Concentric Spherical Capacitor — Breakdown-Limited Charge: 04-BS-9, May 2013, Q1

Question 1: Concentric Spherical Capacitor — Breakdown-Limited Charge (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Value
Inner shell outer radius $a$	10 mm
Dielectric thickness (covers inner shell)	2 mm $\Rightarrow$ dielectric outer radius $b=12$ mm
Outer shell inner radius $d$	15 mm (air gap from $b$ to $d$)
Dielectric relative permittivity $\varepsilon_r$	2.25
Dielectric breakdown field	$10^7$ V/m
Air breakdown field	$10^6$ V/m

Find. The largest charge $Q$ the capacitor can hold before either the dielectric layer or the air gap electrically breaks down.

Concentric spherical shells: dielectric sleeve ($a$ to $b$) then air gap ($b$ to $d$) before the outer shell.

Approach. For a spherical charge distribution, Gauss's law gives $D(r)=Q/(4\pi r^2)$, continuous across the dielectric/air interface (no free charge there); the local field is $E(r)=D(r)/\varepsilon$, which is largest at the smallest radius of each region. Compute the charge that would just reach breakdown in each region separately, then take the smaller value — that is the true (lowest) upper bound.

Field profile from Gauss's law. A concentric spherical Gaussian surface of radius $r$ encloses charge $Q$ regardless of the dielectric, so $$D(r)=\frac{Q}{4\pi r^2}, \qquad E(r)=\frac{D(r)}{\varepsilon(r)}=\frac{Q}{4\pi\varepsilon(r) r^2}.$$ In each region $E(r)$ decreases with $r$, so the breakdown risk is highest at the inner radius of that region: $r=a$ in the dielectric, $r=b$ at the start of the air gap.
Charge limit set by the dielectric ($a\le r\le b$). Breakdown first occurs at $r=a=0.010$ m: $$Q_{\text{diel}}=E_{\text{bd,diel}}\cdot 4\pi\varepsilon_0\varepsilon_r a^2 = (10^7)(4\pi)(8.85\times10^{-12})(2.25)(0.010)^2$$ $$Q_{\text{diel}} \approx \boxed{2.50\times10^{-7}\ \text{C} = 250\ \text{nC}}$$
Charge limit set by the air gap ($b\le r\le d$). Breakdown first occurs at $r=b=0.012$ m: $$Q_{\text{air}}=E_{\text{bd,air}}\cdot 4\pi\varepsilon_0 b^2 = (10^6)(4\pi)(8.85\times10^{-12})(0.012)^2$$ $$Q_{\text{air}} \approx \boxed{1.60\times10^{-8}\ \text{C} = 16.0\ \text{nC}}$$
Governing limit. Since $Q_{\text{air}}<Q_{\text{diel}}$, the air gap breaks down first as charge is added. The lowest upper bound on stored charge is therefore $$Q_{\max}=\boxed{16.0\ \text{nC}}.$$

Quantity	Result
$Q_{\text{diel}}$ (dielectric-limited)	$250$ nC
$Q_{\text{air}}$ (air-limited)	$16.0$ nC
Lowest upper bound $Q_{\max}$	$16.0$ nC

Concept & Theory

Topic. Spherical capacitors with layered dielectrics — dielectric breakdown limits.

Key relations. Gauss's law $\oint \vec D\cdot d\vec A = Q_{\text{enc}}$; for spherical symmetry $D=Q/4\pi r^2$; local field $E=D/\varepsilon$; breakdown occurs when local $E$ reaches the material's dielectric strength.

Why this works. $D$ (and hence $Q$) is the same at every radius because no free charge exists between the shells, but $E$ jumps at the material interface because $\varepsilon$ changes — the field is strongest at the smallest radius of whichever layer has the lower $\varepsilon$/breakdown strength. A layered insulation system is only as good as its weakest region, and here that is the air gap despite the dielectric layer's much higher rated breakdown field, because $\varepsilon_r=2.25$ more than compensates for the dielectric's higher $10^7$ V/m rating.

Common pitfall. Evaluating $E$ at the outer radius of a layer instead of the inner radius (the field is largest where $r$ is smallest, not where the layer starts geometrically); forgetting that $D$, not $E$, is continuous across the dielectric interface.

Source. Sadiku, Elements of Electromagnetics, Ch. 4–5 (Gauss's Law, Dielectrics and Capacitance).

Question 1 of 8: Concentric Spherical Capacitor — Breakdown-Limited Charge

Question 1: Concentric Spherical Capacitor — Breakdown-Limited Charge (20 marks)