Question

Show that Ampere's Law is not necessarily consistent if the surface through which the flux is to be calculated is a closed surface, but that the Maxwell- Ampere Law always is. (Hence, Maxwell's introduction of his law of induction and the displacement current are not optional; they are logically necessary.) Show also that Faraday's Law of Induction does not suffer from this consistency problem.

Short answer

Answer: Ampere's Law can have consistency issues when calculating flux through a closed surface because it does not account for the changing electric field. However, the Maxwell-Ampere Law fixes this issue by including the displacement current due to changing electric fields. Faraday's Law of Induction does not suffer from similar consistency issues because it accounts for time-varying magnetic fields.

Step-by-step solution

Recall Ampere's Law and Maxwell-Ampere Law

Ampere's Law states that the magnetic field B around a closed loop is equal to the total current I flowing through a surface bounded by that loop. It is represented mathematically as: $$\oint_{C}\vec{B}\cdot d\vec{l}=\mu_{0}I$$ The Maxwell-Ampere Law extends Ampere's Law by introducing a time-varying electric field, which generates a displacement current. Mathematically, the Maxwell-Ampere Law is represented as: $$\oint_{C}\vec{B}\cdot d\vec{l}=\mu_{0}(I + \epsilon_{0}\frac{d\Phi}{dt})$$ Here, \(\epsilon_{0}\) is the vacuum permittivity, and \(d\Phi/dt\) is the rate of change of electric flux through the surface.

Show an example to illustrate the consistency issue with Ampere's Law

Consider a capacitor that is being charged with a current I. Since the current flows into the capacitor, at a certain moment, there will be charges on both plates but no current flowing between the plates. If we choose a surface that passes through the capacitor, Ampere's Law would not account for the changing electric field due to the accumulating charges. This shows that Ampere's Law can sometimes fail in accounting for flux when discussing closed surfaces.

Demonstrate why the Maxwell-Ampere Law is always consistent

The inconsistency in Ampere's Law is fixed by the Maxwell-Ampere Law because it takes into account the displacement current due to a changing electric field. In the capacitor example, a changing electric field occurs, which generates a displacement current represented by \(\epsilon_{0}\frac{d\Phi}{dt}\). The Maxwell-Ampere Law includes this displacement current, making it consistent for closed surfaces: $$\oint_{C}\vec{B}\cdot d\vec{l}=\mu_{0}(I+\epsilon_{0}\frac{d\Phi}{dt})$$

Analyze Faraday's Law of Induction

Faraday's Law of Induction states that the electromotive force (EMF) around a closed loop is equal to the negative rate of change of magnetic flux through the surface bounded by that loop, mathematically represented as: $$\oint_{C}\vec{E}\cdot d\vec{l}= -\frac{d\Phi_{B}}{dt}$$ This law essentially shows that a changing magnetic field generates an electric field. Unlike Ampere's Law, Faraday's Law does not suffer from inconsistency with closed surfaces because it already takes into account the time-varying magnetic field enclosed by the surface. In conclusion, we have shown that Ampere's Law can have consistency issues when calculating the flux through a closed surface, while the Maxwell-Ampere Law fixes this issue by including the displacement current due to changing electric fields. Additionally, Faraday's Law does not suffer from similar consistency issues because it accounts for time-varying magnetic fields.