2023 Midterm · Annotated Formula Sheet
Midterm index · Question paper · Full course formula reference
The tables reproduce all 31 relations on the supplied sheet, grouped by topic. Conditions and symbol explanations are added. This is the 2023 sheet; the instructor has said formulas will also be provided for the current exam, but has not supplied its final sheet here.
1. Electrostatics and dipoles
| Supplied relation | How to use it |
|---|---|
| \(\mathbf E=\dfrac{q}{4\pi\epsilon_0r^2}\hat{\mathbf r}\) | A stationary point charge in vacuum; for an extended source integrate vector contributions. |
| \(\epsilon_0\oint_S\mathbf E\cdot d\mathbf A=q_{\rm enc}\) | Closed surface, outward normal. Extracting \(E\) requires sufficient symmetry. |
| \(V_f-V_i=-\int_i^f\mathbf E\cdot d\mathbf s\) | Electrostatic field; fix the integration direction and potential reference. |
| \(\mathbf E=-\nabla V\) | Spatial derivative of a scalar electrostatic potential. |
| \(\mathbf p=q\mathbf d\) | \(\mathbf d\) points from negative to positive charge. |
| \(\boldsymbol\tau_E=\mathbf p\times\mathbf E\) | External field; in a uniform field a dipole has torque but zero net force. |
| \(\nabla\cdot\mathbf E=\rho_q/\epsilon_0\) | Here \(\rho_q\) is charge density, not resistivity. In a dielectric this counts total free plus bound charge. |
For the spherical dielectric in Q2, use \(\mathbf D=\epsilon\mathbf E\) and \(\oint\mathbf D\cdot d\mathbf A=Q_{\rm free}\), with \(\epsilon=\epsilon_0\epsilon_r\). Do not insert only free charge into the vacuum \(\epsilon_0\) formula and ignore polarization.
2. Transport, emf, and Hall effect
| Supplied relation | How to use it |
|---|---|
| \(\rho_{\rm res}=\dfrac1\sigma=\dfrac{m}{ne^2\tau}\) | Resistivity and the one-electron-carrier Drude model; \(\tau\) is mean free time. |
| \(\mathbf E=\rho_{\rm res}\mathbf J\) | Isotropic local Ohm's law for a stationary conductor. |
| \(\mathcal E=dW/dq\) | Source work per unit charge; emf has units V and need not be a scalar-potential difference around a loop. |
| \(P=i^2R\) | Joule heating of a resistor; check which branch current is \(i\). |
| \(R_H=\dfrac{E_H}{BJ}=\dfrac1{nq_c}\) | Single-carrier Hall coefficient with consistent signed transverse directions; \(E_H\) is the Hall field and \(q_c\) the carrier charge. |
The original uses \(\rho\) for both resistivity and charge density on different lines. It also uses \(\tau\) for collision time, while torque is a vector; identify the quantity from its equation and units.
3. Magnetic forces and fields
| Supplied relation | How to use it |
|---|---|
| \(\mathbf F=q\mathbf E+q\mathbf v\times\mathbf B\) | Total force on a moving charge; magnetic force alone does no work. |
| \(\mathbf F_B=i\mathbf L\times\mathbf B\) | Straight segment in uniform \(\mathbf B\); otherwise use \(\int i\,d\boldsymbol\ell\times\mathbf B\). |
| \(\boldsymbol\mu=Ni\mathbf A\) | Oriented area vector, chosen by the right-hand rule for the positive current. |
| \(\boldsymbol\tau_B=\boldsymbol\mu\times\mathbf B\) | Torque on a small loop in an external uniform field. |
| \(d\mathbf B=\dfrac{\mu_0}{4\pi}\dfrac{i\,d\mathbf s\times\hat{\mathbf r}}{r^2}\) | Biot–Savart law; displacement points from source element to observation. |
| \(\oint\mathbf B\cdot d\mathbf s=\mu_0i_{\rm enc}\) | Magnetostatic Ampère law; changing electric flux requires the Maxwell term below. |
| \(\oint_S\mathbf B\cdot d\mathbf A=0\) | No net magnetic flux through a closed surface. |
| \(\nabla\cdot\mathbf B=0\) | Local version; does not imply \(\mathbf B=0\). |
4. Induction, energy, and oscillations
| Supplied relation | How to use it |
|---|---|
| \(\oint\mathbf E\cdot d\mathbf s=-\dfrac d{dt}\int\mathbf B\cdot d\mathbf A\) | Fixed contour and surface, with right-hand orientation. For moving material include motional emf. |
| \(U_C=q^2/(2C)\) | Linear capacitor; also \(U_C=CV^2/2\). |
| \(u_E=\epsilon_0E^2/2\) | Vacuum electric energy density. A uniform linear nondispersive dielectric uses \(\epsilon E^2/2\). |
| \(L=N\Phi_B/i\) | Self-flux linkage divided by current; \(\Phi_B\) is the flux per turn for equivalent turns. |
| \(U_L=Li^2/2\) | Linear inductor; \(L\) fixed. |
| \(u_B=\mu_0B^2/2\) as printed; incorrect | Use \(\boxed{u_B=B^2/(2\mu_0)}\) in vacuum or a nonmagnetic medium. |
| \(M=N_2\Phi_{21}/i_1\) | Flux linkage of coil 2 caused by coil 1; sign depends on chosen winding senses. |
| \(\omega_0=1/\sqrt{LC}\) | Natural angular frequency of an ideal LC circuit. Damping and driven resonance require their stated conditions. |
| \(\omega=2\pi/T\) | \(\omega\) is angular frequency in rad/s; \(f=1/T\) in Hz. |
Magnetic-energy correction. Dividing \(B^2\) by \(\mu_0\) gives J/m³; multiplying by \(\mu_0\) does not. The correct expression also follows from the course's solenoid-energy derivation. This typo is in the supplied image, not a different convention.
For the moving disk, the useful general emf is \(\mathcal E=\oint(\mathbf E+\mathbf u\times\mathbf B)\cdot d\boldsymbol\ell\). Its entire disk flux is constant while its center–rim motional emf is nonzero; choose the actual circuit path.
5. Time-dependent Maxwell equations
| Supplied relation | How to use it |
|---|---|
| \(\nabla\times\mathbf E=-\partial\mathbf B/\partial t\) | An induced field generally cannot be described by \(-\nabla V\) alone. |
| \(\nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\epsilon_0\partial\mathbf E/\partial t\) | Vacuum Ampère–Maxwell law with total current density. In a simple dielectric the free-current form uses \(\epsilon\partial\mathbf E/\partial t\). |
The 2023 page does not list every Chapter 14 wave relation. Be able to obtain or use
for a single traveling vacuum plane wave. These are course supplements, not additional relations transcribed from the old sheet. The full derivation is in Chapter 14.
Constants printed on the sheet
| Constant | Printed value |
|---|---|
| Elementary-charge magnitude \(e\) | \(1.6\times10^{-19}\ \mathrm C\); an electron has charge \(-e\) |
| Electron mass \(m_e\) | \(0.91\times10^{-30}\ \mathrm{kg}\) |
| Boltzmann constant \(k_B\) | \(1.38\times10^{-23}\ \mathrm{J/K}\) |
| Vacuum permittivity \(\epsilon_0\) | \(8.85\times10^{-12}\ \mathrm{C^2/(N\,m^2)}\) |
| Vacuum permeability \(\mu_0\) | \(4\pi\times10^{-7}\ \mathrm{T\,m/A}\) |
符号识别比抄公式更重要：\(R\) 可能是球半径或电阻，\(\sigma\) 可能是电导率、面电荷密度，或 Q3 中的 \(dq/d\theta\)。先定义再代入。