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

Ampere's Law inconsistency occurs when the surface through which the flux is calculated is a closed surface because it does not account for the displacement current in changing electric fields. This can be seen in the charging capacitor example, where Ampere's Law implies there is no magnetic field when the closed loop is considered around the gap between the capacitor plates, despite the wire carrying a current. The introduction of Maxwell-Ampere Law, which includes the displacement current term, resolves this inconsistency.

Step-by-step solution

Ampere's Law

Ampere's Law can be written as: ${\oint }_C\overrightarrow{B}\cdot d\overrightarrow{l}={\mu }_0{I}_{enc}$ where ${\oint }_C\overrightarrow{B}\cdot d\overrightarrow{l}$ is the magnetic field B integrated over a closed loop C, ${\mu }_0$ is the permeability of free space, and $I_{enc}$ is the total current passing through the surface enclosed by the loop.

Consider a charging capacitor

Let's consider the example of a charging capacitor with constant current I and a closed loop C around one of the connecting wires. In this case, Ampere's Law states the following: ${\oint }_C\overrightarrow{B}\cdot d\overrightarrow{l}={\mu }_{0}I$ However, since the current in the wire is charging the capacitor plates, there is no current passing through the surface enclosed by the loop when the loop is considered around the gap between the capacitor plates. Thus, Ampere's Law implies: ${\oint }_C\overrightarrow{B}\cdot d\overrightarrow{l}=0$ This creates a contradiction because the wire is carrying a current, but the loop with the gap between the capacitor plates is not carrying any current. Ampere's Law is not consistent in this case.

Maxwell-Ampere Law

The Maxwell-Ampere Law, which includes the displacement current term, is given by: ${\oint }_C\overrightarrow{B}\cdot d\overrightarrow{l}={\mu }_0(I_{enc}+{ϵ}_0\frac{d{\mathrm{\Phi }}_E}{dt})$ where ${ϵ}_0\frac{d{\mathrm{\Phi }}_E}{dt}$ is the displacement current term, and ${\mathrm{\Phi }}_E$ is the electric flux.

Apply Maxwell-Ampere Law to the charging capacitor

With the charging capacitor scenario, when the loop is around the gap between the capacitor plates, the displacement current term accounts for the changing electric field between the plates. The displacement current is given by: ${ϵ}_0\frac{d{\mathrm{\Phi }}_E}{dt}=I$ Thus, the Maxwell-Ampere Law becomes consistent in this case: ${\oint }_C\overrightarrow{B}\cdot d\overrightarrow{l}={\mu }_0(I_{enc}+{ϵ}_0\frac{d{\mathrm{\Phi }}_E}{dt})={\mu }_{0}I$

Faraday's Law of Induction

Faraday's Law of Induction states that the electromotive force (EMF) induced in a closed loop equals the negative time rate of change of magnetic flux through the loop: ${\oint }_C\overrightarrow{E}\cdot d\overrightarrow{l}=-\frac{d{\mathrm{\Phi }}_B}{dt}$

Faraday's Law consistency

Faraday's Law is consistent because it is based on the conservation of energy and does not suffer from any inconsistency problems like Ampere's Law. The induced electromotive force in a loop is directly proportional to the rate of change of magnetic flux through that loop. The closed surface for calculating flux is not an issue in this case. In conclusion, we demonstrated that Ampere's Law is not necessarily consistent if the surface through which the flux is calculated is a closed surface. We also showed that Maxwell-Ampere Law addresses this issue, making it always consistent. Furthermore, we verified that Faraday's Law of Induction does not suffer from any consistency problems.