JEE Main & Advanced Physics Wave Optics / तरंग प्रकाशिकी Maxwell's Contribution

(1) Ampere's Circuital law : According to this law the line integral of magnetic field along any closed path or circuit is \[{{\mu }_{0}}\] times the total current threading the closed circuit i.e., \[\oint{\overset{\to }{\mathop{B}}\,}.\overset{\to }{\mathop{dl}}\,={{\mu }_{0}}i\]

(2) Inconsistency of Ampere's law : Maxwell explained that Ampere's law is valid only for steady current or when the electric field does not change with time. To see this inconsistency consider a parallel plate capacitor being charged by a battery. During the charging time varying current flows through connecting wires.

Applying Ampere's law for loop \[{{l}_{1}}\] and \[{{l}_{2}}\] \[\oint{\underset{{{l}_{1}}}{\mathop{{}}}\,\overset{\to }{\mathop{B}}\,}.\overset{\to }{\mathop{dl}}\,={{\mu }_{0}}i\]

But \[\oint{\underset{{{l}_{2}}}{\mathop{{}}}\,\overset{\to }{\mathop{B}}\,}.\overset{\to }{\mathop{dl}}\,=0\] (Since no current flows through the region between the plates). But practically it is observed that there is a magnetic field between the plates. Hence Ampere's law fails

i.e.  \[\oint{\underset{{{l}_{1}}}{\mathop{{}}}\,\overset{\to }{\mathop{B}}\,}.\overset{\to }{\mathop{dl}}\,\ne {{\mu }_{0}}i\].

(3) Modified Ampere's Circuital law or Ampere- Maxwell's Circuital law : Maxwell assumed that some sort of current must be flowing between the capacitor plates during charging process. He named it displacement current. Hence modified law is as follows

\[\oint{\overset{\to }{\mathop{B}}\,}.\overset{\to }{\mathop{dl}}\,={{\mu }_{0}}({{i}_{c}}+{{i}_{d}})\ \]or  \[\oint{\overset{\to }{\mathop{B}}\,}.\overset{\to }{\mathop{dl}}\,={{\mu }_{0}}({{i}_{c}}+{{\varepsilon }_{0}}\frac{d{{\varphi }_{E}}}{dt})\] 

where \[{{i}_{c}}=\] conduction current = current due to flow of charges in a conductor and

\[{{i}_{d}}=\] Displacement current = \[{{\varepsilon }_{0}}\frac{d{{\varphi }_{E}}}{dt}\] = current due to the changing electric field between the plates of the capacitor

(4) Maxwell's equations

(i)  \[\oint{\underset{s}{\mathop{{}}}\,\overset{\to }{\mathop{E}}\,}.\overset{\to }{\mathop{ds}}\,=\frac{q}{{{\varepsilon }_{0}}}\]             (Gauss's law in electrostatics)

(ii)  \[\oint{\underset{s}{\mathop{{}}}\,\overset{\to }{\mathop{B}}\,}.\overset{\to }{\mathop{ds}}\,=0\]                  (Gauss's law in magnetism)

(iii)  \[\oint_{{}}{\overset{\to }{\mathop{B}}\,\,.\,\overset{\to }{\mathop{dl}}\,}\,=-\frac{d{{\varphi }_{B}}}{dt}\]                 (Faraday's law of EMI)

(iv) \[\oint{\overset{\to }{\mathop{B}}\,\,\,\overset{\to }{\mathop{dl}}\,}={{\mu }_{o}}({{i}_{c}}+{{\varepsilon }_{o}}\frac{d{{\varphi }_{E}}}{dt}\] (Maxwell- Ampere's Circuital law)