General Physics II · Formula Sheet and Conceptual Checks

Course index · Mathematical toolkit

Use this sheet after reading the derivations. Unless otherwise specified, formulas assume vacuum and the geometric idealizations stated in the corresponding chapters.

1. Charge, fields, and potentials

Relation	Formula	Condition / interpretation
Point-charge field	\(\mathbf E=kq\mathbf R/R^3\)	\(\mathbf R\) points from source to observation; \(R>0\).
Force on charge	\(\mathbf F=q\mathbf E_{\mathrm{ext}}\)	Exclude its own field.
Continuous-source field	\(\mathbf E=k\int\mathbf R\,dq/R^3\)	Sum vectors and use source/observation coordinates consistently.
Flux	\(\Phi_E=\int_S\mathbf E\cdot\hat{\mathbf n}\,dA\)	Closed surfaces use outward normals.
Gauss's law	\(\oint_S\mathbf E\cdot d\mathbf A=Q_{\mathrm{enc}}/\epsilon_0\)	All charges contribute to the field; only enclosed charge determines net flux.
Potential difference	\(V_f-V_i=-\int_i^f\mathbf E\cdot d\boldsymbol\ell\)	Electrostatic field, with a consistent reference.
Localized-source potential	\(V=k\int dq/R\)	Zero at infinity when the integral converges.
Potential energy change	\(\Delta U=q\Delta V\)	Signed test charge, fixed external sources.
Field from potential	\(\mathbf E=-\nabla V\)	Differentiate in observation coordinates.
Pair assembly energy	\(U=\sum_{i<j}kq_iq_j/r_{ij}\)	Each pair once; exclude point-charge self-energy.

2. Standard geometries

Source	Field or potential	Scope
Uniform ring, radius \(R\)	\(E_z=kQz/(R^2+z^2)^{3/2}\); \(V=kQ/\sqrt{R^2+z^2}\)	On axis; finite ring, zero potential at infinity.
Uniform disk, radius \(R\)	\(E_z=\sigma_s[1-z/\sqrt{z^2+R^2}]/(2\epsilon_0)\)	On positive axis \(z>0\).
Same disk potential	$V=\sigma_s(\sqrt{z^2+R^2}-	z
Infinite sheet	$E=	\sigma_s
Two opposite infinite sheets	$E=	\sigma_s
Infinite line	\(\mathbf E=\lambda\hat{\mathbf r}/(2\pi\epsilon_0r)\)	Cylindrical radial distance \(r>0\).
Infinite-line potential difference	\(V(r)-V(r_0)=-\lambda\ln(r/r_0)/(2\pi\epsilon_0)\)	Finite reference radius; do not set zero at infinity.
Uniform insulating ball, inside	\(E_r=kQr/R^3\); \(V=kQ(3-r^2/R^2)/(2R)\)	\(r<R\); total \(Q\) uniformly in volume.
Spherical conductor, inside	\(E=0\); \(V=kQ/R\)	Equilibrium; isolated sphere without other sources.
Either spherical source, outside	\(E_r=kQ/r^2\); \(V=kQ/r\)	Full spherical symmetry, zero potential at infinity.

3. Dipoles and local field equations

\[ \mathbf p=q\mathbf d,\qquad V_{\mathrm{dip}}\simeq\frac{k\mathbf p\cdot\hat{\mathbf r}}{r^2},\qquad \mathbf E_{\mathrm{dip}}\simeq\frac{k}{r^3}[3(\mathbf p\cdot\hat{\mathbf r})\hat{\mathbf r}-\mathbf p]. \]

The finite-dipole approximation requires \(r\gg d\). In a uniform external field,

\[ \mathbf F_{\mathrm{net}}=0,\qquad \boldsymbol\tau=\mathbf p\times\mathbf E,\qquad U=-\mathbf p\cdot\mathbf E. \]

Electrostatics in vacuum obeys

\[ \nabla\times\mathbf E=0,\qquad \nabla\cdot\mathbf E=\frac{\rho_q}{\epsilon_0},\qquad \nabla^2V=-\frac{\rho_q}{\epsilon_0}. \]

4. Conductors and image methods

Result	Formula / condition
Equilibrium in metal	\(\mathbf E=0\), no excess bulk volume charge, \(V=V_c\).
Surface boundary	\(\mathbf E_{\mathrm{vac}}=(\sigma_s/\epsilon_0)\hat{\mathbf n}\), with normal out of metal.
Charged cavity	\(Q_{\mathrm{inner}}=-q\); isolated conductor has \(Q_{\mathrm{outer}}=Q_c+q\).
Grounded plane image	Real \(q\) at \(z=d\); image \(-q\) at \(z=-d\); physical region \(z>0\).
Plane density	\(\sigma_s(s)=-qd/[2\pi(s^2+d^2)^{3/2}]\).
Plane attraction	\(\mathbf F=-kq^2\hat{\mathbf z}/(4d^2)\).
Plane assembly energy	\(U=-kq^2/(4d)\); do not count image as an independent real particle.
Uniform-field sphere	\(\mathbf p_{\mathrm{ind}}=4\pi\epsilon_0R^3\mathbf E_0\); \(\sigma_s=3\epsilon_0E_0\cos\theta\).
Spherical image location	\(b=R^2/a\), \(q'=-qR/a\); image must lie outside physical region.

For the spherical image, distinguish cases: outside a grounded sphere (\(a>R\)), total induced charge equals \(q'\). Inside a spherical cavity (\(a<R\)), inner-wall charge equals \(-q\), not \(q'\). See the full spherical construction.

5. Transport and capacitance

Relation	Formula	Condition
Current	\(i=\int_S\mathbf J\cdot d\mathbf A\)	Oriented section.
Carrier current density	\(\mathbf J=nq_c\mathbf v_d\)	Sum over species if needed.
Electron drift	\(\mathbf v_d=-e\tau\mathbf E/m_e\)	Steady Drude model.
Conductivity	\(\sigma_{\mathrm{cond}}=ne^2\tau/m_e\)	One-carrier free-electron model.
Resistivity	\(\rho_{\mathrm{res}}=1/\sigma_{\mathrm{cond}}\)	Isotropic linear response.
Wire resistance	\(R=\rho_{\mathrm{res}}L/A\)	Uniform one-dimensional axial flow.
Mean free path	\(\ell\sim v_{\mathrm{micro}}\tau\)	Microscopic speed, not drift speed.
Charge conservation	\(\partial_t\rho_q+\nabla\cdot\mathbf J=0\)	Fixed-volume local continuity.
Capacitance	\(C=q/(V_+-V_-)\)	\(q\) is positive plate-charge magnitude.
Parallel plates	\(C=\epsilon_0A/d\)	Neglect fringes; vacuum gap.
Coaxial cylinders	\(C=2\pi\epsilon_0L/\ln(b/a)\)	\(b>a\); long cylinders, vacuum gap.
Parallel capacitors	\(C_{\mathrm{eq}}=\sum_iC_i\)	Same voltage.
Series capacitors	\(1/C_{\mathrm{eq}}=\sum_i1/C_i\)	Neutral floating intermediate nodes, ideal lumped elements.

6. Conceptual checks（自测）

1. Can \(V=0\) where \(\mathbf E\ne0\)? Yes: the perpendicular bisector plane of a dipole is an example.
2. Can \(\mathbf E=0\) where \(V\ne0\)? Yes: the interior of a charged conducting sphere with zero at infinity.
3. Does zero enclosed charge imply zero field? No; it only implies zero net closed-surface flux.
4. Can an equilibrium conductor's charged cavity have a field? Yes. Zero field applies to the metal material; an empty closed cavity is a separate uniqueness result.
5. Does grounding make a conductor neutral? No. It fixes potential while permitting charge exchange.
6. Is zero dipole torque necessarily stable? No. Anti-alignment also has zero torque but is unstable.
7. Is the field radial everywhere outside a sphere in a uniform external field? No. It is normal at the surface, with a generally nonzero exterior angular component.
8. Does a current-carrying resistive wire have zero internal field? No. Its steady current needs an internal field.
9. Are slow electron drift and fast circuit response contradictory? No; drift, microscopic motion, and signal propagation are different quantities.
10. Does impurity scattering always exchange energy? No. Static impurity scattering can be elastic and still relax momentum.
11. Does geometry change leave a capacitor's charge fixed? Only if the electrical connection imposes that condition; a voltage source instead fixes voltage.
12. Can every conductor problem use one image charge? No. The construction must satisfy all source, boundary, and total-charge conditions.

7. Final calculation checklist

Specify the region and references; draw signed sources and normals; use symmetry; keep dimensions consistent; verify source singularities; check total charge and boundary conditions; test limiting cases. For piecewise potentials, match the same additive reference across regions.

References

The six main chapters contain derivations and source attributions. Source map gives the original files and progress boundaries; figure notes explains visualization conventions.