91Ó°ÊÓ

Two capacitors with the same capacitance \(C\) and charge \(Q\) are placed next to each other, as shown in Fig. 3.39. The two positive plates are then connected by a wire. Will charge flow in the wire? Consider two possible reasonings: (A) Before the plates are connected, the potential differences of the two capacitors are the same (because \(Q\) and \(C\) are the same). So the potentials of the two positive plates are equal. Therefore, no charge will flow in the wire when the plates are connected. (B) Number the plates 1 through 4 , from left to right. Before the plates are connected, there is zero electric field in the region between the capacitors, so plate 3 must be at the same potential as plate 2 . But plate 2 is at a lower potential than plate \(1 .\) Therefore, plate 3 is at a lower potential than plate 1 , so charge will flow in the wire when the plates are connected. Which reasoning is correct, and what is wrong with the wrong reasoning?

Step by step solution

01

Analyzing Reasoning A

To analyze the first scenario, recall that the potential difference across the capacitor is given by \(\frac{Q}{C}\) where \(Q\) is the charge on the capacitor and \(C\) is the capacitance. Since both capacitors have the same charge \(Q\) and capacitance \(C\) before connecting, the potential differences for both capacitors are indeed the same. Therefore, when plate 1 and plate 3 are connected, no charge will flow because there is no potential difference. This reasoning seems technically sound, and corresponds to the conservation of energy where no energy is given or taken in this process.

02

Analyzing Reasoning B

Looking at the second scenario, we consider the electric field and potential. There is indeed a zero electric field in the region between the two capacitors, which implies that plate 3 is at the same potential as plate 2. However, we have to remember that plate 2 is negatively charged and has a lower potential than plate 1. This would make plate 3 have a lower potential than plate 1 too. Hence, suggesting that upon connection, charge will flow from plate 1 to plate 3 until both have the same potential, equating the electric field to zero. This reasoning acknowledges that the electric field inside a conductor in electrostatic equilibrium is zero, and the electric potential is constant throughout a conductor in electrostatic equilibrium.

03

Deciding the Correct Reasoning

Both Reasoning A and B seem technically sound and adhere to certain principles of physics. However, from the perspective of electrostatic equilibrium in conductors, Reasoning B has an edge over Reasoning A. While the potential difference of the two capacitors are initially the same, upon connecting the plates 1 and 3, plate 3 would indeed be at a lower potential than plate 1, which would cause a charge flow until electric field and the potential of the conductive plates are uniform, leading to electrostatic equilibrium. As such when considering the principles of elecrtostatic equilibrium in conductors, Reasoning B is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electrostatic Equilibrium

Electrostatic equilibrium describes the state of a conductor when the electric field inside it becomes zero. This happens because the charges in the conductor redistribute themselves in response to any applied electric field until no further charge movement occurs.

In a solid conductor, electrons can move freely. When a conductor reaches electrostatic equilibrium:

• The electric field within the conductor is zero.
• The electric potential is constant throughout the conductor's surface and interior.
• Any excess charge resides on the external surface.

These principles indicate that connecting two plates as mentioned in Reasoning B will cause charges to move until they reach a state where they are evenly distributed. The charges stop moving once the electric field is zero and the potential is the same throughout the conductor. In this scenario, the initially different potentials of the connected plates prompt charge movement to restore balance.

Potential Difference

Potential difference, also known as voltage, represents the work done to move a charge from one point to another in an electric field. It is an essential concept for understanding how electric circuits function because it drives the movement of charge.

• It is measured in volts.
• Potential difference can be calculated with the formula: \( V = \frac{Q}{C} \), where \(Q\) is the charge and \(C\) is the capacitance.
• If there is no potential difference between two points, no current flows.

In the exercise, Reasoning A correctly identifies that the potential difference for both capacitors initially is the same, suggesting no immediate charge flow when they are connected. However, this overlooks adjustments that occur because of the redistribution of charge, as explained in Reasoning B. When wires connect conductors at different potentials, electrons move until the potential difference is nullified, resulting in equilibrium.

Electric Field

An electric field represents the force exerted per unit charge in the surrounding space of a charged particle. It's a vector field that influences how charged objects interact with one another.

• The direction of the electric field shows the direction of the force it would apply to a positive charge, and it's measured in newtons per coulomb (N/C).
• Electric fields point away from positive charges and toward negative charges.

In this exercise, the absence of an electric field between two charged plates when they are in electrostatic equilibrium suggests uniform potential across them. When plates 1 and 3 are connected, initially, an electric field exists due to different potentials, driving charge from a higher to a lower electric potential. The movement continues until the electric field becomes zero once more, reinforcing the state of electrostatic equilibrium. Understanding electric fields is crucial to comprehending how and why charges will move in a given setup and how they will eventually reach equilibrium.