Ch 25: The Electric Potential

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 25: The Electric Potential

Problem 68

Chapter 25, Problem 68

Two metal objects that are in contact must be at the same potential, an assertion we'll prove in the next chapter. Suppose a metal sphere of radius R is charged to 1000 V and a second metal sphere of radius 2R is charged to 2000 V. The two spheres are brought into contact and then separated. Afterward, what is the potential of each sphere?

Verified step by step guidance

Step 1: Understand the concept of charge redistribution. When two conductors are brought into contact, charge redistributes between them until they reach the same electric potential. This is because charges move to minimize potential differences.

Step 2: Write the formula for the potential of a sphere. The potential of a charged 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.

Step 3: Use the principle of charge conservation. The total charge before contact is equal to the total charge after contact. Let \( Q_1 \) and \( Q_2 \) be the initial charges on the spheres, and \( Q_1' \) and \( Q_2' \) be the charges after contact. Then, \( Q_1 + Q_2 = Q_1' + Q_2' \).

Step 4: Set the potentials equal after contact. Since the spheres are in contact, their potentials become equal: \( \frac{Q_1'}{4 \pi \epsilon_0 R} = \frac{Q_2'}{4 \pi \epsilon_0 (2R)} \). Simplify this equation to relate \( Q_1' \) and \( Q_2' \).

Step 5: Solve for the final potential. Use the charge conservation equation and the relationship between \( Q_1' \) and \( Q_2' \) to find the total charge and distribute it between the spheres. Substitute the charges into the potential formula to find the final potential of each sphere.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Potential

Electric potential, measured in volts (V), is the amount of electric potential energy per unit charge at a point in an electric field. It indicates how much work would be done to move a charge from a reference point to a specific point within the field. In this scenario, the potentials of the two spheres before contact are crucial for understanding how they will equalize when brought into contact.

Charge Distribution

When two conductive objects come into contact, charge redistributes between them until they reach the same electric potential. The total charge is conserved, meaning the sum of the charges on both spheres before contact equals the sum after they are separated. This principle is essential for determining the final potential of each sphere after they are disconnected.

Capacitance

Capacitance is the ability of a system to store charge per unit potential difference and is influenced by the geometry of the conductors. For spheres, capacitance is proportional to their radius, meaning a larger sphere can hold more charge at the same potential. Understanding capacitance helps in calculating how the charges will distribute between the two spheres based on their sizes and initial potentials.

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Related Practice

Textbook Question

Two spherical drops of mercury each have a charge of 0.10 nC and a potential of 300 V at the surface. The two drops merge to form a single drop. What is the potential at the surface of the new drop?

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FIGURE P25.72 shows a thin rod with charge Q that has been bent into a semicircle of radius R. Find an expression for the electric potential at the center.

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Textbook Question

FIGURE P25.67 shows two uniformly charged spheres. What is the potential difference between points 1 and 2? Which point is at the higher potential? Hint: The potential at any point is the superposition of the potentials due to all charges.

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FIGURE P25.70 shows a thin rod of length L and charge Q. Find an expression for the electric potential a distance x away from the center of the rod on the axis of the rod.

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The wire in FIGURE P25.74 has linear charge density λ. What is the electric potential at the center of the semicircle?

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Textbook Question

A Van de Graaff generator is a device for generating a large electric potential by building up charge on a hollow metal sphere. A typical classroom-demonstration model has a diameter of 30 cm. What is the electric field strength just outside the surface of the sphere when it is charged to 500,000 V?