Correct Answer - Option 2 :
\(\epsilon{_0}\frac{{d{\phi _E}}}{{dt}}\)
Amperes law:
The line integral of the magnetic field around any closed curve is equal to μ0 times the net current I threading through the area enclosed by the curve.
\(\oint \vec B \cdot \overrightarrow {dl} = {\mu _0}I\)
Drawback: Ampere’s law in this form is not valid if the electric field at the surface varies with time.
Displacement current (ID): It is the current that comes into existence in addition to the conduction current, whenever the electric field and hence the electric flux changes with time.
Modified Ampere's Law:
- To modify Ampere’s law, Maxwell followed a symmetry consideration, i.e. by Faraday’s law, a changing magnetic field induces an electric field, hence a changing electric field must induce a magnetic field.
- As currents are the usual sources of the magnetic field, a changing electric field must be associated with the current. Maxwell called that current as displacement current.
- To maintain the dimensional consistency, the displacement current is added in ampere’s law.
\(\oint \vec B \cdot \overrightarrow {dl} = {\mu _0}I + {\mu _0}{\epsilon_0}\left( {\frac{{d{{\rm{E}}}}}{{dt}}} \right)\)
\(\oint \vec B \cdot \overrightarrow {dl} = {\mu _0}[I + {\epsilon_0}\left( {\frac{{d{{\rm{E}}}}}{{dt}}} \right)]\)
Where:
\({\epsilon_0}\left( {\frac{{d{{\rm{E}}}}}{{dt}}} \right)\) is the displacement current.
Applying Stokes theorem, the above form can be written as:
\(\nabla \times B=\mu _0J+\mu _0\epsilon _0\frac{\partial E}{\partial t}\)
