Conservation of Electric Charge and the Process of Charging: Study Guide (Physics 2 10.2)

Charge is conserved

Pick a system: one object or a group of objects you choose to study. Its net charge is the sum of all the positive and negative charges inside it.

Conservation of charge: the net charge of a system stays constant unless charge moves into or out of the system. Charge is never created from nothing or destroyed. Any change in a system’s net charge is matched by an equal and opposite change in its surroundings.

In everyday solids the protons are locked inside nuclei, which stay in place. The particles that move are electrons. So when an object becomes charged:

• it becomes negative by gaining electrons, and
• it becomes positive by losing electrons.

“Positive charge flowed onto the sphere” is almost always the wrong picture for a solid. Electrons flowed off it.

You can count the electrons with q = Ne from Topic 10.1. A rod that gains 2.5 × 10¹⁰ electrons has charge −(2.5 × 10¹⁰)(1.60 × 10⁻¹⁹ C) = −4.0 × 10⁻⁹ C.

Charging by friction

Rub two different materials together, for example a plastic rod with a wool cloth. Close contact lets electrons move from one material to the other. Which way they move depends on the two materials.

Suppose the rod gains 2.5 × 10¹⁰ electrons from the cloth. Then:

• the rod has charge −4.0 × 10⁻⁹ C,
• the cloth has charge +4.0 × 10⁻⁹ C, and
• the rod-plus-cloth system still has net charge zero.

Rubbing does not make charge. It separates charge that was already there. The two objects always end with equal and opposite charges, as long as no charge escapes to anything else.

Charging by contact

Touch a charged conductor to an uncharged (or differently charged) conductor. Electrons move between them until neither has a reason to push more across. Then separate them.

• The total charge of the two objects is the same before and after.
• For two identical conducting spheres, symmetry means each ends with half the total: q_each = (q₁ + q₂)/2.
• A sphere can change sign. If one sphere has +6 nC and the other −2 nC, the total is +4 nC, so each ends with +2 nC.

If one object is an insulator, little charge moves. Charge on an insulator stays roughly where it was put, because charge carriers cannot move easily through it. On a conductor, extra charge spreads over the whole surface.

Induced charge separation: the neutral object is not passive

A charged object can rearrange the charges inside a nearby object without touching it. This is induced charge separation. It happens even when the nearby object is neutral, and its net charge does not change.

In a conductor. Bring a negatively charged rod near a neutral metal sphere on an insulating stand. The rod repels the sphere’s free electrons, which move to the far side. The near side is left with a shortage of electrons, so it is positive. The sphere as a whole is still neutral: equal amounts of + and − charge, just separated.

In an insulator. Electrons cannot travel across the material. Instead, inside each atom or molecule the electrons shift slightly away from the rod. Every molecule becomes a tiny ±pair, all lined up the same way. This is polarization. The surface nearest the rod ends up slightly positive and the far surface slightly negative.

Why a charged object attracts a neutral one

In both cases in Figure 1, the charge nearer the rod has the opposite sign to the rod, and the charge farther away has the same sign. Coulomb’s law depends on 1/r², so the nearer charges feel a stronger force than the farther ones. The attraction wins, and the neutral object is pulled toward the rod.

This works whatever the sign of the rod. A positive rod pulls electrons to the near side and also attracts the neutral object. So “it was attracted” does not prove an object is charged. Only repulsion proves that two objects carry charge of the same sign.

Grounding

Grounding means connecting an object by a conductor to a very large, roughly neutral system, usually Earth. Earth is so large that it can give or accept huge numbers of electrons while its own charge barely changes.

• A negatively charged metal sphere connected to ground loses its extra electrons to Earth and becomes (nearly) neutral.
• A positively charged metal sphere connected to ground gains electrons from Earth and becomes (nearly) neutral.
• Charge is still conserved. It is just shared with a system so large that the sphere’s share is close to zero.

If a charged rod is nearby while the sphere is grounded, the result is different, as the next section shows.

Charging by induction

You can give a metal sphere a charge using a charged rod that never touches it. With a negatively charged rod:

1. Bring the rod near. Electrons in the sphere are pushed to the far side; the near side becomes positive. Net charge: 0.
2. Ground the sphere (touch it with a wire to Earth, or a finger). The rod’s repulsion drives electrons from the sphere into the ground.
3. Remove the ground connection while the rod is still there. The electrons that left cannot return.
4. Remove the rod. The sphere is left with a shortage of electrons, so it is positive. The positive charge spreads evenly over the surface.

Two rules to remember:

• The sphere ends with charge opposite to the rod’s.
• The rod loses none of its own charge. The charge the sphere gains or loses comes from or goes to the ground.

If you remove the rod before breaking the ground connection, electrons flow back from Earth and the sphere ends neutral.

Worked example 1: three identical spheres

Question. Three identical metal spheres on insulating stands start with A: +9.0 nC, B: −3.0 nC, C: 0. A touches B and they are separated. Then B touches C and they are separated. (a) Find the final charge on each sphere and check that charge is conserved. (b) How many electrons moved during the first contact, and in which direction? (c) A and B are 0.10 m apart before and after the first contact. Compare the forces between them.

1. Total charge at the start: 9.0 + (−3.0) + 0 = +6.0 nC.
2. A and B touch: total +6.0 nC, shared equally, so A = B = +3.0 nC.
3. B and C touch: total +3.0 + 0 = +3.0 nC, so B = C = +1.5 nC.
4. Final: A = +3.0 nC, B = +1.5 nC, C = +1.5 nC. Sum = +6.0 nC. Charge is conserved.
5. (b) B went from −3.0 nC to +3.0 nC, a change of +6.0 nC. B lost electrons: N = 6.0 × 10⁻⁹ C ÷ 1.60 × 10⁻¹⁹ C = 3.75 × 10¹⁰ electrons, which moved from B to A. (A changed by −6.0 nC, which matches.)
6. (c) Before: F = (9.0 × 10⁹)(9.0 × 10⁻⁹)(3.0 × 10⁻⁹) ÷ (0.10)² = 2.43 × 10⁻⁵ N, attractive (opposite signs). After: F = (9.0 × 10⁹)(3.0 × 10⁻⁹)² ÷ (0.10)² = 8.1 × 10⁻⁶ N, repulsive (same sign). The force is 1/3 as large and has reversed direction.

Check. The final charges add up to the starting total, +6.0 nC. Notice that B’s sign changed: contact sharing can do that when the other sphere carries more charge of the opposite sign.

Worked example 2: charging by induction, with numbers

Question. A neutral metal sphere on an insulating stand is charged by induction using a negatively charged rod, following the four steps in Figure 2. At the end the sphere has charge +2.4 nC. (a) How many electrons left the sphere, and where did they go? (b) What is the charge of the rod at the end? (c) A student repeats the experiment but removes the rod before removing the ground wire. Predict the final charge of the sphere and explain.

1. (a) N = 2.4 × 10⁻⁹ C ÷ 1.60 × 10⁻¹⁹ C = 1.5 × 10¹⁰ electrons. They went through the ground connection into Earth in step 2, pushed by the repulsion from the negative rod.
2. (b) The rod never touched the sphere, so no charge moved to or from the rod. Its charge is unchanged.
3. (c) When the rod is removed, nothing repels the electrons any more. With the ground still connected, the sphere attracts electrons back from Earth until it is neutral. Final charge: 0.

Interpretation. Count the system carefully. Sphere + Earth: net charge zero at the start and at the end (+2.4 nC on the sphere, −2.4 nC spread through Earth). Charge was conserved; it was moved, not made.

Common misconceptions

• “Positive charges flow onto an object.” In solids, protons stay in nuclei. Positive objects have lost electrons.
• “Rubbing creates charge.” Rubbing transfers electrons. The two objects end with equal and opposite charges.
• “Two touching spheres always share charge equally.” Only identical spheres share equally. The total is conserved in every case.
• “A neutral object cannot be attracted.” Induced charge separation or polarization makes the near side opposite in sign, and that side is closer.
• “Attraction proves both objects are charged.” A charged object attracts neutral objects too. Only repulsion proves like charges.
• “In induction the rod gives charge to the sphere.” The rod keeps all its charge. The sphere’s charge comes from or goes to ground.
• “Grounding always makes an object neutral.” Only when no other charge is nearby. With a charged rod nearby, a grounded object takes on a charge opposite to the rod’s.
• “Insulators cannot be charged.” They can, by rubbing. The charge simply stays where it was placed instead of spreading.

Where this leads

Topic 10.3 describes the effect of a charge on the space around it as an electric field, and explains why charge on a conductor sits on its surface: see the Topic 10.3 study guide. Go back to the Topic 10.1 study guide if Coulomb’s law is not yet secure. Test yourself with the practice questions, then use the revision notes and the topic checklist to consolidate.