Problem 83 - 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) Electric field: \( E = \frac{kQ_1}{R_1^2} \), potential: \( V = \frac{kQ_1}{R_1} \). (b) Charges: \( Q_1 = \frac{Q R_2}{R_1+R_2} \), \( Q_2 = \frac{Q R_1}{R_1+R_2} \). (c) Same potential: \( V = \frac{kQ}{R_1+R_2} \). (d) Fields: \( E_1 = \frac{kQ_1}{R_1^2} \), \( E_2 = \frac{kQ_2}{R_2^2} \).

Step 1: Determine the Electric Field at the Surface

For a charged conducting sphere, the electric field on the surface is given by the formula:\[ E = \frac{kQ_1}{R_1^2} \]where \(k\) is Coulomb's constant \((8.99 \times 10^9 \, \text{N m}^2/\text{C}^2)\), \(Q_1\) is the charge on the sphere, and \(R_1\) is the radius of the sphere.

Step 2: Determine the Electric Potential at the Surface

The electric potential \(V\) at the surface of a charged sphere is given by:\[ V = \frac{kQ_1}{R_1} \]This is because the potential is constant over the surface of a conductor. The reference potential is zero at an infinite distance.

Step 3: Establish Charge Redistribution in Equilibrium

When a charged sphere is connected to an uncharged sphere by a conducting wire, charge will flow until the potential at the surface of both spheres becomes equal.If \(Q_2\) is the charge on the second sphere after redistribution, then:\[ V_1 = V_2 \rightarrow \frac{kQ_1}{R_1} = \frac{kQ_2}{R_2} \]

Step 4: Solve for Charges on Each Sphere at Equilibrium

Because charge is conserved, the total charge \(Q = Q_1 + Q_2\) remains constant. Using \(Q = Q_1 + Q_2\) and Step 3, solve for \(Q_1\) and \(Q_2\):1. Express \(Q_2\) in terms of \(Q_1\) using \(\frac{Q_1}{R_1} = \frac{Q_2}{R_2}\).2. Combine with \(Q = Q_1 + Q_2\) to find:\[ Q_1 = \frac{QR_2}{R_1+R_2}, \quad Q_2 = \frac{QR_1}{R_1+R_2} \]

Step 5: Calculate the Electric Potential for Each Sphere

At equilibrium, both spheres have the same potential:\[ V = \frac{kQ_1}{R_1} = \frac{kQ_2}{R_2} \]Each calculation should yield \( V = \frac{kQ}{R_1+R_2} \).

Step 6: Determine the Electric Field for Each Sphere

Use the formula for electric field at the surface for each sphere:\[ E_1 = \frac{kQ_1}{R_1^2}, \quad E_2 = \frac{kQ_2}{R_2^2} \]Plug the charges found in Step 4 into these equations to get specific values.

Key Concepts

Electrostatic Equilibrium

When two conducting spheres are connected with a long, thin wire, electrostatic equilibrium occurs. This means that charge flows between the spheres until the potential on both surfaces is equal. This is crucial because it ensures that there is no net movement of charge on the wire, which is important for maintaining equilibrium.
At equilibrium, both spheres will have the same electric potential. The equation for this can be written as:

• \( V_1 = V_2 \) where \( V_1 \) is the potential of the first sphere and \( V_2 \) is the potential of the second sphere.

Using the formula for electric potential of a sphere \( V = \frac{kQ}{R} \), we equate the potentials: \( \frac{kQ_1}{R_1} = \frac{kQ_2}{R_2} \). This equation allows us to determine the distribution of charge across the two spheres after equilibrium is reached.The conservation of charge is another vital principle here. The total charge before and after connecting the spheres remains constant. Thus, if \( Q_1 \) is the initial charge and \( Q \) is the combined charge, we have:

• \( Q = Q_1 + Q_2 \)

Using this relationship, we can solve for the new charges \( Q_1 \) and \( Q_2 \) once electrostatic equilibrium is established.

Electric Field

The electric field at the surface of a charged sphere is determined by how the charge is distributed over the sphere. The formula for the electric field at the surface is given by:

• \( E = \frac{kQ}{R^2} \)

In this equation, \( E \) is the electric field, \( k \) is Coulomb's constant, \( Q \) is the charge, and \( R \) is the radius of the sphere.
When the spheres reach electrostatic equilibrium, the charge distribution changes, affecting the electric fields at the surfaces. For connected spheres with charges \( Q_1 \) and \( Q_2 \), the electric fields become:

• \( E_1 = \frac{kQ_1}{R_1^2} \) for the first sphere
• \( E_2 = \frac{kQ_2}{R_2^2} \) for the second sphere

The change in charge distribution results in variations in the electric field experienced at each sphere's surface. Accurately calculating these fields is vital for understanding the behavior of charged conductors in electric potential problems.

Charge Redistribution

Charge redistribution is an essential concept when dealing with connected conductors. Initially, one sphere is charged, and the other is uncharged. Once they are connected, charge will move from the higher potential sphere to the uncharged one until both have the same potential.
The equations of equilibrium, \( \frac{Q_1}{R_1} = \frac{Q_2}{R_2} \) and \( Q = Q_1 + Q_2 \), allow us to solve for the final distribution of charge. This can be summarized as:

• The charge on the first sphere after redistribution, \( Q_1 = \frac{QR_2}{R_1 + R_2} \)
• The charge on the second sphere after redistribution, \( Q_2 = \frac{QR_1}{R_1 + R_2} \)

With these equations, students can determine how charge distributes between the spheres, depending on their respective radii.
Understanding charge redistribution is crucial because it affects both the electric field and electric potential at the surface of each sphere. This knowledge can be applied to various contexts where conductors are used, such as in networking multiple capacitors in circuitry.