Problem 20 - Physics

Question

Four identical charges \((+2.0 \mu C\) each \()\) are brought from infinity and fixed to a straight line. The charges are located \(0.40 \mathrm{~m}\) apart. Determine the electric potential energy of this group.

Step-by-Step Solution

Answer

389.57 J

Step 1: Understand the Configuration

There are four identical charges, each with a charge of +2.0 μC, placed at equal distances of 0.40 meters along a straight line. We need to calculate the total electric potential energy of this system of charges.

Step 2: Formula for Electric Potential Energy between Two Charges

The electric potential energy between two charges \( q_1 \) and \( q_2 \), separated by a distance \( r \), is given by the formula: \[ U = \frac{k \cdot q_1 \cdot q_2}{r} \]where \( k \) is Coulomb's constant, \( k = 8.99 \times 10^9 \text{ N m}^2/\text{C}^2 \).

Step 3: Calculate Potential Energy for Each Pair

Since potential energy is additive, calculate the potential energy for each unique pair of charges:1. Charges 1 & 2, 1 & 3, 1 & 4, 2 & 3, 2 & 4, and 3 & 4.2. Distances are: 0.40 m (1 & 2), 0.80 m (1 & 3), 1.20 m (1 & 4), 0.40 m (2 & 3), 0.80 m (2 & 4), and 0.40 m (3 & 4).Calculate each separately:- For Charges 1 & 2 or 2 & 3 or 3 & 4 (distance = 0.40 m), \[ U = \frac{(8.99 \times 10^9) \cdot (2.0 \times 10^{-6}) \cdot (2.0 \times 10^{-6})}{0.40} \]- For Charges 1 & 3 or 2 & 4 (distance = 0.80 m),\[ U = \frac{(8.99 \times 10^9) \cdot (2.0 \times 10^{-6}) \cdot (2.0 \times 10^{-6})}{0.80} \]- For Charges 1 & 4 (distance = 1.20 m),\[ U = \frac{(8.99 \times 10^9) \cdot (2.0 \times 10^{-6}) \cdot (2.0 \times 10^{-6})}{1.20} \]

Step 4: Calculate Individual Potential Energies

Calculate each potential energy using the distances:- For distance 0.40 m, potential energy for each pair is:\[ U = \frac{(8.99 \times 10^9) \cdot 4.0 \times 10^{-12}}{0.40} = 89.9 \text{ J} \]- For distance 0.80 m, potential energy is:\[ U = \frac{(8.99 \times 10^9) \cdot 4.0 \times 10^{-12}}{0.80} = 44.95 \text{ J} \]- For distance 1.20 m, potential energy is:\[ U = \frac{(8.99 \times 10^9) \cdot 4.0 \times 10^{-12}}{1.20} = 29.97 \text{ J} \]

Step 5: Sum All Potential Energies

Add the potential energies of all unique pairs:\[ U_{total} = 3(89.9) + 2(44.95) + 29.97 = 269.7 + 89.9 + 29.97 = 389.57 \text{ J} \]

Step 6: Conclusion: Electric Potential Energy Total

The total electric potential energy of the group of charges is 389.57 J.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle used to understand electric forces between charges. It states that the electric force \( F \) between two point charges \( q_1 \) and \( q_2 \) is directly proportional to the product of their charges and inversely proportional to the square of the distance \( r \) that separates them. The equation for the electric force can be expressed as follows:\[ F = \frac{k \cdot q_1 \cdot q_2}{r^2} \]Here, \( k \) represents Coulomb's constant, with a value of approximately \( 8.99 \times 10^9 \) N m²/C².

• It helps to predict how strongly two charges will interact.
• It is essential for calculating electric potential energy, which is the energy due to the position of charges in an electric field.

Coulomb's Law is similar to Newton's law of universal gravitation, though it applies to charges rather than masses. In both laws, the force follows an inverse square law, meaning if the distance doubles, the force reduces to a quarter. It's crucial in physics because it allows us to understand and calculate interactions in fields generated by charges.

Electric Charges

Electric charges are the building blocks of electricity. There are two types of charges: positive and negative. Like charges repel each other, while opposite charges attract. This basic property explains behaviors and phenomena related to electricity and electrostatics.

• Positive charge: Found in protons.
• Negative charge: Found in electrons.
• Neutral atoms have equal amounts of positive and negative charges.

Electric charges are quantified in a unit called the coulomb (C). In the given exercise, each charge is \( +2.0 \mu C = 2.0 \times 10^{-6} \) coulombs. When you bring charges together, they interact via electric forces, leading to potential energy. In these interactions, identical electric charges in proximity will try to move apart due to repulsion, complying with Coulomb's law.Understanding electric charges is key in analyzing electric circuits and fields because they are the foundation of electric current. This is also why conductors (which allow charges to move freely) and insulators (which restrict charge movement) facilitate or block electrical flow.

Superposition Principle

In electrostatics, the superposition principle allows us to calculate the total electric potential energy or electric field generated by multiple charges. According to this principle, the net effect (force, field, or potential energy) at a point in space due to multiple charges is the algebraic sum of the effects produced by each individual charge.

• Simplifies calculations with multiple charges.
• Aids in determining the total force or potential energy in a system of charges.

For instance, in the given exercise with four charges, the total electric potential energy is the sum of potential energies between each distinct pair of charges. The calculation involves evaluating each unique pair for the distance separating them and using formulas separately: 1. Calculate energy between charges 1 & 2, 1 & 3, 1 & 4, etc. 2. Use the superposition principle to sum potentials from individual pairs. 3. Get the total potential energy of the system. This principle is extremely useful for solving complex problems in physics by breaking them down into simpler components. It helps us understand how a group of charges will exert forces and contribute to potential energy, allowing us to predict the behavior of charge systems accurately.