Question

Two metal spheres of radii \(r_{1}=10.0 \mathrm{~cm}\) and \(r_{2}=\) \(20.0 \mathrm{~cm},\) respectively, have been positively charged so that both have a total charge of \(100, \mu C\) a) What is the ratio of their surface charge distributions? b) If the two spheres are connected by a copper wire, how much charge flows through the wire before the system reaches equilibrium?

Short answer

Answer: To find the ratio of the surface charge distributions of the two spheres, follow the steps below: 1. Calculate the surface charge density of each sphere using the formula \(\sigma = \frac{Q}{A}\). Given their radii (\(r_1 = 10.0\) cm, \(r_2 = 20.0\) cm) and charge (\(100\ \mu C\)), find \(\sigma_1\) and \(\sigma_2\). 2. Determine the ratio between the surface charge densities: \(\frac{\sigma_1}{\sigma_2}\). For the charge flow: 1. Find the ratio of charges at equilibrium: \(\frac{Q_1}{Q_2} = \frac{r_1}{r_2}\) 2. Calculate final charges on each sphere by solving \(Q_1 + Q_2 = Q_{total}\) with the given ratio. 3. Determine the charge flow: \(Q_{flow} = \Delta Q = |Q_2 - Q_1|\). Perform these calculations using the given values to obtain the surface charge distribution ratio and the charge flow between the connected spheres.

Step-by-step solution

Find surface charge densities of the spheres

To find the surface charge density (\(\sigma\)) of each sphere, we use the formula: $$\sigma = \frac{Q}{A}$$ Where \(Q\) is the charge on the sphere and \(A\) is the surface area of the sphere, given by: $$A = 4 \pi r^2$$ For each sphere, we can calculate the surface charge density using the given radii (\(r_1 = 10.0\) cm, \(r_2 = 20.0\) cm) and the same charge (\(100\ \mu C\)).

Calculate the ratio of surface charge distributions

Now that we have the surface charge densities for each sphere, we will determine the ratio: $$\frac{\sigma_1}{\sigma_2}$$ Using the surface charge density equations from Step 1, we can plug in the values and calculate the ratio.

Determine the final charges of the spheres at equilibrium

When the two spheres are connected by a wire, they will reach an equilibrium state where both spheres have the same potential. The potential of a charged sphere is given by: $$V = \frac{kQ}{r}$$ Where \(k\) is the electrostatic constant (\(8.9875 \times 10^9 N m^2 C^{-2}\)). At equilibrium, the potentials of the two spheres are equal: $$V_1 = V_2$$ $$\frac{kQ_1}{r_1} = \frac{kQ_2}{r_2}$$ We can then use this equation to determine the ratio of charges on the spheres at equilibrium: $$\frac{Q_1}{Q_2} = \frac{r_1}{r_2}$$

Calculate the charge flow through the wire

Now that we have the ratio of charges at equilibrium, we can determine the final charge on each sphere and thereby find the charge flow between the spheres. Using the charge ratio, we can calculate the charges \(Q_1\) and \(Q_2\): $$Q_1 + Q_2 = Q_{total}$$ Where \(Q_{total}\) is the total charge on both spheres (200 \(\mu\)C). Using the charge ratio and this equation, we can determine the final charges on each sphere and calculate the difference in charges, which represents the charge flow through the wire: $$Q_{flow} = \Delta Q = |Q_2 - Q_1|$$

Key concepts

Electrostatics

Electrostatics is a branch of physics that studies the properties and behaviors of electric charges at rest. It deals with the forces between charges that are not moving, known as electrostatic forces. These forces arise due to the presence of charges and can be attractive or repulsive depending on the types of charges involved.

Understanding the Basics

Every object around us is made up of atoms, and atoms have positively charged protons and negatively charged electrons. When an imbalance occurs between the number of protons and electrons, an object becomes electrically charged. In this exercise, the metal spheres are positively charged, indicating an excess of protons.

The principle of electrostatics is crucial when comprehending phenomena such as the attraction and repulsion between objects, the distribution of electric charges on conductors, and the concept of electric potential, which is the ability to do work because of the position of a charge in an electric field. Applying this to our metal spheres exercise, the charges on the spheres create electrostatic forces and potential differences that can be calculated and analyzed.

Electric Charge

The concept of electric charge is fundamental to understanding electrostatics. An electric charge is a property of subatomic particles, such as protons and electrons, and it determines how those particles interact with each other electromagnetically.

Charge Quantization and Conservation

Electric charges are quantized, which means they can be broken down into a specific, smallest unit named the elementary charge, commonly represented by the symbol 'e'. Also, charges are conserved in an isolated system, meaning the total charge remains constant over time.

In the exercise, we are dealing with two metal spheres each carrying a charge of 100 microcoulombs (\(100 \times 10^{-6} C\)). When we consider the flow of charge between the spheres, we're essentially looking at how the charges redistribute themselves to maintain electrostatic equilibrium. The calculation of surface charge density is key to comparing how the charges are spread out on the surfaces of different-sized spheres before they are connected.

Electric Potential

Electric potential, often simply called potential, is the electric potential energy per unit charge at a certain point in a static electric field. It's analogous to gravitational potential in mechanics and indicates the work done in bringing a unit positive charge from infinity to the point in question.

Electric Potential and Potential Difference

When we talk about a charged sphere's electric potential in the context of our exercise, we're referring to the work that would be necessary to bring a small charge from infinitely far away up to the surface of the sphere. We use the formula \(V = \frac{kQ}{r}\), where 'V' is the electric potential, 'k' is Coulomb's constant, 'Q' is the charge on the sphere, and 'r' is the radius of the sphere.

In the scenario where the two charged spheres are connected by a wire, they reach an equilibrium state where both spheres share the same electric potential. This common potential causes charges to flow from one sphere to the other, redistributing until equilibrium is achieved. The amount of charge that flows can be calculated using their surface charge densities before connection and understanding that at equilibrium, the charge ratios will adjust based on their respective radii.