Question

What is Maxwell's modification of Ampere’s law?

Answer 1

Hint: In order to solve this question, we are going to first write the statement of the Ampere’s law and also the mathematical statement, then considering the shortcomings of the Ampere’s law and the corresponding Maxwell’s modification, we write the final relation that is derived.

Formula used:
The magnitude of electric field is:
\[E = \dfrac{V}{d}\]
Where, \[V\] is the potential difference and \[d\] is the distance between the plates.
The modification in Ampere’s law is given by:
\[\int {\vec B \cdot d\vec l = {\mu _0}\left( {{I_c} + {\varepsilon _0}\dfrac{{d\phi }}{{dt}}} \right)} \]

Complete step-by-step solution:
Ampere’s law states that “the line integral of the total magnetic field along a closed plane curve is equal to\[{\mu _0}\]times the total current crossing the area that is bounded by the closed curve provided that the electric field inside the loop remains constant"
According to Maxwell, between the plates of the capacitor there is an electric field which is directed from the positive plate to the negative plate. The magnitude of electric field is:
\[E = \dfrac{V}{d}\]
Where, \[V\]is the potential difference and \[d\]is the distance between the plates.
Simplifying this, we get

$ E = \dfrac{q}{{{\varepsilon _0}A}} \\
   \Rightarrow EA = \dfrac{q}{{{\varepsilon _0}}} = \phi \\ $

This is the flux.
Now, \[\dfrac{{d\phi }}{{dt}} = \dfrac{1}{{{\varepsilon _0}}}\left( {\dfrac{{dq}}{{dt}}} \right)\]
But \[\dfrac{{dq}}{{dt}}\] is the rate of change of charge on the capacitor plates. It is called displacement current and given by
\[I = \dfrac{{dq}}{{dt}} - {\varepsilon _0}\dfrac{{d\phi }}{{dt}}\]
This term is the missing one in the Ampere’s circuital law. Thus, the modification in Ampere’s law is given by:
\[\int {\vec B \cdot d\vec l = {\mu _0}\left( {{I_c} + {\varepsilon _0}\dfrac{{d\phi }}{{dt}}} \right)} \]

Note: The Ampere’s law is not valid for the steady currents. So, it was modified by Maxwell and he included the time-varying electric fields. For the Ampere’s law to be correct, there has to be some amount of current between the plates of the capacitor. There is current outside the plates due to the flow of electrons.