Maxwell equations in integral form

A point charge and a spherical enclosure

Given

A point charge $Q=6.0\,\mathrm{nC}$ lies at the centre of an imaginary sphere of radius $r=0.15\,\mathrm m$ in vacuum, with no other sources.

Find

Determine total electric flux and field strength on the sphere.

1. 1

   Approach: First obtain flux from enclosed charge, then use spherical symmetry.

   $$\begin{aligned}\Phi_E&=\frac{Q}{\varepsilon_0}\\E&=\frac{\Phi_E}{4\pi r^2}\end{aligned}$$
2. 2

   Using $\varepsilon_0\approx8.854\times10^{-12}\,\mathrm{F/m}$ gives the flux balance.

   $$\Phi_E=\frac{6{,}0\times10^{-9}}{8{,}854\times10^{-12}}\,\mathrm{V\,m}\approx677{,}7\,\mathrm{V\,m}$$
3. 3

   The field is radial everywhere on the sphere and has constant magnitude.

   $$E=\frac{677{,}7}{4\pi(0{,}15)^2}\,\mathrm{V/m}\approx2{,}40\times10^3\,\mathrm{V/m}$$

Result: Flux is approximately 678 V·m; the field is approximately 2.40 kV/m and points outward.

Check: Check: $E\,4\pi r^2\approx\Phi_E$. Doubling the sphere radius leaves flux unchanged and reduces field magnitude to one quarter. Moving the charge off centre keeps total flux unchanged but removes constant E on the sphere.

Magnetic field inside and outside the plate radius

Given

An idealized circular vacuum capacitor has plate radius $R=30\,\mathrm{mm}$ and is charging at $I=3.0\,\mathrm{mA}$. Plate separation is small compared with R. Geometry and excitation are modeled as rotationally symmetric; additional contributions from actual supply and return wiring are neglected. Axial E increases uniformly within the plate radius; its contribution outside is neglected in the simplified gap model. Field variation is slow compared with light travel across the relevant geometry; fringing and radiation lie outside this approximation.

Find

Find $\frac{\mathrm dE}{\mathrm dt}$ and B magnitude in the idealized gap plane on concentric circles of radii $r_1=10\,\mathrm{mm}$ and $r_2=50\,\mathrm{mm}$.

1. 1

   Approach: In the gap, count enclosed displacement current; the circles are concentric with the symmetry axis.

   $$\begin{aligned}I&=\varepsilon_0\pi R^2\frac{\mathrm dE}{\mathrm dt}\\B\,2\pi r&=\mu_0I_{d,\mathrm{enc}}\end{aligned}$$
2. 2

   Determine field growth from total plate current.

   $$\frac{dE}{dt}=\frac{3{,}0\times10^{-3}}{(8{,}854\times10^{-12})\pi(0{,}030)^2}\,\frac{\mathrm V}{\mathrm{m\,s}}\approx1{,}20\times10^{11}\,\frac{\mathrm V}{\mathrm{m\,s}}$$
3. 3

   The inner circle encloses only the area fraction; the outer circle encloses the full current term.

   $$\begin{aligned}I_{d,1}&=I\frac{r_1^2}{R^2}=\frac I9\\B_1&=\frac{\mu_0 I r_1}{2\pi R^2}\approx6{,}67\,\mathrm{nT}\\B_2&=\frac{\mu_0 I}{2\pi r_2}\approx12{,}0\,\mathrm{nT}\end{aligned}$$

Result: In the idealized gap, B grows linearly up to the plate radius and decreases inversely with radius outside it.

Check: At $r=R$, both approximation formulas give $B=20.0\,\mathrm{nT}$. For the inner circle the total source term is only $I/9$. An alternative spanning surface gives the same total current-plus-field-change term, but not necessarily the full wire current alone. If its total charge-current term on that surface is I, its oriented displacement-current contribution must here supply $-8I/9$. Only for a loop enclosing the entire associated plate flux can a pure wire cut alone give I in the simplified model. Real fringing and asymmetric excitation limit the local B values.

Calculate · Foundation

An enclosure contains total charge $Q=-3.0\,\mathrm{nC}$. Find net electric flux and interpret its sign.

Learning goal: Interpret the sign of net flux through a closed surface.

Rearrange · Application

A fixed circle of radius $r=4.0\,\mathrm{cm}$ encloses a uniform +z field increasing at $dB/dt=0.30\,\mathrm{T/s}$. Positive circulation is counterclockwise when viewed from +z. Find emf and tangential E in the rotationally symmetric model.

Learning goal: Translate changing flux into oriented field circulation.

Interpret · Application

Magnetic net flux through an imaginary cube is zero. Does this imply $B=0$ inside the cube?

Learning goal: Distinguish net flux from local field strength.

Error analysis · Application

Consider a loop enclosing the full plate flux of an idealized charging capacitor. In the model, the wire-cut surface has $I_A=I$, $I_d\approx0$, while the gap surface has $I_A=0$, $I_d=I$. A calculation adds both full contributions to obtain $2\mu_0I$. What is wrong?

Learning goal: Detect double counting caused by switching spanning surfaces.

Transfer · Challenge

Across a fixed enclosing boundary, $10\,\mathrm{mA}$ of conventional current enters and $4\,\mathrm{mA}$ leaves, both constant during the next $2\,\mathrm{ms}$. Find $\frac{\mathrm dQ_V}{\mathrm dt}$, the change rate of net electric flux and the change of enclosed Q over that interval.

Learning goal: Connect charge conservation to electric flux balance.

Induced field outside a magnetic region

Given

In a rotationally symmetric quasistatic model, $B_z$ grows uniformly within $R=0.0200\,\mathrm m$ at $\dot B=0.400\,\mathrm{T/s}$; this field contribution outside is negligible. A fixed circular loop has $r=0.0500\,\mathrm m$; positive circulation is counterclockwise from +z.

Find

Find emf and tangential E on the larger loop.

1. 1

   Approach: active flux area is the magnetic region, not the entire spanning area. Then divide by loop circumference.

   $$\begin{aligned}\mathcal E&=-\pi R^2\dot B\\E_\varphi(r)&=\frac{\mathcal E}{2\pi r}=-\frac{R^2\dot B}{2r}\end{aligned}$$
2. 2

   Substitution: both signs refer to the specified positive orientation.

   $$\begin{aligned}\mathcal E&=-\pi(0.0200)^2(0.400)\,\mathrm V\approx-0.502655\,\mathrm{mV}\\E_\varphi&=-\frac{(0.0200)^2(0.400)}{2(0.0500)}\,\mathrm{V/m}=-1.600\,\mathrm{mV/m}\end{aligned}$$

Result: Electric circulation is clockwise. Tangential field on the specified loop is −1.60 mV/m.

Check: Check: $E_\varphi2\pi r=\mathcal E$. Doubling the outer loop radius leaves enclosed flux unchanged and halves field magnitude. Inside the magnetic region, $E_\varphi=-\frac r2\dot B$; both expressions match at $r=R$.