Question

A parallel plate capacitor (fig.) made of circular plates each of radius $R=6.0cm$ has a capacitance $C=100\mu F$. The capacitor is connected to a $230V$ ac supply with a (angular) frequency of $300$ rad $s^{-1}$. Determine the amplitude of $B$ at a point $3.0cm$ from the axis between the plates.?

Answer 1

Given: A parallel plate capacitor made of circular plates each of radius R=6.0 cm has a capacitance $C=100\mu F$. The capacitor is connected to a 230 V ac supply with a (angular) frequency of $300rad{s}^{-1}.$

To find the amplitude of B at a point 3.0 cm from the axis between the plates

Solution:

Radius of plates, $R=6cm=6\times {10}^{-2}m$

Capacitance of capacitor, $C=100\mu F=100\times {10}^{-6}F={10}^{-4}F$

Voltage of capacitor, $V=230V$

Frequency of capacitance, $\omega =300rad/s$

By definition of the displacement current,

$I_d={\epsilon }_0\frac{d{\varphi }_E}{dt}⟹I_d={\epsilon }_0\frac{d}{dt}\left(EA\right)⟹I_d=\epsilon A\frac{dE}{dt}$

But $E=\frac{\sigma }{{\epsilon }_0}=\frac{Q}{{\epsilon }_{0}A}$

Substituting this in the abve equation, we get

$I_d={\epsilon }_{0}A\frac{d}{dt}\left(\frac{Q}{{\epsilon }_{0}A}\right)⟹I_d={\epsilon }_{0}A.\frac{1}{{\epsilon }_{0}A}.\frac{dQ}{dt}⟹I_d=\frac{dQ}{dt}=I$

The distance of point from the axis between the plates

$r=3cm=0.03m$

Radius of the plates, $R=6cm=0.06m$

The magnetic field at a point between the plates

$B=\frac{{\mu }_0}{2\pi R^2}.r.I_d⟹B=\frac{{\mu }_{0}rI}{2\pi R^2}$

If $I=I_0$, maximum value of the current then

$I=\sqrt{2}{I}_{r}ms$

and

The rms value of the current

$I_{rms}=\frac{V_{rms}}{X_C}...........\left(i\right)$

And $X_C=\frac{1}{\omega C}$

Substituting this in eqn(i), we get

$I_{rms}=\frac{V_{rms}}{\frac{1}{\omega C}}⟹I_{rms}=Vrms\times \omega \times C⟹I_{rms}=230\times 300\times {10}^{-4}⟹I_{rms}=69\times {10}^3\times {10}^{-4}=6.9A$

Therefore $B=\frac{{\mu }_{0}r}{2\pi R^2}\sqrt{2}{I}_{rms}⟹B=\frac{4\pi \times {10}^{-7}\times 0.03\times \sqrt{2}\times 6.9}{2\pi \times 0.06\times 0.06}⟹B=1.63\times {10}^{-5}T$

is the amplitude of B at a point 3.0 cm from the axis between the plates.