What is the maxwell�s modification of ampere�s law?

Displacement Current

Maxwell’s modification, applied to the above situation, consists of considering a mathematical equivalent current to pierce the surface S1. If E is the electric field in the region between the plates of the capacitor, then

\(E=frac{sigma }{{{in }_{0}}}=frac{Q/A}{{{in }_{0}}}=frac{Q}{{{in }_{0}}A}\) … (3)

where, \(sigma =Q/A\) is the surface charge density on the capacitor plate whose area is A.

Clearly, \(Q={{in }_{0}}EA={{in }_{0}}{{phi }_{E}}\) … (4)

where, \({{phi }_{E}}=EA\) is the electric flux between the plates.

Thus, \(I=frac{dQ}{dt}{{in }_{0}}frac{d{{phi }_{E}}}{dt}\) … (5)

The right-hand side of eqn. (5) contains the derivative of electric flux piercing S1, while I is the current piercing S2.

That is, \({{in }_{0}}frac{d{{phi }_{E}}}{dt}\) is mathematically equivalent current for surface S1 to the current I, which pierces surface S2.

We define this effective current, called the displacement currentId to be:

\({{I}_{d}}={{in }_{0}}frac{d{{phi }_{E}}}{dt}\) … (6)

Displacement current is thus the current which comes into existence due to the rate of change of electric flux with respect to time (when the applied electric field is varying).

It is to be noted carefully that I pierces S2 and the displacement current Id pierces S1 and that Id = I. To distinguish between I and Id, I is called the conduction current (or real or true current).

Ampere-Maxwell’s Law: Generalized Ampere Law:

The general form of Ampere’s law, as modified by Maxwell, can now be stated. To the term for true current (I) linking a closed path, we add the displacement current (Id) linking the path and the Ampere’s law takes the form:

\(oint{overrightarrow{B}}.,overrightarrow{dl}={{mu }_{0}}left( I+{{I}_{d}} right)={{mu }_{0}}left( I+{{in }_{0}}frac{d{{phi }_{E}}}{dt} right)\) … (7)

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