Question

What is Coulomb' law? Explain how the potential energy of two charged particles degends on the distance between the charged particles and on the magnitude and sign of their charges.

Short answer

Coulomb's Law states that the force between two point charges is proportional to the product of their charges and inversely proportional to the square of the distance (\( F = k \frac{|q_1q_2|}{r^2} \)). The potential energy depends on the distance and the product of the charges, with attractions having negative energy (\( U = k \frac{q_1q_2}{r} \)).

Step-by-step solution

Define Coulomb's Law

Coulomb's Law describes the electrostatic interaction between two electrically charged particles. It states that the force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. The formula is expressed as: \( F = k \frac{|q_1q_2|}{r^2} \) where \( F \) is the electrostatic force, \( q_1 \) and \( q_2 \) are the charges, \( r \) is the distance between the charges, and \( k \) is Coulomb's constant.

Express Potential Energy in terms of Charge and Distance

The potential energy (\( U \)) of a system consisting of two point charges in vacuum can be calculated using the formula: \( U = k \frac{q_1q_2}{r} \). This states that the potential energy is directly proportional to the product of the charges and inversely proportional to the distance between them. If one charge is positive and the other is negative, the potential energy is negative, indicating an attractive force. If both charges are of the same sign, the potential energy is positive, indicating a repulsive force.

Analyze the Effect of Distance and Magnitude of Charge

From the formula for potential energy, it can be observed that if the distance between two charges increases, the potential energy decreases, meaning the force between them weakens. Conversely, if the charges themselves increase while the distance remains constant, the potential energy between them increases as well, leading to a stronger force.

Key concepts

Electrostatic Force

The concept of electrostatic force is at the heart of Coulomb's Law, which is a fundamental principle in physics. This force is the interaction between particles that are charged, either negatively or positively. In essence, it's what causes charged particles to either pull towards each other or push away.

According to the law, two charged objects exert a force on each other that's directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The mathematical representation of this law is a cornerstone in understanding electric interactions: \[\[\begin{align*}F = k \frac{|q_1q_2|}{r^2}\end{align*}\]\]Here,

• F represents the magnitude of the electrostatic force,
• q_1 and q_2 are the magnitudes of the charges,
• r is the distance between the charges, and
• k is Coulomb's constant (approximately \(8.988 \times 10^9\) Nm^2/C^2).

It's essential to note that the force is attractive if the charges have opposite signs, causing them to pull together. If the charges have the same sign, the force is repulsive, and the particles push away from each other.

Potential Energy of Charged Particles

Potential energy in electrostatics describes the energy that's stored between charged particles due to their positions. It's a vital concept for understanding how charged particles will behave when free to move. The potential energy (\[\[\begin{align*}U\end{align*}\]\]) between two point charges can be calculated from their distance and magnitudes, similar to how we calculate the force but without the square of the distance between the charges.The formula given in Coulomb's Law for potential energy is\[\[\begin{align*}U = k \frac{q_1q_2}{r}\end{align*}\]\]where

• U is the potential energy,
• k is again Coulomb's constant,
• q_1 and q_2 are the charges of the particles, and
• r is the distance between them.

The potential energy is significant because it predicts the work needed to bring charges together or the energy released when they move apart. For example, if you have a positive and a negative charge, they will have negative potential energy, indicating that work must be done to pull them apart. Conversely, if the charges are alike, they have positive potential energy, and energy is released as they push away from each other. The sign of potential energy thus gives us an intimate understanding of the nature of the electrostatic interaction.

Magnitude and Sign of Charges

The magnitude and sign of charges are pivotal in Coulomb's Law—they determine the strength and direction of the electrostatic force and the potential energy of a system. In everyday terms, the magnitude refers to how much charge a particle has, and the sign indicates whether the charge is positive or negative.

Charges with greater magnitudes will exert a stronger force on each other and will have higher potential energy when close together. To visualize this, imagine the difference in effort it takes to push together the opposite poles of small fridge magnets versus large, powerful industrial magnets—the larger the magnet (and thus the larger the magnitude of its magnetic 'charge'), the more force it takes to push them together.Similarly, the sign of the charges plays a crucial role. When two charges are of the same sign, they repel each other; when they are of opposite signs, they attract. This simple rule of attraction and repulsion is a fundamental aspect of electrostatics and is extensively used in fields ranging from chemistry to electrical engineering.

• Identical signs lead to positive potential energy and repulsive force,
• Opposite signs lead to negative potential energy and attractive force.

Understanding the magnitude and sign of charges helps students predict the behavior of charged particles in complex systems, whether it's determining the stability of atomic structures or diagnosing the cause of electrical malfunctions.