( y \)-axis.

Step 2: Determine the Electric Field Expressions

The electric field due to a point charge is given by \( \mathbf{E} = \frac{kq}{r^2} \hat{r} \). The total electric field on the positive \( x \)-axis from all three charges will be a vector sum of the fields from each charge. For a point \( (x, 0) \), the distances to each charge are \( r_1 = \sqrt{x^2 + a^2} \), \( r_2 = x \), and \( r_3 = \sqrt{x^2 + a^2} \), for the charges at \( y = a, 0, -a \) respectively.

Step 3: Calculate Electric Field from Each Charge at Positive x-axis

For the charge \( q \) at \( y = a \), the x-component of its electric field at \( (x, 0) \) is \( E_{1,x} = \frac{kq(x)}{(x^2 + a^2)^{3/2}} \). Similarly, for the charge \( q \) at \( y = -a \), the x-component is given by the same expression: \( E_{3,x} = \frac{kq(x)}{(x^2 + a^2)^{3/2}} \). For the charge \( -2q \) at the origin, the x-component is \( E_{2,x} = \frac{-2kq}{x^2} \).

Step 4: Sum up the x-components of Electric Field

The total electric field at \( x \) is the sum of the x-components: \[ E_{\text{total, x}} = \frac{kq(x)}{(x^2 + a^2)^{3/2}} + \frac{kq(x)}{(x^2 + a^2)^{3/2}} - \frac{2kq}{x^2} \]. Simplifying, \[ E_{\text{total, x}} = \frac{2kqx}{(x^2 + a^2)^{3/2}} - \frac{2kq}{x^2} \].

Step 5: Use Binomial Expansion for Approximation

For \( x \gg a \), approximate \( (x^2 + a^2)^{-3/2} \approx x^{-3} (1 - \frac{3a^2}{2x^2}) \) using binomial expansion. Substitute this back into the expression for \( E_{\text{total, x}} \): \[ E_{\text{total, x}} \approx \frac{2kqx}{x^3}(1 - \frac{3a^2}{2x^2}) - \frac{2kq}{x^2} = \frac{2kq}{x^2}(-\frac{3a^2}{2x^2}) \]. The dominant term as \( x \rightarrow \infty \) behaves as \( -\frac{3kqa^2}{x^4} \), indicating a quartic decrease with distance, unlike the monopole \( (\frac{kq}{x^2}) \) or dipole \( (\frac{kqa}{x^3}) \).

Key Concepts

Electric Field

An electric field is a fundamental concept in physics that describes the region around a charged object where electric forces operate. If a charge is placed within an electric field, it will experience a force. The strength and direction of this force depend on both the charge creating the field and the charge being influenced by it.

• The electric field \( \mathbf{E} \) due to a point charge is calculated using Coulomb's law: \( \mathbf{E} = \frac{kq}{r^2} \hat{r} \), where \( k \) is Coulomb's constant, \( q \) is the magnitude of the charge, and \( r \) is the distance from the charge.
• The direction of the electric field is away from positive charges and towards negative charges.

In the context of an electric quadrupole, which involves three-point charges, the electric field is determined by calculating the contributions from each charge and summing them. This involves understanding vector components because the field has both magnitude and direction.

Binomial Expansion

Binomial expansion is a mathematical method used for approximating expressions that involve terms raised to a power, especially useful when the terms involve large or small values.

• The binomial theorem declares that \( (1 + x)^n = 1 + nx + \frac{n(n-1)}{2}x^2 + \ldots \), where \( n \) is a real number and \( x \) is small compared to 1.
• In physics, we often use binomial expansion to simplify terms like \( (x^2 + a^2)^{-3/2} \) when \( x \) is much larger than \( a \).

For the electric quadrupole, using the binomial expansion helps find an approximate expression for the electric field when the observation point is far from the charge configuration. This approximation shows how the electric field behaves differently compared to simpler systems like monopoles or dipoles.

Point Charge

A point charge refers to a charged entity that is considered to be infinitesimally small, allowing us to analyze electric fields as if the charge is centralized at a single point. This simplification is essential for mathematical computations and theoretical physics.

• Point charges are hypothetical but useful in calculations where the actual size of the charge does not affect the problem.
• The electric field around a point charge decreases with the square of the distance from the charge, in accordance with the equation \( \mathbf{E} = \frac{kq}{r^2}\hat{r} \).

In the given exercise, the charges \( q \) and \( -2q \) are considered as point charges, which allows us to apply the formula for the electric field to each charge individually and then sum the results to find the total field.

Dipole

An electric dipole consists of two equal and opposite charges separated by some fixed distance. The significance of a dipole lies in its ability to model complex charge systems simply and analyze fields and potentials. Helpful in understanding molecules and more complex charge interactions.

• The electric field of a dipole at a point in space diminishes with the cube of the distance from the dipole rather than with the square, as seen in monopoles.
• The field is expressed by the formula \( \mathbf{E} = \frac{kqa}{r^3} \hat{p} \), where \( a \) is the separation distance and \( \hat{p} \) is the vector pointing from negative to positive charge.

In an electric quadrupole, the dipole model is extended by adding another pair of charges, further complicating the field behavior and making the use of approximations such as binomial expansions more important to understand the drop-off with distance as different from both dipoles and monopoles. This explains why the electric field drops off more rapidly (\( x^4 \) compared to \( x^3 \) or \( x^2 \) for simpler configurations).