A capacitor is discharging through a resistor R. Consider in time t1 the energy stored in the capacitor reduces to half of its initial value and in time t2, the charge stored in the capacitor reduces to half of its initial value and in time t2 the charge stored reduces to one eight of its initial value. The ratio t1/t2 will be
According to Law of discharging of capacitor, we can write
<p>According to Law of discharging of capacitor, we can write</p><p><math><mrow><mi>Q</mi><mo>=</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mi>t</mi></mrow><mrow><mi>τ</mi></mrow></mfrac></mrow></msup><mo>⇒</mo><mfrac><mrow><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>8</mn></mrow></mfrac><mo>=</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mi>τ</mi></mrow></mfrac></mrow></msup><mo>⇒</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub><mn>3</mn><mi>τ</mi><mi mathvariant="script">l</mi><mi>n</mi><mrow><mo> (</mo><mrow><mn>2</mn></mrow><mo>)</mo></mrow><mo>, </mo><mtext> </mtext><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi></mrow></math></p><p><math><mrow><mi>U</mi><mo>=</mo><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><mi>C</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mn>2</mn><mi>C</mi></mrow></mfrac><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>2</mn><mi>t</mi></mrow><mrow><mi>τ</mi></mrow></mfrac></mrow></msup><mo>⇒</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><mo> (</mo><mrow><mfrac><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mn>2</mn><mi>C</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mn>2</mn><mi>C</mi></mrow></mfrac><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>2</mn><mi>t</mi></mrow><mrow><mi>τ</mi></mrow></mfrac></mrow></msup><mo>⇒</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mi mathvariant="script">l</mi><mi>n</mi><mrow><mo> (</mo><mrow><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></p><p><math><mrow><mo>⇒</mo><mfrac><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></math></p>
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