Question

A negative charge, \(-q\), is fixed at the coordinate (0,0) It is exerting an attractive force on a positive charge, \(+q,\) that is initially at coordinate \((x, 0)\). As a result, the positive charge accelerates toward the negative charge. Use the binomial expansion \((1+x)^{n} \approx 1+n x,\) for \(x \ll 1,\) to show that when the positive charge moves a distance \(\delta \ll x\) closer to the negative charge, the force that the negative charge exerts on it increases by \(\Delta F=2 k q^{2} \delta / x^{3}\) .

Short answer

Answer: The approximate change in force exerted by the fixed negative charge on the positive charge after the positive charge moves a small distance closer to the negative charge is given by the expression: \(\Delta F = \frac{2kq^2\delta}{x^3}\).

Step-by-step solution

Write down the expression for Coulomb's force

The force between two point charges can be calculated using Coulomb's law: \(F = k\frac{q_1q_2}{r^2}\). In this case, \(q_1 = -q\), \(q_2 = +q\), and the distance between the charges is \(r\). The charges have opposite signs, resulting in an attractive force.

Determine the initial force, \(F_{initial}\), between the charges

Initially, the positive charge is at coordinate \((x, 0)\), and the distance between the two charges is \(r = x\). The initial force, \(F_{initial}\), between the charges can be calculated using the formula from Step 1: \(F_{initial} = k\frac{-q^2}{x^2}\).

Determine the final force, \(F_{final}\), after moving the positive charge a distance \(\delta\) closer to the negative charge

The positive charge moves \(\delta\) closer to the negative charge, so its new coordinate is \((x-\delta,0)\). Thus, the new distance between the charges is \(r = x-\delta\). The final force, \(F_{final}\), can also be calculated using the formula from Step 1: \(F_{final} = k\frac{-q^2}{(x-\delta)^2}\).

Use the binomial expansion to approximate \(F_{final}\)

Since \(\delta \ll x\), \((x-\delta) \approx x\left(1 - \frac{\delta}{x}\right)\). Using the binomial expansion with \(n=-2\) and \(x = -\frac{\delta}{x}\), we get: \(F_{final} = k\frac{-q^2}{x^2(1-\frac{\delta}{x})^{-2}} \approx k\frac{-q^2}{x^2(1+2\frac{\delta}{x})}\)

Calculate the change in force, \(\Delta F\)

The change in force, \(\Delta F\), is given by the difference between the final and initial forces: \(\Delta F = F_{final} - F_{initial}\). Substituting the expressions for \(F_{final}\) and \(F_{initial}\) from Steps 2 and 4, we get: \(\Delta F = k\frac{-q^2}{x^2(1+2\frac{\delta}{x})} - k\frac{-q^2}{x^2} = k\frac{q^2}{x^2}\left(1-\frac{1}{1+2\frac{\delta}{x}}\right) = k\frac{q^2}{x^2}\left(\frac{2\frac{\delta}{x}}{1+2\frac{\delta}{x}}\right)\) Since \(\delta \ll x\), we can approximate the denominator as \(1+2\frac{\delta}{x} \approx 1\): \(\Delta F = k\frac{q^2}{x^2}\left(2\frac{\delta}{x}\right) = \frac{2kq^2\delta}{x^3}\)

Key concepts

Electric Force

Electric force is a fundamental concept in physics that describes the interaction between electric charges. It is governed by Coulomb's Law. This law states that the magnitude of the force \[ F \] between two point charges \( q_1 \) and \( q_2 \) is proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance \( r \) between them. Mathematically, it is expressed as: \[ F = k \frac{q_1 q_2}{r^2} \]where \( k \) is Coulomb's constant.

Electric force plays a crucial role in numerous physical phenomena and has a directional property. This means that it can either be attractive or repulsive depending on the signs of the charges involved:

• If the charges have opposite signs, the force is attractive.
• If the charges have the same signs, the force is repulsive.

Understanding electric force is essential as it explains how two charged particles exert forces on one another, impacting how they move and interact.

Charge Interaction

Charge interaction refers to the behavior of electric charges when they come into proximity with one another. This concept underpins much of electrostatics and helps explain electrical attraction or repulsion.

When discussing charge interaction, it is important to consider:

• Nature of Charges: Positive and negative charges interact with each other, where like charges repel and opposite charges attract.
• Distance Effect: The interaction becomes stronger when the charges are closer together. This is evident in Coulomb's Law showing how force increases as the separation distance decreases.
• Fixed and Moving Charges: In scenarios where one charge is fixed, like in the problem exercise, the dynamics of movement involve one charge moving while influenced by the fixed charge's electric field.

Understanding charge interaction is vital for solving problems in electrostatics, as it aids in predicting how charged particles will behave under various conditions.

Binomial Expansion

The binomial expansion is a mathematical technique used to simplify expressions raised to a power under certain conditions. It becomes particularly useful in physics when dealing with small quantities.

In this exercise, binomial expansion is applied to approximate expressions involving a small change in distance \( \delta \) as compared to the original distance \( x \). The formula \((1 + x)^n \approx 1 + nx\) is used here when \( x \ll 1 \).

For example, when recalculating the distance \((x - \delta)\) which is altered slightly by \(\delta\), the expression simplifies using the approximation:\[ (x - \delta)^{-2} \approx x^{-2} (1 + 2\frac{\delta}{x}) \]This simplification allows us to find how small changes in distance affect other physical quantities, like the electric force in this scenario, without extensive calculations. It thus becomes a crucial tool in theoretical physics whenever minor adjustments or interactions need to be considered.