A parallel plate capacitor is filled by a dielectric whose relative permittivity varies with the applied voltage (U) as ε= αU where α = 2V^-1. A similar capacitor with no dielectric is charged to U₀= 78 V. It is then connected to the uncharged capacitor with the dielectric. Find the final voltage on the capacitors.

<p><span data-teams="true">This is a long answer type question as classified in NCERT Exemplar</span></p><p>To lift the disc ther must be some electrostatic force is reqired.</p><p>F=qE……. (1), E=V/d………. (2)</p><p>But the charge required during the process is -<span contenteditable="false"> <math> <mo>∈</mo> </math> </span>_{0 <span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> <mrow> <mrow> <mi>d</mi> </mrow> </mrow> </mfrac> <mi>π</mi> </math> </span>}r², using this relation in 1 and 2</p><p>F=-<span contenteditable="false"> <math> <mo>∈</mo> </math> </span>_{0<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> <mrow> <mrow> <mi>d</mi> </mrow> </mrow> </mfrac> <mi>π</mi> </math> </span>}r²V/d…… (3)</p><p>Also to balance something F=mg……. (4)</p><p>From equation 3 and 4</p><p>Mg= -<span contenteditable="false"> <math> <mo>∈</mo> </math> </span>_{0<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> <mrow> <mrow> <mi>d</mi> </mrow> </mrow> </mfrac> <mi>π</mi> </math> </span>}r²V/d, so V =<span contenteditable="false"> <math> <mo>√</mo> <mfrac> <mrow> <mrow> <mi>m</mi> <mi>g</mi> <mi>d</mi> </mrow> </mrow> <mrow> <mrow> <mo>∈</mo> <mn>0</mn> <mi mathvariant="normal">π</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </mfrac> </math> </span>₂ this the requird solution.</p>

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