# Conductors and Capacitance

Physics II · Electromagnetism · https://tryals.app/learn/physics-ii/conductors-and-capacitance

## Equilibrium Forces the Field Out

A conductor in **electrostatic equilibrium** has no moving charge, and that single fact settles everything else. If any field remained inside, free charges would move, so $\mathbf{E} = \mathbf{0}$ throughout the interior. Consequently:

- All excess charge sits on the **surface**, since $\nabla \cdot \mathbf{E} = 0$ inside forces $\rho = 0$ there.
- The whole conductor is an **equipotential**: with no interior field there is no potential difference between any two of its points.
- The surface field is **perpendicular** to the surface, and equals $\sigma/\varepsilon_0$ just outside. A tangential component would drive surface currents.

The charge experiences an outward **electrostatic pressure** $\sigma^2/2\varepsilon_0$, the reason a heavily charged conductor can physically tear.

**Capacitance** measures how much charge a conductor system holds per volt: $C = Q/V$, in farads. It depends only on geometry and the medium, never on how much charge you happen to have put there. For a parallel-plate capacitor

$$C = \frac{\varepsilon_0 \varepsilon_r A}{d}$$

Combinations follow from which quantity the elements share:

| Arrangement | Shared | Result |
|---|---|---|
| Parallel | Voltage | $C = C_1 + C_2$ |
| Series | Charge | $1/C = 1/C_1 + 1/C_2$ |

Note this is the opposite of resistors, and for the same reason: series capacitors share charge while series resistors share current.

With several conductors, **electrostatic influence** means each one's potential depends on all the charges, through a matrix of capacitance and influence coefficients. A conductor fully enclosing another shields it completely, the basis of the Faraday cage.

> **Common pitfall:** thinking capacitance grows when you add charge. $C$ is a fixed property of the geometry: doubling $Q$ doubles $V$ and leaves $C$ untouched. It changes only if you alter the plate area, the separation, or the dielectric.

## Practice questions

7 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. Why must the electric field vanish inside a conductor in electrostatic equilibrium?

A. Any interior field would drive the free charges, contradicting equilibrium
B. The outward electrostatic pressure mechanically cancels the interior field
C. The surface charge density is inherently too small to sustain an inner field
D. Mobile charges neutralise each other completely, leaving no charge carriers

**Answer:** A. Any interior field would drive the free charges, contradicting equilibrium

**Why:** Equilibrium is defined by charges having stopped. A residual interior field would keep pushing them, so the charges rearrange precisely until they cancel it, which is why the interior field is exactly zero, not merely small.

Page: https://tryals.app/practice/physics-ii/conductors-and-capacitance/why-must-the-electric-field-vanish-inside-a-conductor-in

### 2. A parallel-plate capacitor has plate area 0.02 m$^2$, separation 1.0 mm, and vacuum between the plates. Compute its capacitance in pF, using $\varepsilon_0 = 8.854 \times 10^{-12}$ F/m. Give the answer to the nearest whole number.

**Answer:** 177 (within ±3)

**Why:** $C = \varepsilon_0 A/d = 8.854 \times 10^{-12} \times 0.02/10^{-3} = 1.77 \times 10^{-10}$ F $= 177$ pF. Note how large an area is needed for even this modest capacitance.

Page: https://tryals.app/practice/physics-ii/conductors-and-capacitance/a-parallel-plate-capacitor-has-plate-area-0-02-m-separation-1-0-mm

### 3. The capacitance of a capacitor increases when more charge is placed on its plates.

**Answer:** False

**Why:** False, $C = Q/V$ is fixed by geometry and the dielectric. Adding charge raises $V$ in exact proportion, so the ratio is unmoved. Only area, separation or the medium can change it.

Page: https://tryals.app/practice/physics-ii/conductors-and-capacitance/the-capacitance-of-a-capacitor-increases-when-more-charge-is-placed

### 4. Which are true of a conductor in electrostatic equilibrium?

A. All excess charge resides on the surface
B. The surface field has a tangential component
C. The entire conductor is at one potential
D. The interior field is zero

**Answer:** A. All excess charge resides on the surface; C. The entire conductor is at one potential; D. The interior field is zero

**Why:** Zero interior field forces surface-only charge and a single potential throughout. A tangential surface field would drive charge along the surface, which contradicts equilibrium, so the surface field is purely perpendicular.

Page: https://tryals.app/practice/physics-ii/conductors-and-capacitance/which-are-true-of-a-conductor-in-electrostatic-equilibrium

### 5. An isolated conductor is brought into equilibrium after having charge added to it. Because the whole body forms an equipotential, what must follow for the electrostatic forces acting on the surface charges?

A. An inward restorative force counters electrostatic pressure
B. A normal outward force acts on them without lateral drift
C. A purely tangential force directs charges across the surface
D. Zero net force acts because equilibrium requires cancellation

**Answer:** B. A normal outward force acts on them without lateral drift

**Why:** Equipotential surfaces forbid tangential fields, preventing lateral drift along the conductor. However, the surface charge experiences an outward normal electrostatic pressure proportional to the square of surface charge density, meaning the local normal force does not vanish.

Page: https://tryals.app/practice/physics-ii/conductors-and-capacitance/an-isolated-conductor-is-brought-into-equilibrium-after-having-charge

### 6. Match each capacitor arrangement to the quantity its elements share and the resulting combination.

**Answer:**

- Two capacitors in parallel → Share voltage; capacitances add directly
- Two capacitors in series → Share charge; reciprocals add
- Plates moved further apart → Capacitance falls, as one over the gap
- Dielectric inserted between the plates → Capacitance rises by the relative permittivity

**Why:** Parallel elements share voltage so their charges add, giving a larger capacitance; series elements share charge and split the voltage, giving a smaller one. Geometry and the dielectric change $C$ directly, since $C = \varepsilon_0\varepsilon_r A/d$.

Page: https://tryals.app/practice/physics-ii/conductors-and-capacitance/match-each-capacitor-arrangement-to-the-quantity-its-elements-share

### 7. Arrange these consequences of electrostatic equilibrium in the order they logically follow from one another.

**Answer:**

1. Charges have stopped moving, so equilibrium holds
2. The field inside the conductor must be zero
3. The charge density inside must be zero, so charge sits on the surface
4. The whole conductor is at a single potential

**Why:** Equilibrium means nothing moves, which forces zero interior field; zero field forces zero interior charge density by Gauss’s law, pushing all charge to the surface; and with no interior field there is no potential difference anywhere inside.

Page: https://tryals.app/practice/physics-ii/conductors-and-capacitance/arrange-these-consequences-of-electrostatic-equilibrium-in-the-order