Problem 76 - University Physics with Modern Physics

Question

A metal sphere with radius \(R_1\) has a charge \(Q_1\) . Take the electric potential to be zero at an infinite distance from the sphere. (a) What are the electric field and electric potential at the surface of the sphere? This sphere is now connected by a long, thin conducting wire to another sphere of radius \(R_2\) that is several meters from the first sphere. Before the connection is made, this second sphere is uncharged. After electrostatic equilibrium has been reached, what are (b) the total charge on each sphere; (c) the electric potential at the surface of each sphere; (d) the electric field at the surface of each sphere? Assume that the amount of charge on the wire is much less than the charge on each sphere.

Step-by-Step Solution

Answer

(a) \(E_1 = \frac{1}{4\pi\varepsilon_0}\frac{Q_1}{R_1^2}; V_1 = \frac{1}{4\pi\varepsilon_0}\frac{Q_1}{R_1}\). (b) \(Q_1' = \frac{R_1}{R_1 + R_2} Q_1; Q_2' = \frac{R_2}{R_1 + R_2} Q_1\). (c) \(V' = \frac{1}{4\pi\varepsilon_0}\frac{Q_1}{R_1 + R_2}\). (d) \(E_1' = \frac{1}{4\pi\varepsilon_0}\frac{Q_1}{R_1(R_1 + R_2)}; E_2' = \frac{1}{4\pi\varepsilon_0}\frac{Q_1}{R_2(R_1 + R_2)}\).

Step 1: Electric Field at the Surface of the Sphere

The electric field at the surface of a conducting sphere with charge \(Q_1\) and radius \(R_1\) is given by Coulomb's law: \[E_1 = \frac{1}{4\pi\varepsilon_0}\frac{Q_1}{R_1^2}\] This formula calculates the electric field just outside the sphere, assuming the sphere is conductively isolated.

Step 2: Electric Potential at the Surface of the Sphere

The electric potential \(V\) at the surface of a sphere with radius \(R_1\) is given by: \[V_1 = \frac{1}{4\pi\varepsilon_0}\frac{Q_1}{R_1}\] This results from the integration of the electric field from infinity to the surface of the sphere.

Step 3: Charge Redistribution After Connection

When the two spheres are connected, charge will redistribute until the potentials on both spheres are equal because charge moves to equalize potential. Let the charges be \(Q_1'\) and \(Q_2'\) after equilibrium. The total charge remains conserved: \[Q_1 + 0 = Q_1' + Q_2'\] and the potentials satisfy: \[\frac{1}{4\pi\varepsilon_0}\frac{Q_1'}{R_1} = \frac{1}{4\pi\varepsilon_0}\frac{Q_2'}{R_2}\] Hence: \[\frac{Q_1'}{R_1} = \frac{Q_2'}{R_2}\]

Step 4: Solving for Charges on Each Sphere

Using the conservation of charge, \(Q_1' + Q_2' = Q_1\), and the equality of potentials, we find: \[Q_1' = \frac{R_1}{R_1 + R_2} Q_1\] \[Q_2' = \frac{R_2}{R_1 + R_2} Q_1\] These give the charges on spheres \(1\) and \(2\) respectively, after equilibrium.

Step 5: Electric Potential at Each Sphere's Surface After Equilibrium

Since potentials are equal after equilibrium and given by \[V' = \frac{1}{4\pi\varepsilon_0}\frac{Q_1'}{R_1} = \frac{1}{4\pi\varepsilon_0}\frac{Q_2'}{R_2}\]Plug in \(Q_1' = \frac{R_1}{R_1 + R_2} Q_1\) to find \[V' = \frac{1}{4\pi\varepsilon_0}\frac{Q_1}{R_1 + R_2}\] at both spheres.

Step 6: Electric Field at Each Sphere's Surface After Equilibrium

The electric field for each sphere after charge redistribution can be calculated using the surface charge and size of each sphere. Thus, \[E_1' = \frac{1}{4\pi\varepsilon_0}\frac{Q_1'}{R_1^2} = \frac{1}{4\pi\varepsilon_0}\frac{R_1}{(R_1 + R_2)R_1^2}Q_1\] and \[E_2' = \frac{1}{4\pi\varepsilon_0}\frac{Q_2'}{R_2^2} = \frac{1}{4\pi\varepsilon_0}\frac{R_2}{(R_1 + R_2)R_2^2}Q_1\] giving the electric field for spheres \(1\) and \(2\), respectively.

Key Concepts

Electric Field

The electric field describes the force per unit charge experienced by a small test charge placed in the vicinity of another charge. In the case of a conducting sphere with charge, this field can be specifically calculated using Coulomb's law. Just outside the surface of a sphere of radius \(R_1\) with charge \(Q_1\), the electric field \(E_1\) is represented by:

• \(E_1 = \frac{1}{4\pi\varepsilon_0}\frac{Q_1}{R_1^2}\)

This equation springs from the observation that the electric field created by a uniformly charged sphere behaves as if the entire charge is concentrated at its center. The field strength decreases with the square of the distance from the center, hence the \(R_1^2\) in the denominator. After two spheres are connected, charges adjust till the potential is the same on both, affecting how fields are calculated around each sphere. Understanding the electric field is key to describing how charges influence one another.

Electric Potential

Electric potential is the work done to bring a unit positive charge from infinity to a point in space in an electric field, without acceleration. For a conducting sphere, this depends only on the charge and the radius. The potential \(V_1\) at the surface of a sphere of radius \(R_1\) and charge \(Q_1\) is:

• \(V_1 = \frac{1}{4\pi\varepsilon_0}\frac{Q_1}{R_1}\)

This formula results from the integration of the electric field from infinity to the point of interest. When two spheres at different initial potentials are connected, the charge rearranges so that the potential is uniform across both surfaces. This balance and uniformity throughout connected conducting equipment simplify the analysis of complex charge systems.

Charge Distribution

Charge distribution refers to how electric charge is spread over a structure. In electrostatics, charges will move around until they reach equilibrium, particularly in conductors. Upon connecting the charged sphere and an initially uncharged one with a conducting wire, charge redistributes between them:

• Total charge stays constant: \(Q_1 + 0 = Q_1' + Q_2'\)
• Equalize potentials: \(\frac{Q_1'}{R_1} = \frac{Q_2'}{R_2}\)

Using these relations, the final charges \(Q_1'\) and \(Q_2'\) are found:

• \(Q_1' = \frac{R_1}{R_1 + R_2} Q_1\)
• \(Q_2' = \frac{R_2}{R_1 + R_2} Q_1\)

This shows how charge equalizes across two connected spheres based on their sizes, ensuring both surfaces have identical electric potentials after the charge transfer.

Conservation of Charge

The principle of conservation of charge is a fundamental concept in physics, stating that the total charge in an isolated system remains constant. This is crucial in electrostatic interactions like those involving two spheres connected by a wire. When these spheres are connected, charge moves to equalize potential across the spheres, but the total amount of charge does not change:

• Start with \(Q_1\) on the first sphere and 0 on the second.
• After equilibrium, \(Q_1' + Q_2' = Q_1\).

Even as charges shuffle between them, the sum remains conserved. Understanding conservation of charge helps track how electricity behaves and redistributes in connected systems, foundational to circuit design and electrostatic applications.