Conductors and Capacitors

Mobile charges in a conductor respond to internal electric fields. In electrostatic equilibrium the field inside the conducting material is zero and the whole connected conductor is at one potential. Excess charge resides on surfaces. The field immediately outside a conductor is perpendicular to the surface; a tangential component would move charges.

A cavity containing no charge has zero electrostatic field when enclosed by a conductor in equilibrium. If a charge is inside a cavity, induced inner-surface charge sums to its negative; the conductor’s net charge determines the remaining outer charge. Grounding allows charge exchange with a large reservoir, so grounded and isolated conductors are different constraints.

A capacitor stores separated equal-magnitude opposite charges. Its capacitance is C=QΔV, where Q is the magnitude on either plate, not the sum of both plates. For large parallel plates with negligible edge effects, C=ε₀Ad in vacuum. Increasing area increases C; increasing separation decreases C.

The approximately uniform gap field is E=ΔVd=Q(ε₀A). State the idealizations: plate dimensions much larger than separation, nearly uniform field, and negligible fringing. This course’s quantitative capacitor work uses parallel plates rather than derivations for cylindrical or spherical capacitors.

Moving additional charge onto a capacitor takes increasing work because the potential difference grows with charge. Integrating dW=(qC)dq gives U=Q²(2C)=C(ΔV)²2=QΔV2. These expressions are equivalent, but choose the one that keeps the experiment’s fixed quantity explicit.

For a vacuum gap, energy density is u=ε₀E²2. Multiplying by volume Ad recovers the capacitor energy. This field description helps explain why separating charged plates requires work when Q is fixed: the field-filled volume increases.

If a charged capacitor is disconnected from its battery and leakage is negligible, Q stays constant. Doubling plate spacing halves C, doubles ΔV, and doubles stored energy; external work supplies that energy. The gap field remains Q(ε₀A) in the ideal model.

If the battery remains connected, ΔV stays constant. Doubling spacing halves C and Q, halves E, and halves capacitor energy. Charge and energy flow between the capacitor and battery while an external agent moves the plates. Looking only at capacitor energy is not a full accounting of external work.

A dielectric placed in the gap polarizes, reducing the field due to the same free plate charge. If it completely fills the gap, C=κε₀Ad. κ is the dielectric constant and is greater than one for the ordinary media considered here.

For fixed Q, dielectric insertion increases C and decreases V and U. For fixed V, it increases Q and U; the battery supplies charge and energy. Do not say a dielectric always decreases stored energy without stating which quantity is held fixed.

Parallel capacitors share the same potential difference, so their capacitances add. In an ideal series chain with initially neutral isolated interior nodes, charge magnitudes match and potential differences add, giving 1C_eq=Σ1C_i. The smaller series capacitance takes the larger voltage.

After reconnecting charged capacitors, conserve total charge on each isolated conductor group and require a common final potential for connected nodes. Electrostatic energy need not be conserved during redistribution: resistance and electromagnetic radiation can carry energy away even when the connecting wires are treated as ideal for the final state.

A conductor contains mobile charges. If a nonzero electric field persisted inside its conducting material, charges would continue to move. In electrostatic equilibrium the internal field is zero, the conductor is equipotential, and excess charge resides on surfaces. This statement concerns equilibrium; a current-carrying resistive wire can have an internal electric field.

The field immediately outside an ideal conducting surface is normal to that surface, because a tangential component would drive further surface motion. Surface charge need not be uniformly distributed on an irregular object. Greater curvature can be associated with stronger local fields, but use the geometry supplied rather than assuming every conductor is a sphere.

A cavity and the conducting material around it are not the same region. Charges placed inside a cavity can induce charges on its wall. A Gaussian surface within the metal, where E=0, constrains the net charge it encloses. Apply that constraint with the actual enclosed cavity charge.

Capacitance C=QΔV relates the magnitude of separated charge to the potential difference in the linear capacitor model. For ideal parallel plates, C=εAd, neglecting fringing. Larger area permits more charge for the same voltage; greater separation reduces capacitance. C is not simply “how much charge is currently present.”

A dielectric polarizes in response to the field, changing the relation between free charge and field. In the ideal linear model its insertion increases capacitance. The resulting charge and voltage changes depend on whether the capacitor remains connected to a battery or is isolated.

A battery fixes voltage approximately, allowing charge to enter or leave. An isolated capacitor fixes its free charge approximately, allowing voltage to change. Start every geometry-change question by identifying that constraint. Using both constant Q and constant V when C changes is inconsistent.

Parallel capacitors share the same potential difference, and their charges add, giving an equivalent capacitance equal to the sum. Series capacitors in the standard initially uncharged isolated-node arrangement carry equal charge magnitudes, while potential differences add, giving reciprocal addition.

Stored energy can be written as ½CV², Q²(2C) or ½QV. These forms are equivalent but highlight different fixed quantities. Increasing C at constant Q decreases stored energy, whereas increasing C at constant V increases it. In the battery-connected case, energy exchanged with the battery must be included when discussing the whole process.

Do not decide a force or work direction from capacitor energy alone without stating the external electrical constraint. Mechanical work, field energy and battery exchange can all participate. A consistent system energy account resolves apparent contradictions.

For a simple parallel-plate idealization, the field is approximately uniform away from edges and potential changes linearly across the gap. Fringing matters near edges and when separation is not small compared with plate dimensions. A schematic’s uniform arrows communicate an approximation rather than an exact field everywhere.

If an isolated capacitor’s plate spacing doubles, ideal capacitance halves, voltage doubles and stored field energy doubles. External work is required to separate attracting plates. If the same change occurs at fixed battery voltage, charge decreases instead. These two experiments have different energy exchanges despite identical final plate geometry.

When several dielectric regions are present, identify whether they divide the plate area side by side or divide the gap along the field. The corresponding parallel or series model follows from shared voltage or charge constraints, not simply from the number of materials shown.

For large parallel plates with equal and opposite charge densities, neglect fringing and use the approximately uniform field E=σε₀ between them. The potential difference magnitude is Ed. With Q=σA, capacitance C=QΔV becomes ε₀Ad. The geometry determines C in this ideal model; changing Q alone changes voltage proportionally.

A dielectric changes the relation between free charge and field through polarization. In the simple linear model, inserting a dielectric with relative permittivity κ fully between plates multiplies capacitance by κ. Distinguish bound polarization charge from free charge supplied by a battery.

Check whether the battery remains attached. At fixed voltage, increased capacitance draws additional charge; at fixed charge, the voltage decreases. The stored-energy formulas Q²(2C) and C(ΔV)²2 are equivalent, but each makes a different constraint easy to see. Using the wrong fixed quantity reverses an energy prediction.

Inside conducting material at electrostatic equilibrium, the electric field is zero and potential is constant. This does not require the potential to be zero. If a charge lies inside a cavity, induced charge appears on the cavity wall; the cavity itself is not conducting material and can contain a field.

Capacitors in parallel share the same potential difference, so their charges add and capacitances add. Capacitors in series carry equal charge magnitudes on ideal initially neutral intermediate conductors; their voltage differences add and reciprocal capacitances add. Trace actual nodes rather than deciding from the visual orientation of symbols.

After finding an equivalent capacitance, return to the individual components if asked for charge, voltage, or energy. Equivalent reduction preserves the terminal relation but does not make each internal capacitor carry the total terminal charge. Check that individual series voltages sum to the source voltage and parallel charges sum to the total.