Ncert Exercises & Solutions

Show that the magnetic field B at a point in between the plates of the parallel capacitor during charging is $\frac{ε_{0} μ_{0} r}{2} \frac{dE}{dt}$ (symbols having usual meaning).

Hint: The magnetic field in between the plates depends on the displacement current.

Step 1: Find the magnetic field between the plates.

Consider the figure given below to prove that the magnetic field B at a point in between the plate of a parallel-plate capacitor during charging is $\frac{ε_{0} μ_{0} r}{2} \frac{dE}{dt}$ .

$Let I_{d} be the displacement current in the region between two plates of parallel plate capaciter in the figure .$

The magnetic field induction at a point in a region between two plates of the capacitor at a perpendicular distance r from the axis of plates is:

$B = \frac{μ_{0} 2 I_{d}}{4 πr} = \frac{μ_{0}}{2 πr} I_{d} = \frac{μ_{0}}{2 μr} \times ε_{0} \frac{{dϕ}_{E}}{dt} [∴ I_{d} = \frac{ε_{0} {dϕ}_{E}}{dt}]$
$= \frac{μ_{o} ε_{0}}{2 πr} \frac{d}{dt} ({Eπr}^{2}) = \frac{μ_{0} ε_{0}}{2 πr} {πr}^{2} \frac{dE}{dt}$
$B = \frac{μ_{0} ε_{0} r}{2} \frac{dE}{dt} [∵ ϕ_{E} = {Eπr}^{2}]$

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