Short on time? This page is the recap. For the derivations, Figure 1 and the worked examples, use the full study guide. This topic is part of the calculus-based course.
Recap
- Conductors in electrical contact become one conductor, so they end at one potential. The flow is effectively instant.
- Total charge is conserved. Electrons move towards higher potential (positive charge “flows” from high V to low V).
- Equal charges result only for identical conductors.
- A charged conductor joined to a hollow conductor that surrounds it loses all its charge to the outer surface.
- Ground is an ideal reference at V = 0 that can absorb or supply any amount of charge without changing its potential.
- Grounding an isolated conductor makes it neutral. Grounding a conductor near a charge leaves an induced charge of opposite sign.
- To keep an induced charge: break the ground first, then remove the nearby charge.
Key relationships
| Situation | Result |
|---|---|
| Any conductors in contact | V₁ = V₂; q₁ + q₂ = Q_total |
| Distant spheres joined by a thin wire | q₁/R₁ = q₂/R₂; q₁ = QR₁/(R₁ + R₂) |
| Same pair: densities and fields | σ₁/σ₂ = E₁/E₂ = R₂/R₁; E = V/R at each surface |
| Grounded sphere (radius R), point charge q at distance d from centre | Q′ = −qR/d |
| Grounded shell around a charge q | outer surface charge → 0; no field outside |
Assumptions
- “Far apart”: each sphere’s potential comes only from its own charge.
- “Thin wire”: the wire holds negligible charge.
- Ground is ideal: its potential never changes.
Mistakes to avoid
- Sharing equally between different-sized conductors. Use equal V.
- Bigger sphere → bigger surface field. It is the smaller sphere.
- Grounding always gives zero charge. Not with a charge nearby.
- Wrong order of steps. Remove the ground before the inducing charge.
- Using the uneven spread of Q′ to block the calculation. At the centre, all of Q′ is at distance R, so its potential is kQ′/R.
Quick self-check
- Distant spheres of radii 1.0 cm and 4.0 cm share 10 nC through a wire. Charges? (2.0 nC and 8.0 nC)
- Which of those spheres has the stronger surface field, and by what factor? (The 1.0 cm sphere, 4 times stronger)
- A grounded sphere of radius 0.10 m is 0.50 m (centre to charge) from +2.0 nC. Induced charge? (−0.40 nC)
Next: practice questions.