Problem 20 - Physics

Question

Four identical charges \((+2.0 \mu \mathrm{C}\) 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

The total electric potential energy of this group is 0.2685 J.

Step 1: Identify the Problem

We want to find the total electric potential energy of four identical charges arranged linearly. Each charge is +2.0 µC and they're 0.40 meters apart.

Step 2: Understanding Electric Potential Energy

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

Step 3: Calculate Pairwise Potential Energy

Since we have four charges, we need to calculate the potential energy for each pair of charges. The potential energy for each pair with distance \(r\) can be calculated and summed for all pairs.

Step 4: List All Charge Pairs and Distances

Arrange the charges as A, B, C, D. Then, calculate the potential energy between pairs: \(AB\) (same for \(BC\) and \(CD\)), distance = 0.40 m; \(AC\), distance = 2 \(\times\) 0.40 m; \(AD\) and \(BD\), distance = 3 \(\times\) 0.40 m; \(BC\), \(CD\), distance = 0.40 m.

Step 5: Calculate Energies for 0.40 m Pairs

The potential energy between charges \((AB, BC, CD)\) is \(U_{AB} = U_{BC} = U_{CD} = \frac{k \times (+2.0 \times 10^{-6})^2}{0.40} = 0.0896 \, ext{J}\).

Step 6: Calculate Energies for 0.80 m Pairs

The potential energy between charges \((AC, BD)\) is \(U_{AC} = U_{BD} = \frac{k \times (+2.0 \times 10^{-6})^2}{0.80} = 0.0448 \, ext{J}\).

Step 7: Calculate Energy for 1.20 m Pair

The potential energy between the charges \((AD)\) is \(U_{AD} = \frac{k \times (+2.0 \times 10^{-6})^2}{1.20} = 0.0299 \, ext{J}\).

Step 8: Sum All Potential Energies

Add the potential energies from all relevant pairs to find the total potential energy: \(3 \times 0.0896 + 2 \times 0.0448 + 1 \times 0.0299 = 0.2685 \, ext{J}\).

Key Concepts

Coulomb's Law

Coulomb's Law is crucial for understanding electric forces between charged objects. It describes how two point charges interact with each other. The law states that the magnitude of the force \(F\) between two point charges, \(q_1\) and \(q_2\), is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance \(r\) between them. The formula is:\[F = k \frac{|q_1 q_2|}{r^2}\]where \(k\) is Coulomb's constant, approximately \(8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\). This law helps understand both the direction and magnitude of the force. If both charges are of the same sign, they repel each other, while opposite charges attract. In the problem of four charges on a line, Coulomb's Law lets us calculate each pairwise interaction's electric potential energy.

Point Charges

Point charges are idealized charges considered to have no dimensions, meaning their size is negligible compared to the distances between them. This simplification allows us to apply Coulomb's Law straightforwardly. Real charges have size and structure, but assuming them to be point charges makes calculations feasibleand helps learn core concepts.

• Point charges can be positive or negative.
• They interact via electric forces in empty space.
• Potential energy between two point charges is determined by their magnitudes and separation distance.

For the given exercise with four identical charges of \(+2.0 \, \mu \text{C}\), considering them as point charges simplifies the calculation of electric potential energy by focusing solely on charge values and distances.

Linear Charge Configuration

A linear charge configuration is when multiple charges are arranged in a straight line. It is a simplified model in electrostatics to study interactions between multiple charges. Calculating the total electric potential energy for such a configuration involves accounting for every possible pair of interactions. In this scenario, understanding the following will make these calculations easier:

• Every adjacent pair of charges has a different distance than non-adjacent pairs.
• Non-adjacent pairs contribute significantly less to the sum of potential energy because of larger distances.
• The symmetry of a linear arrangement can sometimes simplify calculations.

To solve the given problem, each pair of charges along the line contributes to the total electric potential energy. Distances like \(0.40 \text{ m}, 0.80 \text{ m},\) and \(1.20 \text{ m}\) were utilized to compute pairwise potential energies, which were then summed to find the system's total electric potential energy.