Question

Each of the following pairs of charges are separated by a distance \(d\). Which pair has the highest potential energy? a) \(+5 \mathrm{C}\) and \(+3 \mathrm{C}\) d) \(-5 \mathrm{C}\) and \(+3 \mathrm{C}\) b) \(+5 \mathrm{C}\) and \(-3 \mathrm{C}\) e) All pairs have the \(\begin{array}{ll}\text { c) }-5 \mathrm{C} \text { and }-3 \mathrm{C} & \text { same potential energy. }\end{array}\)

Short answer

Answer: e) All pairs have the same potential energy.

Step-by-step solution

Calculate potential energy for each pair

To calculate the potential energy for each pair, we will use the formula \(U = k\frac{|q_1q_2|}{d}\), where \(k\) is a constant and can be ignored since we are comparing the values. a) \(U_a = \frac{|(+5C)(+3C)|}{d} = \frac{15C^2}{d}\) d) \(U_d = \frac{|(-5C)(+3C)|}{d} = \frac{15C^2}{d}\) b) \(U_b = \frac{|(+5C)(-3C)|}{d} = \frac{15C^2}{d}\) c) \(U_c = \frac{|(-5C)(-3C)|}{d} = \frac{15C^2}{d}\)

Compare potential energies

Comparing the potential energy calculated for all the pairs, we can see that they have the same potential energy, which is \(\frac{15C^2}{d}\). Thus, the answer is: e) All pairs have the same potential energy.

Key concepts

Coulomb's law

Coulomb's law is a fundamental principle in electrostatics, governing the interaction between electric charges. It describes how the electric force between two charges is dependent on:

• The magnitude of each charge
• The distance separating them

This law is expressed by the equation \[ F = k \frac{|q_1 q_2|}{r^2}, \]where

• \(F\) is the force of attraction or repulsion between the charges,
• \(q_1\) and \(q_2\) are the charges,
• \(r\) is the distance between the centers of the two charges,
• \(k\) is Coulomb's constant, approximately \(8.99 \times 10^9 \text{ N m}^2/\text{C}^2\).

The electric force is

• directly proportional to the product of the magnitudes of the charges, and
• inversely proportional to the square of the distance between them.

This means that if the distance increases, the force decreases, and vice versa. Coulomb's law is crucial for understanding how charged particles interact with one another.

Charge interaction

Charge interaction is an essential concept that helps us understand how charged objects influence each other. Charged particles can either attract or repel each other based on their properties:

• Like charges repel each other, meaning two positive or two negative charges will push away from each other.
• Opposite charges attract each other, so a positive and a negative charge will pull toward each other.

This behavior is due to the electric field surrounding each charge, a region where other charges experience a force. Magnitude and sign of the charge determine the electric field's strength and direction.
Electric interactions don't require physical contact, which allows them to act over a distance. These interactions help describe phenomena like how protons and electrons in atoms attract each other, keeping the atom stable.
Furthermore, the energy stored in the interaction between these charges is called electric potential energy. When charges are moved closer or further apart, work is done on or by them, which changes the electric potential energy.

Electrostatics

Electrostatics is the area of physics that studies forces, fields, and potentials arising from static charges, or charges at rest. Unlike electrodynamics, which deals with moving charges, electrostatics focuses on situations where charges are stationary.

• Charges can be positive or negative, representing protons and electrons, respectively.
• Electric fields arise from these charges, resulting in forces between them.
• The field direction is away from positive charges and towards negative charges.

One important concept in electrostatics is electric potential energy, which is the energy due to the position of a charge in an electric field. This energy depends on:

• The magnitude of the charges.
• The distance between the charges.

Sources such as rubbing a balloon against your hair can create static electricity, which is a practical example of electrostatic forces at work. Understanding these principles can help explain everyday occurrences, like why your hair stands up when it encounters a statically charged object.