Capacitor flash cards

1. 1.Define capacitance of a conductor/capacitor.

   Capacitance $C = Q/V$ is the charge stored per unit potential difference. It depends only on geometry and the surrounding medium, not on $Q$ or $V$.

   Hint: Ratio of charge to potential.

2. 2.What is the SI unit of capacitance and its definition?

   The farad (F): $1\,\text{F} = 1\,\text{C/V}$. One farad is a very large unit; practical values are in $\mu\text{F}$, $n\text{F}$ and $p\text{F}$.

   Hint: Coulomb per volt.

3. 3.Give the capacitance of an isolated spherical conductor of radius $R$.

   $C = 4\pi\varepsilon_0 R$. It behaves like a sphere-and-infinity capacitor; capacitance grows linearly with radius.

   Hint: Use $V = \frac{Q}{4\pi\varepsilon_0 R}$.

4. 4.State the parallel plate capacitor formula (vacuum) and name each symbol.

   $C = \dfrac{\varepsilon_0 A}{d}$, where $A$ is plate area, $d$ the separation, $\varepsilon_0$ the permittivity of free space.

   Hint: Area over gap.

5. 5.How does the capacitance of a parallel plate capacitor change if plate separation $d$ is doubled (area fixed)?

   $C \propto 1/d$, so doubling $d$ halves $C$.

   Hint: Inverse dependence on $d$.

6. 6.What field exists between the plates of a parallel plate capacitor and its value?

   A nearly uniform field $E = \dfrac{\sigma}{\varepsilon_0} = \dfrac{Q}{\varepsilon_0 A} = \dfrac{V}{d}$, directed from + to − plate.

   Hint: $\sigma/\varepsilon_0$.

7. 7.Why must the two plates of a capacitor carry equal and opposite charges $+Q$ and $-Q$?

   Charge $+Q$ on one plate induces $-Q$ on the facing plate. The field is confined between them, and this configuration is what stores energy; the 'charge on capacitor' means the magnitude $Q$.

   Hint: Induction between facing plates.

8. 8.Define dielectric constant (relative permittivity) $K$.

   $K = \varepsilon/\varepsilon_0 = C_{\text{medium}}/C_{\text{vacuum}}$. It is the factor by which capacitance increases when the gap is filled with the dielectric ($K \geq 1$).

   Hint: Ratio of permittivities.

9. 9.A parallel plate capacitor is completely filled with a dielectric of constant $K$. New capacitance?

   $C = \dfrac{K\varepsilon_0 A}{d} = K C_0$. Capacitance increases $K$ times.

   Hint: Multiply vacuum value by $K$.

10. 10.Why does inserting a dielectric increase capacitance?

    The dielectric polarizes, producing bound charges that partially cancel the field, reducing $E$ (and $V$) for the same $Q$. Since $C = Q/V$, lower $V$ means higher $C$.

    Hint: Polarization reduces internal field.

11. 11.What is dielectric strength and why does it matter for capacitors?

    Dielectric strength is the maximum field a material can withstand before breakdown (conduction). It sets the maximum safe voltage $V_{\max} = E_{\text{breakdown}}\,d$ a capacitor can hold.

    Hint: Breakdown field limit.

12. 12.Derive capacitance of a parallel plate capacitor with a dielectric slab of thickness $t$ ($t<d$) and constant $K$.

    $C = \dfrac{\varepsilon_0 A}{d - t + t/K}$. The slab acts like reducing the effective air gap by $t(1 - 1/K)$.

    Hint: Series of air and dielectric regions.

13. 13.A conducting slab of thickness $t$ is inserted (parallel to plates, $t<d$). New capacitance?

    $C = \dfrac{\varepsilon_0 A}{d - t}$. A conductor is the $K\to\infty$ limit, so it just removes thickness $t$ from the gap.

    Hint: Set $K\to\infty$ in the slab formula.

14. 14.For a partial dielectric slab, does the answer depend on where (which position) the slab sits in the gap?

    No. For a slab parallel to the plates, only its thickness $t$ matters, not its distance from either plate (as long as it stays fully inside).

    Hint: Series capacitors add reciprocally regardless of order.

15. 15.Two capacitors $C_1, C_2$ in series: equivalent capacitance and charge relation?

    $\dfrac{1}{C_{eq}} = \dfrac{1}{C_1} + \dfrac{1}{C_2}$. Same charge $Q$ on each; voltages add: $V = V_1 + V_2$.

    Hint: Reciprocals add; equal charge.

16. 16.Two capacitors $C_1, C_2$ in parallel: equivalent capacitance and voltage relation?

    $C_{eq} = C_1 + C_2$. Same voltage $V$ across each; charges add: $Q = Q_1 + Q_2$.

    Hint: Capacitances add; equal voltage.

17. 17.For series capacitors, across which capacitor is the voltage largest?

    Across the smallest capacitor. Since $Q$ is common, $V = Q/C$, so smaller $C$ takes larger $V$.

    Hint: $V \propto 1/C$ at fixed $Q$.

18. 18.For two capacitors in series, give $C_{eq}$ as a single fraction.

    $C_{eq} = \dfrac{C_1 C_2}{C_1 + C_2}$ (product over sum).

    Hint: Analogous to parallel resistors.

19. 19.Energy stored in a charged capacitor — give the three equivalent forms.

    $U = \dfrac{1}{2}CV^2 = \dfrac{Q^2}{2C} = \dfrac{1}{2}QV$.

    Hint: Use $Q = CV$ to switch forms.

20. 20.Why is the energy stored $\frac{1}{2}QV$ and not $QV$?

    Charge is transferred gradually; the potential rises from $0$ to $V$ as charging proceeds, so the average work per unit charge is $V/2$. Integrating $\int V\,dq = Q^2/2C$.

    Hint: Average potential during charging.

21. 21.Define energy density of the electric field between capacitor plates.

    $u = \dfrac{1}{2}\varepsilon_0 E^2$ (vacuum), or $\dfrac{1}{2}K\varepsilon_0 E^2$ in a dielectric. It is energy per unit volume of the field.

    Hint: Half epsilon E squared.

22. 22.Capacitance of a spherical capacitor (inner radius $a$, outer $b$, vacuum).

    $C = 4\pi\varepsilon_0\,\dfrac{ab}{b - a}$.

    Hint: Integrate field between shells for $V$.

23. 23.In the spherical capacitor formula, what limit recovers the isolated sphere result?

    Let $b \to \infty$: $C = 4\pi\varepsilon_0\,\dfrac{ab}{b-a} \to 4\pi\varepsilon_0 a$, the isolated sphere of radius $a$.

    Hint: Push the outer shell to infinity.

24. 24.Capacitance of a cylindrical capacitor (length $L$, radii $a$ inner and $b$ outer).

    $C = \dfrac{2\pi\varepsilon_0 L}{\ln(b/a)}$, assuming $L \gg b$ (edge effects ignored).

    Hint: Field of a line charge, integrate for $V$.

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