You are given the equation(s) used to solve a problem. Finish the solution of the problem: (9.0×10⁹Nm²/C²)q₁q₂/0.030m =90×10⁻⁶J; q₁+q₂=40nC.

Verified step by step guidance

1. Step 1: Start by identifying the given information. The first equation relates the electric potential energy between two charges to their magnitudes and separation distance: \( U = \frac{k q_1 q_2}{r} \), where \( U = 90 \times 10^{-6} \; \text{J} \), \( k = 9.0 \times 10^9 \; \text{Nm}^2/\text{C}^2 \), and \( r = 0.030 \; \text{m} \). The second equation is \( q_1 + q_2 = 40 \; \text{nC} \).
2. Step 2: Rearrange the first equation to solve for \( q_1 q_2 \): \( q_1 q_2 = \frac{U r}{k} \). Substitute the known values \( U = 90 \times 10^{-6} \; \text{J} \), \( r = 0.030 \; \text{m} \), and \( k = 9.0 \times 10^9 \; \text{Nm}^2/\text{C}^2 \) into this equation.
3. Step 3: Use the second equation \( q_1 + q_2 = 40 \; \text{nC} \) to express one charge in terms of the other. For example, let \( q_1 = 40 \; \text{nC} - q_2 \). Substitute this expression for \( q_1 \) into the equation \( q_1 q_2 = \frac{U r}{k} \).
4. Step 4: Expand the equation \( (40 \; \text{nC} - q_2) q_2 = \frac{U r}{k} \) to form a quadratic equation in terms of \( q_2 \). Simplify the terms to get \( -q_2^2 + 40 \; \text{nC} \cdot q_2 - \frac{U r}{k} = 0 \).
5. Step 5: Solve the quadratic equation for \( q_2 \) using the quadratic formula \( q_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -1 \), \( b = 40 \; \text{nC} \), and \( c = -\frac{U r}{k} \). Once \( q_2 \) is found, use \( q_1 = 40 \; \text{nC} - q_2 \) to find \( q_1 \).

Key Concepts

Coulomb's Law

Coulomb's Law describes the electrostatic force between two charged objects. It states that the 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 relationship is mathematically expressed as F = k(q₁q₂/r²), where k is Coulomb's constant. Understanding this law is essential for solving problems involving electric forces and potential energy.

Electric Potential Energy

Electric potential energy is the energy stored in a system of charged particles due to their positions relative to each other. It can be calculated using the formula U = k(q₁q₂/r), where U is the potential energy, k is Coulomb's constant, and r is the distance between the charges. This concept is crucial for understanding how energy is transferred in electric fields and is directly related to the work done in moving charges.

Charge Conservation

Charge conservation is a fundamental principle stating that the total electric charge in an isolated system remains constant over time. In the context of the problem, the equation q₁ + q₂ = 40 nC indicates that the sum of the two charges is fixed. This principle is vital for solving problems involving multiple charges, as it allows for the determination of unknown charges when one or more values are provided.