The magnitude of the magnetic field B(r) inside the capacitor as a function of distance from the axis is [tex]\mathbf{B(r) =\dfrac{\mu_o I r}{2 \pi a^2} }[/tex]
A charged capacitor usually cause changes between the plates of electric field E and the electric flux Φ
As seen in Ampere's law which is extended by Maxwell:
[tex]\mathbf{\oint B^{\to} .dl^{\to} = \mu_o \Big( I+\dfrac{d \phi }{dt} \Big)}[/tex]
To meet the equilibrium equation of electric charge, Maxwell modified Ampere's law by incorporating the displacement current through into electric current component.
∴
The displacement current can be expressed by the relation;
[tex]\mathbf{I = \varepsilon_o \dfrac{dE}{dt} }[/tex]
From the given information, the displacement current density through area A can now be expressed as:
[tex]\mathbf{I = \varepsilon_o A \dfrac{dE}{dt} }[/tex]
Replacing the value of displacement current density into Maxwell modified Ampere Law, we have:
[tex]\mathbf{\oint B .dl= \mu_o I_{enclosed} }[/tex]
[tex]\mathbf{B \times (2 \pi r) = \mu_o I\Big (\dfrac{\pi r^2}{\pi a^2} \Big) }[/tex]
[tex]\mathbf{B(r) =\dfrac{\mu_o I r}{2 \pi a^2} }[/tex]
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