Metal sphere 1 has a positive charge of 6.0 nC. Metal sphere 2, which is twice the diameter of sphere 1, is initially uncharged. The spheres are then connected together by a long, thin metal wire. What are the final charges on each sphere?

검증된 단계별 안내

Understand the concept: When two conductive spheres are connected by a wire, charge redistributes between them until they reach the same electric potential. The electric potential on a sphere is given by \( V = \frac{Q}{4 \pi \epsilon_0 R} \), where \( Q \) is the charge, \( R \) is the radius, and \( \epsilon_0 \) is the permittivity of free space.

Relate the radii of the spheres: Since sphere 2 has twice the diameter of sphere 1, its radius \( R_2 \) is twice that of sphere 1, i.e., \( R_2 = 2R_1 \).

Set up the condition for equilibrium: At equilibrium, the electric potentials of the two spheres are equal. Using \( V_1 = V_2 \), substitute the formula for potential: \( \frac{Q_1}{4 \pi \epsilon_0 R_1} = \frac{Q_2}{4 \pi \epsilon_0 R_2} \). Simplify to find \( \frac{Q_1}{R_1} = \frac{Q_2}{R_2} \).

Express the relationship between charges: Using \( R_2 = 2R_1 \), substitute into \( \frac{Q_1}{R_1} = \frac{Q_2}{R_2} \) to get \( Q_2 = 2Q_1 \). This means the charge on sphere 2 is twice that on sphere 1 after redistribution.

Conserve total charge: The total charge before and after redistribution must be the same. Initially, the total charge is \( Q_{total} = 6.0 \text{ nC} \). After redistribution, \( Q_1 + Q_2 = Q_{total} \). Substitute \( Q_2 = 2Q_1 \) into this equation to solve for \( Q_1 \) and \( Q_2 \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Charge Conservation

Charge conservation is a fundamental principle in physics stating that the total electric charge in an isolated system remains constant over time. When two conductive objects are connected, the total charge is redistributed between them, but the sum of their charges before and after the connection remains the same.

Conductors and Charge Distribution

Conductors allow electric charges to move freely. When two conductors are connected, charges will redistribute until they reach the same electric potential. The larger conductor (sphere 2) will hold more charge due to its greater surface area, leading to an unequal distribution of charge based on their sizes.

Electric Potential

Electric potential is the amount of electric potential energy per unit charge at a point in an electric field. When two spheres are connected, they will equalize their electric potentials. The final charge on each sphere can be determined by ensuring that the potential is the same across both spheres, taking into account their sizes and initial charges.