Question

Two positive charges, each equal to \(Q\), are placed a distance \(2 d\) apart. A third charge, \(-0.2 Q\), is placed exactly halfway between the two positive charges and is displaced a distance \(x \ll d\) perpendicular to the line connecting the positive charges. What is the force on this charge? For \(x \ll d\), how can you approximate the motion of the negative charge?

Short answer

The motion of the negative charge can be approximated as a simple harmonic motion when x ≪ d.

Step-by-step solution

Calculate the electric field due to positive charges at the location of the negative charge

Since both positive charges have the same magnitude, we only need to calculate the electric field due to one of them and then multiply by 2. Let's consider the positive charge located on the left. By symmetry, the electric fields created by the positive charges have the same magnitudes but opposite directions. Let \(E_1\) be the electric field created by this charge at the location of the negative charge: \(E_1 = \frac{kQ}{(d + x)^2}\), where \(k\) is the electrostatic constant. Now, we can find the total electric field at the location of the negative charge by multiplying \(E_1\) by 2 and taking the vertical component: \(E_{total} = 2E_1 \sin\theta = 2\frac{kQ}{(d + x)^2} \cdot \frac{x}{\sqrt{d^2 + x^2}}\)

Calculate the net force on the negative charge

To calculate the net force on the negative charge (\(-0.2Q\)), we multiply the total electric field by the negative charge: \(F = (-0.2Q) \cdot E_{total} = -0.2Q \cdot 2\frac{kQ}{(d + x)^2} \cdot \frac{x}{\sqrt{d^2 + x^2}}\)

Approximate the motion of the negative charge for \(x \ll d\)

When \(x \ll d\), we can approximate \((d + x)^2\) as \(d^2\) and \(\sqrt{d^2 + x^2}\) as \(d\). Then, the net force simplifies to: \(F \approx -0.2Q \cdot 2\frac{kQ}{d^2} \cdot \frac{x}{d} = -\frac{0.4kQ^2x}{d^3}\) This force is linearly proportional to the displacement \(x\) and acts opposite to it, which indicates that the negative charge experiences a restoring force, and its motion can be approximated as a simple harmonic motion.

Key concepts

Coulomb's Law

Coulomb's Law is a fundamental principle that describes the force between two point charges. It states that the magnitude of the electrostatic force between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, it is represented as:

\[ F = k \cdot \frac{{|q_1 \cdot q_2|}}{{r^2}} \]

where \( F \) is the force, \( k \) is the Coulomb's constant (approximately \( 8.987 \times 10^9 N \cdot m^2/C^2 \)), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges. In the given exercise, Coulomb's Law helps determine the electric field created by the positive charges which in turn is used to calculate the force on the negative charge. As we can observe, the electric field \( E_1 \) due to a single positive charge and the distance \( d \) are substituted into this equation to derive the force.

Electrostatics

Electrostatics is the branch of physics that studies electric charges at rest. It encompasses the behavior of charges, the forces between them, and their effects in materials. A key aspect of electrostatics is the electric field, which is a vector field representing the force a charge would experience per unit of charge. It is typically denoted by \( E \).

In our scenario, the electric fields are caused by stationary positive charges, and the resultant field at the location of the negative charge is determined by vector addition of the fields from each positive charge. Additionally, the exercise highlights the great importance of considering the direction of the electric field, not just its magnitude, as they combine to result in the total electric field at the point where the charge is placed.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It can be visualized as the motion of a mass attached to a spring that is stretched or compressed.

This motion is described by the equation:\[ F = -kx \]

where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position. In the context of our problem, when the negative charge is displaced by a small \( x \), it feels a restoring force that is proportional to \( x \), which suggests that the charge undergoes SHM. The exercise skillfully guides us to this understanding by simplifying the expression for the force in terms of \( x \) and drawing parallels to the SHM equation.

Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electric field. Charges are quantized and come in two types: positive and negative. Like charges repel each other, while opposite charges attract.

In our exercise, we have two types of charges: two positive charges placed at a fixed distance and a negative charge that is free to move. The behavior of the negative charge in the presence of the positive charges, governed by electrostatic forces, is a direct consequence of the properties of electric charges. This concept underlies the entire problem, as it is the interaction between the negative and positive charges due to their electric properties that leads us to the solution involving electric field calculations and simple harmonic motion.