Short on time? This page is the recap. For the explanations, Figures 1 and 2 and the worked examples, use the full study guide. This version of the topic is for the calculus-based course.
Recap
- A dielectric is an insulator. Its charges are bound, so a field only shifts them slightly: the material polarizes.
- Non-polar molecules gain induced dipoles; polar molecules partly line up.
- Bound charge appears on the faces: negative next to the positive plate, positive next to the negative plate.
- The field of this bound charge opposes the applied field, so the net field is smaller, but not zero.
- κ = ε/ε₀ has no unit. κ = 1 for a vacuum, about 1.0006 for air, larger for other insulators.
- Filling the gap multiplies C by κ, for any capacitor shape.
Key relationships
| Quantity | Result |
|---|---|
| Dielectric constant | κ = ε/ε₀ |
| Field in a filled, isolated capacitor | E = E₀/κ |
| Induced surface charge density | σᵢ = σ(1 − 1/κ) |
| Capacitance, parallel plates | C = κC₀ = κε₀A/d = εA/d |
| Isolated (Q fixed), slab inserted | ΔV ÷ κ, E ÷ κ, U ÷ κ |
| Connected (ΔV fixed), slab inserted | Q × κ, E unchanged, U × κ |
Assumptions behind the results
- The dielectric fills the whole gap and is uniform.
- The plates are large compared with the gap, so edge effects are ignored.
- κ is a constant for the material: it does not depend on the field.
Mistakes to avoid
- Saying the dielectric adds free charge. Bound charge stays in the dielectric.
- Treating a dielectric like a conductor. It reduces E by κ; it does not make E zero.
- “E always falls.” Only when Q is fixed.
- “U always rises with C.” With Q fixed, U = Q²/(2C) falls.
- Dividing C by κ. C goes up.
Quick self-check
- A 40 pF air capacitor is filled with a material of κ = 2.5. New C? (100 pF)
- An isolated capacitor at 30 V is filled with κ = 3.0. New ΔV? (10 V)
- With a battery connected, what happens to E when a slab is inserted? (No change)
Next: practice questions.