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Ch 26: Potential and Field
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 26: Potential and FieldProblema 51
Capitolo 26, Problema 51

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?

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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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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.
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Conservation of Charge

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.
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Electric Fields in Conductors

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.
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Electric Potential
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