10.2 Redistribution of Charge Between Conductors

When two conductors touch or connect by a wire, charge moves until both reach the same electric potential, and the total charge stays constant. Ground acts as an unlimited charge reservoir at zero potential, so grounding a conductor can drain its charge or, with a nearby charged object, leave behind an induced charge. These ideas help you reason through charge sharing, induction, and conductor behavior in electrostatic equilibrium.

Why This Matters for the AP Physics C: E&M Exam

This topic builds the reasoning you use across Unit 10 and into circuits in Unit 11. You connect three big ideas: charge is conserved, contacting conductors share a common potential, and ground can absorb or supply charge freely. The free-response section includes a Qualitative/Quantitative Translation question that asks you to make a claim, support it with reasoning, and then derive matching equations. Charge sharing and grounding are clean setups for that kind of question because you have to explain what happens and back it with the potential of an isolated sphere, $V = \frac{kQ}{R}$. Expect to use diagrams, compare before-and-after charge values, and justify your answer with physical principles rather than plugging into one formula.

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Key Takeaways

• When conductors are in electrical contact, charge redistributes until every connected surface sits at the same electric potential.
• Total charge is conserved during contact; the charges just rearrange between the conductors.
• For identical spheres that touch and separate, the total charge splits equally between them.
• Ground is an idealized point at zero potential that can absorb or supply any amount of charge without changing its own potential.
• Grounding a charged conductor (with no external field nearby) brings it to zero potential, draining its excess charge.
• Grounding a conductor while an external charge is nearby can trap an induced charge if you remove the ground before removing the external charge.

How Charge Redistributes

Contact Between Conductors

When two conductors are connected electrically, free electrons move until the potential is the same everywhere on the connected surfaces. Electrons flow from lower potential toward higher potential (since electrons carry negative charge), and the movement stops once the potentials match. At that point the field inside each conductor is zero again, which is the electrostatic equilibrium condition.

How the final charge splits depends on the geometry of the conductors. For an isolated sphere, potential is $V = \frac{kQ}{R}$, so a larger sphere can hold more charge at the same potential. Two identical spheres that touch share their total charge equally because matching potential means matching charge when the radii are equal.

Ground as a Reference

Ground is treated as an idealized conductor at zero potential with effectively unlimited capacity to take in or release charge. Connecting a conductor to ground lets charge flow until the conductor also reaches zero potential.

• If the conductor sits above zero potential, electrons flow from ground onto it.
• If the conductor sits below zero potential, electrons flow off it into ground.

With no other charges around, grounding a charged conductor drains its excess charge and leaves it neutral.

Charging by Induction Using Ground

Bring a charged object near a grounded conductor without touching it. The external field pushes the conductor's free electrons toward one side, and because the conductor is grounded, extra charge can flow in or out through the ground connection.

The order of steps matters:

1. Bring the external charge near the grounded conductor.
2. Disconnect the ground while the external charge is still present.
3. Remove the external charge.

A positive object nearby pulls electrons toward the near side and draws more electrons up from ground, so the conductor ends up with a net negative charge after grounding is removed. A negative object nearby pushes electrons into ground, leaving the conductor net positive. The induced charge is opposite in sign to the external charge.

Practice Problem 1: Charge Redistribution Between Spheres

Solution

When conductors touch, charge redistributes until they share the same potential. Because the spheres are identical, equal potential means equal charge.

Charge is conserved, so add the starting charges:

$Q_{total} = Q_1 + Q_2 = 6.0 \text{ μC} + (-2.0 \text{ μC}) = 4.0 \text{ μC}$

Split that total equally between the two identical spheres:

$Q_{final} = \frac{Q_{total}}{2} = \frac{4.0 \text{ μC}}{2} = 2.0 \text{ μC}$

Each sphere ends with a final charge of +2.0 μC.

Practice Problem 2: Grounded Conductor

Solution

Connecting a charged conductor to ground lets charge flow until the conductor reaches zero potential, the same as ground.

The sphere starts positive, so electrons flow from ground onto the sphere and neutralize the excess positive charge:

$Q_{final} = 0 \text{ C}$

Ground can supply as many electrons as needed, so the sphere ends neutral and at zero potential.

How to Use This on the AP Physics C: E&M Exam

Problem Solving

For charge-sharing problems, start with conservation of total charge, then use the equal-potential condition to split it. For identical spheres, equal potential means equal charge. For spheres of different radii, use $V = \frac{kQ}{R}$ and set the potentials equal to find how the charge divides.

Free Response

If a question asks you to explain before deriving, state the claim in words first: charge moves until potentials are equal, and total charge is conserved. Then back it with the potential relationship and your before-and-after charge values. This mirrors the structure of the Qualitative/Quantitative Translation question, where you make a claim, support it with reasoning, and then derive the supporting equation.

Common Trap

Watch the sequence in induction problems. The conductor only keeps an induced charge if the ground is removed before the external charge is taken away. If you remove the external charge first, the conductor goes back to neutral.

Common Misconceptions

• Charge does not always split evenly. Equal sharing only happens for identical conductors. Different geometries split charge differently because the equal quantity is potential, not charge.
• Grounding does not always make a conductor neutral. With a nearby external charge, grounding can leave an induced charge once the ground is disconnected in the right order.
• Conductors in contact match potential, not charge. They reach the same electric potential, and the resulting charges depend on their shapes and sizes.
• Ground is not a fixed amount of charge. It is an idealized reservoir that supplies or absorbs whatever charge is needed while staying at zero potential.
• Induced charge is opposite in sign to the nearby external charge, not the same sign. A positive object nearby leaves a grounded conductor negative after the correct sequence of steps.

Related AP Physics C: E&M Guides

• 10.1 Electrostatics with Conductors
• 10.3 Capacitors
• 10.4 Dielectrics