Redistribution of Charge | AP Physics C: E&M Unit 3 Study Guide

Getting Started

Consider a system of two isolated, charged conducting spheres, one large and one small, initially held far apart. Because of their different charges and sizes, they exist at different electric potentials. The core question we will explore is: what happens to the charge, the potential, and the electric field of this system when the two spheres are connected by a thin conducting wire?

What You Should Be able to Do

After studying this section, you will be able to:

Calculate the final charge on each conductor and their common final potential after they are brought into electrical contact, applying the principles of charge conservation and equipotential.
Determine the direction of net charge flow between conductors by comparing their initial electric potentials.
Analyze the process of charging by induction, explaining the role of an external field and a ground connection in establishing a net charge on a conductor.
Relate the final charge distribution between connected conductors to their respective capacitances.

Key Concepts & Mechanisms

This topic is best understood through the lens of Dynamics and Fields as Cause. The process of charge redistribution is driven by the fundamental tendency of a system to reach electrostatic equilibrium. An initial difference in electric potential creates an electric field, which in turn exerts a force on charges, causing them to move until that driving potential difference is eliminated.

System & Preconditions:
Our system consists of two or more conductors, which we assume to be ideal. In an ideal conductor, an unlimited supply of free charges can move without resistance. In electrostatic equilibrium, all net charge resides on the surface of a conductor, and the electric field inside the conductor is zero ( $E_{in} = 0$ ). This implies that the entire volume and surface of a single conductor must be at the same electric potential. We analyze these systems in a quasi-static framework, focusing on the initial and final equilibrium states, assuming the transition between them happens very quickly.
Key Steps / Relations:
1. Initial State & The Driver: The system begins with at least two conductors, say Conductor 1 and Conductor 2, with initial charges $Q_{1}$ and $Q_{2}$ . Due to their charges and geometries, they have initial, distinct electric potentials, $V_{1}$ and $V_{2}$ . If $V_{1} \neq = V_{2}$ , a potential difference $Δ V = V_{1} - V_{2}$ exists. This potential difference is the driver of the process; it signifies the presence of a non-zero electric field in the space between the conductors. The relationship is given by $E = - \nabla V$ , meaning the field points from regions of higher potential to regions of lower potential.
2. The Process: Making Contact: When the conductors are connected by a conducting wire (or touch directly), they effectively become a single, larger conductor. The free charges are now able to move throughout the entire combined system. The electric field associated with the potential difference exerts a force ( $F = q E$ ) on these free charges, causing a net flow. By convention, we speak of positive charge flowing from higher potential to lower potential.
3. Final State: Equilibrium: The net flow of charge continues until the potential is uniform everywhere across the connected conductors. At this point, the potential difference between them is zero, the electric field driving the current vanishes, and the system has reached a new electrostatic equilibrium. The final state is defined by two core principles:
  - Equipotential Condition: The final potentials are equal: $V_{1}^{'} = V_{2}^{'} = V_{f}$ .
  - Conservation of Charge: For an isolated system, the total charge before and after contact is the same: $Q_{t o t a l} = Q_{1} + Q_{2} = Q_{1}^{'} + Q_{2}^{'}$ .
Outputs & Effects:
The result is a redistribution of the initial total charge. The final charges, $Q_{1}^{'}$ and $Q_{2}^{'}$ , are generally different from the initial charges. For any conductor, its potential is proportional to the charge it holds, a relationship defined by its capacitance, $C$ , where $C = Q / V$ . Using this, the final state equations become:
$V_{f} = \frac{Q _{1}^{'}}{C _{1}} = \frac{Q _{2}^{'}}{C _{2}}$ and $Q_{1}^{'} + Q_{2}^{'} = Q_{t o t a l}$ .
These can be solved to find the final charge on each conductor. For example, for two connected spherical conductors with radii $R_{1}$ and $R_{2}$ , their capacitances are $C_{1} = 4 π ϵ_{0} R_{1}$ and $C_{2} = 4 π ϵ_{0} R_{2}$ . The final charges will be distributed in proportion to their radii (and thus their capacitances): $\frac{Q _{1}^{'}}{Q _{2}^{'}} = \frac{R _{1}}{R _{2}}$ .
Regulation & Limits: The Role of Ground
The concept of ground is an important idealization. Ground is defined as an infinitely large conductor whose potential is fixed at zero volts ( $V_{g ro u n d} = 0$ ), regardless of how much charge is added to or removed from it. When a conductor is connected to ground, charge will flow between them until the conductor's potential is also zero. This is the key mechanism in charging by induction:
1. An external field from a charge (e.g., $+ Q_{e x t}$ ) is brought near a neutral conductor, polarizing it.
2. The conductor is connected to ground. To bring the conductor's potential to zero in the presence of $+ Q_{e x t}$ , negative charge flows from ground onto the conductor.
3. The ground connection is removed, trapping the net negative charge on the conductor.
4. The external charge $+ Q_{e x t}$ is removed, and the trapped negative charge spreads out over the conductor's surface.

Key Models & Diagrams

The process of charge redistribution between two conductors can be visualized with the following flowchart:

Initial State

(Conductors Separate)

Conductor 1: Charge $Q_{1}$ , Potential $V_{1}$
Conductor 2: Charge $Q_{2}$ , Potential $V_{2}$
Condition: $V_{1} \neq = V_{2}$
↓

Process: Connection

(Conducting path established)

A potential difference $Δ V = V_{1} - V_{2}$ drives a net flow of charge.
Charge moves from higher potential to lower potential.
↓

Governing Principles

(Applied to the combined system)

Conservation of Charge: $Q_{t o t a l} = Q_{1}^{'} + Q_{2}^{'} = Q_{1} + Q_{2}$
Equilibrium Condition: The final system is a single equipotential surface.
↓

Final State

(Equilibrium Reached)

Conductor 1: Charge $Q_{1}^{'}$ , Potential $V_{f}$
Conductor 2: Charge $Q_{2}^{'}$ , Potential $V_{f}$
Condition: $V_{1}^{'} = V_{2}^{'} = V_{f}$

Key Components & Evidence

Conductor: A material containing mobile charges (e.g., electrons in a metal) that are free to move throughout the material in response to an electric field.
Electric Potential (V): A scalar field representing the potential energy per unit charge. Its gradient is the negative of the electric field vector ( $E = - \nabla V$ ). Differences in potential drive charge flow. Units: Volts (V).
Conservation of Charge: A fundamental law stating that the net charge of an isolated system cannot change.
Equipotential: A region or surface in space where every point has the same electric potential. The surface of a conductor in electrostatic equilibrium is an equipotential.
Ground: An idealized electrical reference point with a potential defined as zero volts ( $V = 0$ ). It can act as a nearly infinite source or sink of charge without its potential changing.
Capacitance (C): A geometric property of a conductor (or system of conductors) that quantifies the amount of charge stored per unit of electric potential, $C = Q / V$ . Units: Farads (F).
Electric Field ( $E$ ): The force experienced per unit of positive charge, $F = q E$ . In static equilibrium, $E = 0$ inside a conductor. Units: Newtons per Coulomb (N/C) or Volts per meter (V/m).
Charging by Induction: A method to give a conductor a net charge without physical contact with another charged object, utilizing an external field and a temporary connection to ground.

Skill Snapshots

Causation

Driver: A potential difference ( $Δ V \neq = 0$ ) between two connected conductors. → Change: Net charge flows from the region of higher potential to the region of lower potential until $Δ V = 0$ .
Driver: An external positive charge is brought near a grounded conductor. → Change: Negative charge flows from ground onto the conductor to maintain the conductor's potential at zero.
Driver: The ground connection to a polarized conductor is severed while an external field is still present. → Change: The induced charge is trapped on the conductor, resulting in a net charge after the external field is removed.

Comparison

Charging by Contact vs. Charging by Induction: In contact, the total charge of the initial conductors is shared, and the final objects typically have charges of the same sign. In induction, charge is transferred from ground, and the final charge on the conductor is opposite in sign to the inducing charge.
Isolated Conductor vs. Grounded Conductor: An isolated conductor maintains a constant total charge, and its potential varies depending on nearby charges. A grounded conductor maintains a constant potential (V=0), and its total charge varies as needed to do so.
Connecting Equal vs. Unequal Spheres: When two identical conducting spheres touch, they share the total charge equally ( $Q_{1}^{'} = Q_{2}^{'}$ ). When two spheres of unequal radii touch, they achieve the same final potential, which results in the larger sphere (larger capacitance) holding a greater amount of the final charge.

Change, Continuity, and Conservation (CCOT)

Consider charging a neutral sphere by induction using a positive rod.

Baseline: The isolated sphere is electrically neutral ( $Q_{n e t} = 0$ ).
Change 1: The positive rod is brought near, polarizing the sphere. Then, the sphere is grounded. Electrons flow from ground onto the sphere, giving it a net negative charge ( $Q_{n e t} < 0$ ) to keep its potential at $V = 0$ .
Change 2: The ground wire is removed, then the rod is removed. The net negative charge is now trapped and redistributes uniformly over the sphere's surface.
Continuity: The positive charge on the inducing rod remains constant and is never transferred to the sphere.

Common Misconceptions & Clarifications

Misconception: When charged conductors touch, they always split the total charge evenly.
Clarification: Conductors in contact reach the same electric potential, not necessarily the same charge. Charge is shared equally only if the conductors have identical size and shape (i.e., equal capacitance). A conductor with a larger capacitance will hold more charge at the same potential.
Misconception: Ground is an object that "absorbs" or "neutralizes" charge.
Clarification: Ground is an idealized, infinite charge reservoir defined to be at zero potential. It can source or sink any amount of charge required to bring a connected conductor to zero potential, without its own potential changing.
Misconception: An object with zero net charge (neutral) must be at zero potential.
Clarification: A neutral conductor will only be at zero potential if it is infinitely far from all other charges or if it is physically located on a V=0 equipotential surface of an external field. In general, a neutral object placed in an external E-field will have a non-zero potential. Grounding is the process that forces its potential to zero.

One-Paragraph Summary

The redistribution of charge between conductors is governed by the system's evolution towards electrostatic equilibrium, where all connected conducting surfaces form a single equipotential. When conductors with an initial potential difference are connected, charge flows from higher to lower potential until a common final potential is reached. For an isolated system, this process is constrained by the conservation of total charge. The final distribution of charge is determined by the capacitance of each conductor, with larger conductors holding more charge at the same potential. This principle is extended in charging by induction, where a connection to ground—an ideal zero-potential reservoir—is used in the presence of an external field to induce and trap a net charge on a conductor without direct contact.

Redistribution of Charge Between Conductors - AP Physics C: Electricity and Magnetism Study Guide