Midterm · Transfer Practice
Revision plan · 2023 paper · Formula sheet
Use the formula sheet, then write a governing equation and its conditions before opening each answer. These are new practice questions, not additional 2023 exam questions or predictions of this year's paper.
1. Zero divergence is not enough
In a charge-free region propose \(\mathbf E=A(y\hat{\mathbf x}-x\hat{\mathbf y})\), \(A\ne0\). Can stationary charges produce this as an electrostatic field? Does zero divergence imply zero electric field?
Knowledge: curl and local Gauss law
Answer
It satisfies the charge-free divergence condition but fails electrostatic circulation. It cannot be the electrostatic field of stationary charges. Nonzero fields can have zero divergence; a uniform field is an electrostatic example. 散度决定局部源，旋度决定局部环流，两者不能互相替代。
2. Grounded versus isolated at \(d=2R\)
A charge \(q>0\) is distance \(2R\) from a conducting sphere's center. Compare an initially neutral isolated sphere with a grounded one. Find the images, sphere potential, and force on \(q\) in both cases. Let \(\hat{\mathbf n}\) point from center to \(q\).
Answer
Both cases use \(-q/2\) at \(R/2\) toward the real charge. Grounding needs no other image:
Isolation requires zero net represented sphere charge, so add \(+q/2\) at the center:
Both attract, but with different magnitudes. An isolated neutral conductor has zero total charge, not zero polarization or zero potential.
3. What changes the leakage time?
For the lossy spherical capacitor, keep the uniform \(\epsilon,\sigma\) fixed and change \(a,b\). Does the charge-decay time change? When has half the initial field energy been dissipated?
Answer
is geometry-independent in this model, even though \(R\) and \(C\) separately change. Since \(U(t)=U_0e^{-2t/\tau}\), half-energy dissipation occurs at
Half-charge time is instead \(\tau\ln2\). 先判断题目问的是电荷、能量还是功率，再使用指数。
4. A loaded RC circuit: initial and final states
A battery of emf \(\mathcal E\) and series resistance \(r\) feeds a parallel resistor \(R\) and capacitor \(C\). The initially uncharged capacitor is connected at \(t=0\). Find its voltage \(V_C(t)\) and the battery current immediately after connection and at long times.
Knowledge: Kirchhoff and RC response
Answer
The node rule is \((\mathcal E-V_C)/r=V_C/R+C\dot V_C\). Thus
With \(V_C(0^+)=0\), \(I_{\rm battery}(0^+)=\mathcal E/r\). At long times, \(I_C=0\), \(V_C=\mathcal E R/(r+R)\), and \(I_{\rm battery}(\infty)=\mathcal E/(r+R)\). 初始电容电压连续；最终电容支路断流，但电阻支路仍通电。
5. Crossed fields with an electron
Let \(\mathbf E=E_0\hat{\mathbf y}\), \(\mathbf B=B_0\hat{\mathbf z}\), \(E_0,B_0>0\). An electron starts at rest. Find its initial acceleration, the mean drift over a cyclotron period, and the constant velocity that would produce no force from the outset.
Knowledge: Lorentz force and crossed fields
Answer
Initially magnetic force is zero, so \(\mathbf a(0)=-eE_0\hat{\mathbf y}/m_e\). The drift is
The same transverse velocity satisfies \(\mathbf E+\mathbf v_D\times\mathbf B=0\) for either charge sign. Any added constant component along \(\mathbf B\) also produces no force. 初始偏转与平均漂移是不同问题。
6. Replace the charged shell by a solid insulator
A uniformly volume-charged insulating ball, radius \(R\) and total charge \(Q\), rotates rigidly at \(\boldsymbol\omega=\omega\hat{\mathbf z}\). Find its magnetic dipole moment and center field. Explain why the 2023 surface-shell coefficients cannot be retained.
Knowledge: ring and shell superposition
Answer
A spherical layer at radius \(s\) has \(dQ=3Qs^2ds/R^3\). Each layer uses the derived surface-shell results:
Therefore
Charge now occupies all radii, requiring a volume integral. The surface-shell answers \(Q\omega R^2/3\) and \(\mu_0Q\omega/(6\pi R)\) describe a different distribution.
7. Induction where the magnetic field is zero
A long solenoid confines \(\mathbf B(t)=(B_0t/t_0)\hat{\mathbf z}\) to radius \(a\), during \(0<t<t_0\). A fixed coaxial loop of radius \(b>a\) lies outside it. Find the loop emf and the electric field at its wire. Is \(B=0\) at the wire enough to rule out induction?
Knowledge: induced electric field
Answer
Choose counterclockwise circulation seen from \(+z\) as positive. The linked flux uses solenoid area \(\pi a^2\), not loop area \(\pi b^2\):
The emf is clockwise even though the local magnetic field at the wire is zero. Faraday's law concerns changing flux through the loop, and the induced electric field extends outside the solenoid.
8. A source holding current fixed can absorb energy
Two linear coils have \(L_1,L_2\) and positive mutual inductance \(M\) for the chosen positive winding senses. Initially \(i_1=I_1>0\), \(i_2=0\). Hold \(i_1\) fixed while a second source changes \(i_2\) to \(-I_2\), \(I_2>0\). Find each source's work in this stage and the final magnetic energy.
Knowledge: mutual-inductance work
Answer
Adding the initial stored energy,
Source 1 absorbs energy despite constant current. Physical inductances satisfy \(M^2\leq L_1L_2\), so this quadratic energy cannot become negative. 互感项的符号由电流与绕向共同决定。
9. Double the resonant drive frequency
A series RLC circuit is driven by peak emf \(\mathcal E_m\cos\omega t\), with \(R>0\) and \(\omega_0=1/\sqrt{LC}\). At \(\omega=2\omega_0\), find the current amplitude, its phase relative to the source, and average resistor power. Contrast with resonance.
Knowledge: impedance and phase
Answer
Current is \(i=I_m\cos(2\omega_0t-\phi)\), with \(\phi=\tan^{-1}[3\omega_0L/(2R)]>0\): it lags the source. Average power is \(\langle P\rangle=I_m^2R/2\). At resonance \(X=0\), \(I_m=\mathcal E_m/R\), and the current is in phase. 峰值电流平方求平均功率时要除以 2。
10. A traveling electromagnetic wave
In vacuum, \(\mathbf E=60\cos(kx-\omega t)\hat{\mathbf y}\ \mathrm{V/m}\), with frequency \(f=50\ \mathrm{MHz}\). Find propagation direction, wavelength, \(\mathbf B\), and mean intensity. Use \(c=3.00\times10^8\ \mathrm{m/s}\) and \(\epsilon_0=8.85\times10^{-12}\ \mathrm{F/m}\).
Answer
Constant phase moves toward \(+x\), so \(\hat{\mathbf k}=\hat{\mathbf x}\). Then
\(\mathbf E\times\mathbf B\) points along \(+x\), agreeing with propagation. The two fields are in phase; \(60\ \mathrm{V/m}\) is peak amplitude, not RMS.
11. Dielectric and magnetic response
(a) Completely fill a vacuum parallel-plate capacitor with a linear dielectric of relative permittivity \(\kappa\). Compare \(C,V,E\) for fixed free plate charge and for fixed battery voltage. (b) For a weak-field paramagnet obeying Curie's law, what happens to magnetization if absolute temperature doubles at fixed \(B\)? (c) Can the same reversible linear law describe a ferromagnet's entire hysteresis loop?
Knowledge: polarization · Magnetic materials
Answer
(a) \(C=\kappa C_0\) in both cases. With fixed free \(q\), \(V=V_0/\kappa\) and \(E=E_0/\kappa\). At fixed \(V\), \(E=E_0\) and the free charge rises to \(\kappa q_0\). For the uniform linear medium \(\mathbf P=\epsilon_0(\kappa-1)\mathbf E\); \(\mathbf D\) counts free charge via its flux.
(b) In the course's weak-field moment model, \(M\simeq n\mu^2B/(k_BT)\), so doubling \(T\) halves \(M\). Independent permanent moments align more weakly relative to thermal agitation. Diamagnetism instead produces an induced moment opposing the applied field; it is not this Curie alignment model.
(c) No. A hysteresis loop includes history dependence, remanence, and coercivity; one constant \(\chi_m\) cannot describe the whole loop. Do not extend a weak-field linear formula beyond its material and field regime.
12. Displacement current during dielectric relaxation
An isolated parallel-plate capacitor contains a uniform weakly conducting dielectric of \(\epsilon,\sigma\). Ignore edges and initial magnetic fields. As it discharges, the uniform gap field is \(\mathbf E(t)=E_0e^{-\sigma t/\epsilon}\hat{\mathbf z}\). Find conduction and displacement current densities. Must the decreasing electric field generate a nonzero circulating magnetic field in this ideal gap model?
Answer
The total source in Ampère–Maxwell cancels. With the stated symmetry, no imposed magnetic field, and negligible edges, \(\mathbf B=0\) is consistent; the displacement term alone cannot determine \(\mathbf B\). Local energy conservation is also satisfied:
Stored electric energy becomes heat locally. 不能只看到电场随时间变，就漏掉同时存在的传导电流。