EM Waves flash cards

1. 1.What inconsistency in Ampere's circuital law did Maxwell discover?

   Ampere's law $\oint \vec{B}\cdot d\vec{l}=\mu_0 I$ gives different results for the same loop of a charging capacitor depending on the surface chosen: a flat surface cutting the wire encloses current $I$, but a bulging surface passing between the plates encloses no conduction current, yet $\vec{B}$ around the loop is the same. Ampere's law was therefore incomplete.

   Hint: Think of two surfaces bounded by the same Amperian loop near a capacitor.

2. 2.Define displacement current and give its expression.

   Displacement current is the current-like quantity arising from a time-varying electric field: $I_d=\varepsilon_0\dfrac{d\Phi_E}{dt}$, where $\Phi_E$ is the electric flux. It is not a flow of charge but produces a magnetic field just like conduction current.

   Hint: It depends on the rate of change of electric flux.

3. 3.Write Ampere-Maxwell law (the corrected form).

   $\oint \vec{B}\cdot d\vec{l}=\mu_0(I_c+I_d)=\mu_0 I_c+\mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$, where $I_c$ is conduction current and $I_d=\varepsilon_0\,d\Phi_E/dt$ is displacement current.

   Hint: Ampere's law plus a term with $d\Phi_E/dt$.

4. 4.In the region between the plates of a charging capacitor, which current flows and how large is it?

   There is no conduction current between the plates, only displacement current $I_d=\varepsilon_0\,d\Phi_E/dt$. Its magnitude equals the conduction current $I_c$ in the connecting wires, so charge conservation and continuity of current are preserved.

   Hint: $I_d=I_c$ ensures continuity of the total current.

5. 5.Show that displacement current between capacitor plates equals the conduction current.

   For a parallel-plate capacitor, $\Phi_E=\dfrac{q}{\varepsilon_0}$ (field $E=q/\varepsilon_0 A$, flux $EA$). Then $I_d=\varepsilon_0\dfrac{d\Phi_E}{dt}=\varepsilon_0\cdot\dfrac{1}{\varepsilon_0}\dfrac{dq}{dt}=\dfrac{dq}{dt}=I_c$.

   Hint: Use $\Phi_E=q/\varepsilon_0$ and differentiate.

6. 6.What is the total 'current' in Maxwell's generalization, and why is it always continuous?

   Total current $I=I_c+I_d$. Where conduction current stops (e.g. capacitor gap), displacement current takes over so that $I_c+I_d$ is continuous throughout the circuit. This makes the total current a closed, continuous quantity.

   Hint: Sum of conduction and displacement current is unbroken.

7. 7.State Gauss's law for electricity (Maxwell's equation I).

   $\oint \vec{E}\cdot d\vec{A}=\dfrac{q_{enc}}{\varepsilon_0}$. The net electric flux through any closed surface equals the enclosed charge divided by $\varepsilon_0$. It expresses that electric field lines begin and end on charges.

   Hint: Electric flux out of a closed surface relates to enclosed charge.

8. 8.State Gauss's law for magnetism (Maxwell's equation II) and its physical meaning.

   $\oint \vec{B}\cdot d\vec{A}=0$. The net magnetic flux through any closed surface is zero. Physically this means isolated magnetic monopoles do not exist; magnetic field lines form closed loops.

   Hint: No magnetic monopoles.

9. 9.State Faraday's law of induction (Maxwell's equation III).

   $\oint \vec{E}\cdot d\vec{l}=-\dfrac{d\Phi_B}{dt}$. A changing magnetic flux induces an electric field whose line integral (emf) equals the negative rate of change of magnetic flux. A time-varying $\vec{B}$ creates $\vec{E}$.

   Hint: Changing magnetic flux produces an electric field.

10. 10.State the Ampere-Maxwell law (Maxwell's equation IV).

    $\oint \vec{B}\cdot d\vec{l}=\mu_0 I_c+\mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$. Both conduction current and a time-varying electric field (displacement current) produce a magnetic field.

    Hint: Changing electric flux (plus current) produces a magnetic field.

11. 11.Which two of Maxwell's equations together imply electromagnetic waves, and why?

    Faraday's law (changing $\vec{B}$ makes $\vec{E}$) and the Ampere-Maxwell law (changing $\vec{E}$ makes $\vec{B}$). A changing $\vec{E}$ produces $\vec{B}$ and that changing $\vec{B}$ produces $\vec{E}$, so the fields regenerate each other and propagate as a self-sustaining wave.

    Hint: The mutual generation of $\vec{E}$ and $\vec{B}$.

12. 12.How are electromagnetic waves produced by a source?

    An accelerating (or oscillating) electric charge produces EM waves. The oscillating charge creates a time-varying electric field, which produces a time-varying magnetic field, and the two fields regenerate each other and radiate outward at frequency equal to the charge's oscillation frequency.

    Hint: Acceleration of charge is essential; a charge at rest or in uniform motion does not radiate.

13. 13.Does a charge moving with constant velocity radiate EM waves?

    No. A charge at rest produces only a static electric field; a charge in uniform (unaccelerated) motion produces steady $\vec{E}$ and $\vec{B}$ but no radiation. Only an accelerating charge radiates electromagnetic waves.

    Hint: Radiation requires acceleration.

14. 14.Why could Maxwell's waves not be produced in a laboratory in his time, and who first produced them?

    Producing EM waves needs charges oscillating at high (radio) frequencies. Heinrich Hertz first experimentally produced and detected EM waves (around 1887) using a spark-gap oscillator, confirming Maxwell's prediction.

    Hint: Hertz's spark-gap experiment.

15. 15.What is the transverse nature of electromagnetic waves?

    In an EM wave, $\vec{E}$ and $\vec{B}$ are both perpendicular to each other and to the direction of propagation. There is no field component along the direction of travel, so EM waves are transverse and can be polarized.

    Hint: $\vec{E}\perp\vec{B}\perp$ direction of propagation.

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