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 65

Chapter 25, Problem 65

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?

Verified step by step guidance

Determine the relationship between the charge, potential, and radius of a spherical conductor. The potential at the surface of a spherical conductor is given by the formula:

V = \frac{k}{q} r

, where

k

is Coulomb's constant,

q

is the charge, and

r

is the radius of the sphere.

Calculate the radius of each original drop using the given potential and charge. Rearrange the formula for potential to solve for radius:

r = \frac{k}{q} V

. Substitute the given values for

q

and

V

Determine the total charge of the new drop after the two drops merge. Since charge is conserved, the total charge is the sum of the charges of the two original drops:

q_{total} = q_1 + q_2

Calculate the radius of the new drop. When two spherical drops merge, the volume is conserved. The volume of a sphere is given by

V = \frac{4}{3} π r^3

. Set the total volume of the two original drops equal to the volume of the new drop and solve for the new radius:

r_{new} = (\frac{2}{1} r_{original}^3)^{1/3}

Calculate the potential at the surface of the new drop using the formula for potential:

V_{new} = \frac{k}{q_{total}} r_{new}

. Substitute the total charge and the new radius into the formula to find the potential.

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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 is the amount of electric potential energy per unit charge at a point in an electric field. It is measured in volts (V) and indicates how much work would be done to move a charge from a reference point to a specific point in the field. In this problem, the potential of the mercury drops is crucial for determining the potential of the merged drop.

Charge Conservation

Charge conservation is a fundamental principle stating that the total electric charge in an isolated system remains constant over time. When the two mercury drops merge, their individual charges combine to form a single charge for the new drop. This principle is essential for calculating the total charge of the new drop after merging.

Capacitance of Spheres

The capacitance of a spherical conductor is defined as the ability to store electric charge per unit potential. For a sphere, the capacitance is directly proportional to its radius. When two spherical drops merge, the radius of the new drop changes, affecting its capacitance and, consequently, the potential at its surface, which can be calculated using the relationship between charge, capacitance, and potential.

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Related Practice

Textbook Question

Two 10-cm-diameter electrodes 0.50 cm apart form a parallel-plate capacitor. The electrodes are attached by metal wires to the terminals of a 15 V battery. After a long time, the capacitor is disconnected from the battery but is not discharged. What are the charge on each electrode, the electric field strength inside the capacitor, and the potential difference between the electrodes after the original electrodes (not the modified electrodes of part b) are expanded until they are 20 cm in diameter?

Textbook Question

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?

Textbook Question

Electrodes of area A are spaced distance d apart to form a parallel-plate capacitor. The electrodes are charged to ±q. What is the infinitesimal increase in electric potential energy dU if an infinitesimal amount of charge dq is moved from the negative electrode to the positive electrode?

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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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Textbook Question

The potential 1.0 cm from the surface of a metal sphere is 8000 V. The potential 3.0 cm from the surface is 4000 V. What is the radius of the sphere?

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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?