27 similar drops of mercury are maintained at 10V each. All these spherical drops combine into a single big drop. The potential energy of the bigger drop is……… times that of a smaller drop.

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alok kumar singh | Contributor-Level 10

a month ago

With the help of conservation of volume, we can write

$27 \times \frac{4}{3} π r^{3} = \frac{4}{3} π R^{3} \Rightarrow R = 3 r . . . . . . . (1)$

With the help of conservation of charge, we can write

Q = 27 q.(2)

Potential energy of single drop = U₁ = $\frac{q^{2}}{8 π ε_{0} r}$

Potential energy of bigger drop = $U_{2} = \frac{Q^{2}}{8 π ε_{0} R} = \frac{27 \times 27 \times q^{2}}{8 π ε_{0} (3 r)} = 243 (\frac{q^{2}}{8 π ε_{0} r}) = 243 U_{1}$

$\Rightarrow \frac{U_{2}}{U_{1}} = 243$

With the help of conservation of volume, we can write  <math> <mrow> <mn>2</mn> <mn>7</mn> <mo>×</mo> <mfrac> <mrow> <mn>4</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mi>π</mi> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mi>π</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>⇒</mo> <mi>R</mi> <mo>=</mo> <mn>3</mn> <mi>r</mi> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </math>                            With the help of conservation of charge, we can writeQ = 27 q.(2)Potential energy of single drop = U1 =   <math> <mrow> <mfrac> <mrow> <msup> <mrow> <mi>q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mi>π</mi> <msub> <mrow> <mi>ε</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mi>r</mi> </mrow> </mfrac> </mrow> </math> Potential energy of bigger drop =   <math> <mrow> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msup> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mi>π</mi> <msub> <mrow> <mi>ε</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mi>R</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mn>7</mn> <mo>×</mo> <mn>2</mn> <mn>7</mn> <mo>×</mo> <msup> <mrow> <mi>q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mi>π</mi> <msub> <mrow> <mi>ε</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mi>r</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mn>4</mn> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mi>q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mi>π</mi> <msub> <mrow> <mi>ε</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mn>4</mn> <mn>3</mn> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </math> <math> <mrow> <mo>⇒</mo> <mfrac> <mrow> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mn>4</mn> <mn>3</mn> </mrow> </math>

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27 similar drops of mercury are maintained at 10V each. All these spherical drops combine into a single big drop. The potential energy of the bigger drop is……… times that of a smaller drop.

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