Problem 78 - 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 on the x-axis for large x is \( E \approx \frac{-3kqa^2}{x^4} \), decaying faster than a point charge or dipole field.

Step 1: Identify the Problem

We need to find the electric field generated by three point charges along the positive x-axis. These charges form an electric quadrupole with charges at different points on the y-axis.

Step 1: Understand the Charge Arrangement

We have three point charges: +q at y = +a, -2q at y = 0, and +q at y = -a. We are interested in the electric field on the positive x-axis.

Step 2: Electric Field Due to a Point Charge

The electric field due to a point charge q at a distance r is given by \[ E = \frac{k q}{r^2} \] where k is Coulomb's constant (8.99 × 10^9 Nm²/C²) and r is the distance from the charge.

Step 3: Calculate Distance to Each Charge

For a point (x,0) on the x-axis, the distances to the charges (x,±a) and (x,0) are: - From +q at (0,a): \[ r_1 = \sqrt{x^2 + a^2} \] - From -2q at (0,0): \[ r_2 = x \] - From +q at (0,-a): \[ r_3 = \sqrt{x^2 + a^2} \]

Step 4: Calculate Electric Fields from Each Charge

The magnitude of electric fields at (x,0) due to each charge is: - From +q at (0,a): \[ E_1 = \frac{k q}{r_1^2} \] - From -2q at (0,0): \[ E_2 = \frac{-2k q}{x^2} \] - From +q at (0,-a): \[ E_3 = \frac{k q}{r_3^2} \]

Step 5: Find Net Electric Field

Because we are on the x-axis, the vertical components of the electric field due to the charges at (0,a) and (0,-a) will cancel out. The net electric field on the x-axis is: \[ E = E_1 + E_3 + E_2 \] \[ E = \frac{k q}{\sqrt{x^2 + a^2}^2} + \frac{k q}{\sqrt{x^2 + a^2}^2} - \frac{2 k q}{x^2} \]

Step 6: Simplify for Large x (Binomial Approximation)

For \(x \gg a\), use the approximation \( (1 + \frac{a^2}{x^2})^{-3/2} \approx 1 - \frac{3a^2}{2x^2} \):\[ E \approx \frac{2kq}{x^2} \left(1 - \frac{3a^2}{2x^2}\right) - \frac{2kq}{x^2} \]\[ E \approx \frac{-3kqa^2}{x^4} \]This is much weaker than the \(1/x^2\) dependence of a single charge and highlights the weak field for distant points characteristic of quadrupoles.

Step 7: Compare with Point Charge and Dipole

The electric field of a quadrupole decays as \( \frac{1}{x^4} \), which is faster than a point charge’s \( \frac{1}{x^2} \) decay and a dipole’s \( \frac{1}{x^3} \) decay.

Key Concepts

Binomial Expansion

The binomial expansion is a powerful tool in mathematics used to approximate expressions that involve powers of binomials. Using this technique can simplify calculations, especially when dealing with powers raised to high exponents.

In the context of an electric quadrupole, the binomial expansion comes into play when simplifying the expression for the electric field at a point far away from the charges. Typically, we approximate terms like \((1 + \frac{a^2}{x^2})^{-3/2}\) using the binomial expansion formula:

• \((1 + y)^n \approx 1 + ny + \frac{n(n-1)}{2!}y^2 + \ldots\)

For our scenario, when \(x \gg a\), this simplifies to just \(1 - \frac{3a^2}{2x^2}\), making our calculations manageably concise.

This approach highlights how an electric quadrupole's field strength diminishes faster than simpler charge arrangements.

Electric Field

An electric field is a vector field that surrounds electric charges and exerts a force on other charges within the field. The strength and direction of the electric field created by a point charge can be calculated using the formula:

• \(E = \frac{kq}{r^2}\)

Here, \(E\) represents the electric field magnitude, \(k\) is Coulomb's constant, \(q\) is the charge, and \(r\) is the distance from the charge to the point of interest.

The direction of the electric field is always directed away from positive charges and toward negative charges. When dealing with multiple charges, you must consider the vector sum of each individual field to find the net electric field at a particular point. For an electric quadrupole, the field becomes markedly weaker as you move further away, diminishing more rapidly than an isolated charge or dipole due to the arrangement's symmetry.

Coulomb's Constant

Coulomb's constant, denoted as \(k\), is a fundamental value in electrostatics aiding in measuring the force between two charges. It appears in the equation for calculating the electric field as well:

• \( k = 8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2 \)

This constant stems from Coulomb's law, which states that the electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

In electric quadrupole problems, Coulomb's constant facilitates calculating individual forces and fields created by each charge. This constant is crucial for precise, accurate computations of electric interactions, allowing us to predict electric field behavior even in complex configurations.

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a small distance. It has a significant impact on electric field behavior, especially when studying charge distributions like a quadrupole.

The electric field of a dipole diminishes more quickly with distance than a single point charge and can be expressed as:

• \(E \propto \frac{1}{x^3}\)

This rapid decay is due to the opposite charges effectively neutralizing each other at a distance.

Unlike a dipole, an electric quadrupole contains four charges arranged symmetrically, leading to an even faster decay of its electric field at large distances, reducing as \(\frac{1}{x^4}\). This stark contrast showcases how charge configuration affects field strength and decay, crucial for understanding electric field interactions in different setups.