CBSE Class 12 · 2016 · Delhi · EM WavesShort Answer

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Write Maxwell's generalization of Ampere's circuital law. Show that, in the process of charging a capacitor, the current produced within the plates of the capacitor is

i = ε_{0} \frac{d Φ _{E}}{d t}

, where

Φ_{E}

is the electric flux produced during charging of the capacitor plates.

Solution

Maxwell's generalization (the Ampere--Maxwell law):

\oint B \cdot d l = μ_{0} i_{c} + μ_{0} ε_{0} \frac{d Φ _{E}}{d t}

During charging, if q is the instantaneous charge and A the plate area, the field between the plates is

E = σ / ε_{0} = q / (ε_{0} A)

.
The electric flux between the plates is

Φ_{E} = E A = \frac{q}{ε _{0}} \Rightarrow q = ε_{0} Φ_{E}

Differentiating, the current within the plates is

i = \frac{d q}{d t} = ε_{0} \frac{d Φ _{E}}{d t}

which is the displacement current.

i = ε_{0} d Φ_{E} / d t

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CBSE Class 12 · 2016 · Delhi · EM WavesShort Answer

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