Question

A solid conducting sphere of radius \(R_{1}=1.206 \mathrm{~m}\) has a charge of \(Q=1.953 \mu \mathrm{C}\) evenly distributed over its surface. A second solid conducting sphere of radius \(R_{2}=0.6115 \mathrm{~m}\) is initially uncharged and at a distance of \(10.00 \mathrm{~m}\) from the first sphere. The two spheres are momentarily connected with a wire, which is then removed. What is the charge on the second sphere?

Short answer

Solution: To find the final charge on the second sphere, we first determined the initial surface charge density of the first sphere and then found the proportionality between surface charge densities for both spheres. Finally, we solved for the charge on the second sphere using the equation \(Q_1 = \sigma_2 \cdot 4 \pi R_{2}^2 \cdot \frac{R_{1}^2}{R_{2}^2}\). After plugging in the given values, the charge on the second sphere can be calculated.

Step-by-step solution

Determine the initial surface charge density for the first sphere

To determine the initial surface charge density for the first sphere, we can use the following formula: \(\sigma_{1} = \frac{Q_1}{4 \pi R_{1}^2}\) Given that \(Q=1.953 \mu \mathrm{C}\) and \(R_{1}=1.206 \mathrm{~m}\), we can plug in these values to find \(\sigma_{1}\).

Find the proportionality between surface charge densities for both spheres

In electrostatic equilibrium, the surface charge densities of both spheres are proportional to their radii, meaning: \(\frac{\sigma_1}{\sigma_2} = \frac{R_1}{R_2}\) We can substitute step 1 result in the equation above, which is like writing: \(\frac{\frac{Q_1}{4 \pi R_{1}^2}}{\sigma_2} = \frac{R_1}{R_2}\)

Solve for the charge on the second sphere

To find the charge on the second sphere, we can multiply both sides of the equation in step 2 by \(\sigma_2\) and then by \(4 \pi R_{2}^2\), \(Q_1 = \sigma_2 \cdot 4 \pi R_{2}^2 \cdot \frac{R_{1}^2}{R_{2}^2}\) Now, simply plug in the given values in this equation and solve for \(Q_2 = \sigma_2 \cdot 4 \pi R_{2}^2\). \(Q = 1.953 \cdot 10^{-6} \mathrm{C}\), \(R_{1} = 1.206 \mathrm{~m}\), and \(R_{2} = 0.6115 \mathrm{~m}\) After solving for \(Q_2\), you will obtain the charge on the second sphere after both spheres reach electrostatic equilibrium.

Key concepts

Electrostatic Equilibrium

Understanding electrostatic equilibrium is crucial when dealing with electrical charges in conductors. Electrostatic equilibrium is the state where the charges within a conductor are at rest, and there is no net movement of charge within the material. This occurs when the electric field within the conductor ceases to exist, and any excess charge resides on the surface, resulting in a uniform surface charge distribution.

For example, when two conducting spheres are connected by a wire, they share charges until they reach electrostatic equilibrium. In this state, the potential difference between the spheres is zero, and the surface charge density is adjusted accordingly. This concept is fundamental in explaining how charges distribute themselves on conductors and why, when two conductive spheres of different sizes are connected, the surface charge density changes to maintain equilibrium.

Conducting Spheres

Conducting spheres are a classic model in electrostatics to illustrate the principles of charge distribution and electrical potential. In conductors, charges are free to move until they are distributed uniformly. When a conducting sphere is charged, the charges distribute themselves evenly across the surface, creating a uniform surface charge density that is inversely proportional to the square of the radius of the sphere.

Whenever two conducting spheres come into contact, or are connected through a wire, they will share their charges until they reach electrostatic equilibrium. The surface charge density, therefore, becomes a key factor in determining the final charge distribution over both spheres. The sphere with the larger radius will end up with a lower surface charge density, compared to a smaller sphere, because it has a greater surface area over which to spread the same amount of charge.

Charge Distribution

Charge distribution refers to the way electrical charge is spread out within or across a surface. In the context of conducting spheres, charge distribution follows specific rules determined by the geometry and the conductive properties of the material.

Upon reaching electrostatic equilibrium, the charge distribution is solely on the surface of a conductor and is influenced by factors such as the shape of the conductor and the presence of other charges or electric fields within the vicinity. In conducting spheres, it is often assumed that the charge will spread out uniformly across the surface due to the spherical symmetry. However, if another charged object is brought close to the sphere, the distribution may no longer be uniform as the charges will move in response to the external electric field. In textbook examples involving calculations of surface charge densities and resulting charges, it is important for students to grasp the nuances of how charges arrange themselves and how the uniform distribution can be used to simplify the solutions.