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A) When a negatively charged electroscope is touched with a finger, the excess electrons on the electroscope are repelled by each other and flow through the human body to the Earth. This process,

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  1. a) When a negatively charged electroscope is touched with a finger, the excess electrons on the electroscope are repelled by each other and flow through the human body to the Earth. This process, called earthing or grounding, neutralizes the electroscope, causing its leaves to collapse.

  2. b)

  • Given values: C1=1 μFC_1 = 1 \, \mu\text{F} C2=2 μFC_2 = 2 \, \mu\text{F} C4=4 μFC_4 = 4 \, \mu\text{F} C5=5 μFC_5 = 5 \, \mu\text{F} V=8.0 VV = 8.0 \, \text{V}

  • Circuit Analysis: The 1 µF and 4 µF capacitors are in parallel. This combination is in series with the 5 µF capacitor. This entire series-parallel branch is in parallel with the 2 µF capacitor. The total combination is connected to an 8.0 V battery.

  • (i) The total capacitance (CtotalC_{total}):

  1. Calculate the equivalent capacitance of the 1 µF and 4 µF capacitors in parallel.

    $$
    C_{14} = C_1 + C_4
    $$
    $$
    C_{14} = 1 \, \mu\text{F} + 4 \, \mu\text{F} = 5 \, \mu\text{F}
    $$
    
  2. Calculate the equivalent capacitance of the 5 µF capacitor in series with C14C_{14}.

    $$
    \frac{1}{C_{5-14}} = \frac{1}{C_5} + \frac{1}{C_{14}}
    $$
    $$
    \frac{1}{C_{5-14}} = \frac{1}{5 \, \mu\text{F}} + \frac{1}{5 \, \mu\text{F}} = \frac{2}{5 \, \mu\text{F}}
    $$
    $$
    C_{5-14} = \frac{5}{2} \, \mu\text{F} = 2.5 \, \mu\text{F}
    $$
    
  3. Calculate the total capacitance by adding C5−14C_{5-14} in parallel with the 2 µF capacitor.

    $$
    C_{total} = C_{5-14} + C_2
    $$
    $$
    C_{total} = 2.5 \, \mu\text{F} + 2 \, \mu\text{F} = 4.5 \, \mu\text{F}
    $$
    The total capacitance is $\boxed{\text{4.5 µF}}$.
    
    • (ii) The total charge (QtotalQ_{total}):

      The total charge is given by Qtotal=Ctotal×VQ_{total} = C_{total} \times V.

      Qtotal=4.5 μF×8.0 VQ_{total} = 4.5 \, \mu\text{F} \times 8.0 \, \text{V} Qtotal=4.5×10−6 F×8.0 VQ_{total} = 4.5 \times 10^{-6} \, \text{F} \times 8.0 \, \text{V} Qtotal=36×10−6 C=36 μCQ_{total} = 36 \times 10^{-6} \, \text{C} = 36 \, \mu\text{C}

      The total charge is 36 µC\boxed{\text{36 µC}}.

    • (iii) Voltage across 4µF (V4V_4):

      The voltage across the parallel combination (C5−14C_{5-14}) is equal to the battery voltage because it is in parallel with the 2 µF capacitor, and this entire parallel arrangement is connected directly across the battery.
      So, the voltage across the series branch containing C5C_5 and C14C_{14} is V=8.0 VV = 8.0 \, \text{V}.
      The charge in this series branch is Q5−14=C5−14×VQ_{5-14} = C_{5-14} \times V.

      Q5−14=2.5 μF×8.0 V=20 μCQ_{5-14} = 2.5 \, \mu\text{F} \times 8.0 \, \text{V} = 20 \, \mu\text{C}

      This charge is the same for both C5C_5 and the parallel combination C14C_{14}. So, Q14=20 μCQ_{14} = 20 \, \mu\text{C}.
      Now, find the voltage across C14C_{14}:

      V14=Q14C14V_{14} = \frac{Q_{14}}{C_{14}} V14=20 μC5 μF=4.0 VV_{14} = \frac{20 \, \mu\text{C}}{5 \, \mu\text{F}} = 4.0 \, \text{V}

      Since the 1 µF and 4 µF capacitors are in parallel, the voltage across them is the same as V14V_{14}.
      Therefore, the voltage across the 4 µF capacitor is V4=V14V_4 = V_{14}.

      V4=4.0 VV_4 = 4.0 \, \text{V}

      The voltage across the 4µF capacitor is 4.0 V\boxed{\text{4.0 V}}.

    • (iv) Energy stored in 4µF (E4E_4):

      The energy stored in a capacitor is given by E=12CV2E = \frac{1}{2} C V^2.
      Using C4=4 μFC_4 = 4 \, \mu\text{F} and V4=4.0 VV_4 = 4.0 \, \text{V}:

      E4=12C4V42E_4 = \frac{1}{2} C_4 V_4^2 E4=12(4×10−6 F)(4.0 V)2E_4 = \frac{1}{2} (4 \times 10^{-6} \, \text{F}) (4.0 \, \text{V})^2 E4=12(4×10−6)(16)E_4 = \frac{1}{2} (4 \times 10^{-6}) (16) E4=2×10−6×16E_4 = 2 \times 10^{-6} \times 16 E4=32×10−6 J=32 μJE_4 = 32 \times 10^{-6} \, \text{J} = 32 \, \mu\text{J}

      The energy stored in the 4µF capacitor is 32 µJ\boxed{\text{32 µJ}}.

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