Final Answer:
The capacitance of the plane is
, the capacitance of the needle tip is
and the potential across the combination is
Explaination:
To model the process of charge leakage in the given scenario, we can use the concept of capacitance and potential. The capacitance
of a conducting sphere is given by:
![\[ C = 4\pi\epsilon_0 \left( (r_1 r_2)/(r_1 + r_2) \right) \]](https://img.qammunity.org/2024/formulas/physics/high-school/rj2s2oi6kw6lwa08b3c1ycqcdp25kinr0l.png)
where:
is the vacuum permittivity, approximately

is the radius of the first sphere (representing the plane).
is the radius of the second sphere (representing the tip of the needle).
The potential
across a capacitor with charge
and capacitance
is given by:
![\[ V = (Q)/(C) \]](https://img.qammunity.org/2024/formulas/physics/high-school/hvc0p0hx79fz378p9vwz20kwcc9qyuf9bs.png)
Now, let's calculate the capacitance of each sphere and the potential across the combination.
1. Capacitance of the plane (\( C_1 \)):
![\[ C_1 = 4\pi\epsilon_0 \left( (r_1^2)/(r_1 + r_2) \right) \]](https://img.qammunity.org/2024/formulas/physics/high-school/mdnkxwutesku84f8fqka3l0eqp1wwyer1r.png)
2. Capacitance of the needle tip

![\[ C_2 = 4\pi\epsilon_0 \left( (r_1 r_2)/(r_1 + r_2) \right) \]](https://img.qammunity.org/2024/formulas/physics/high-school/tu3jjao5d983q3zvtu7lvmww4klr10v87o.png)
3. Total capacitance

![\[ C_{\text{total}} = C_1 + C_2 \]](https://img.qammunity.org/2024/formulas/physics/high-school/1on78nykr4g8z4x7lao1eq4rg8np68o2a5.png)
4. Potential across the combination

![\[ V = \frac{Q}{C_{\text{total}}} \]](https://img.qammunity.org/2024/formulas/physics/high-school/s66y7ulip55g4fdbpv0o83tr6qv15yl7ic.png)
Let's substitute the given values into these equations:
![\[ r_1 = 6.00 \, \text{m} \]\[ r_2 = 0.02 \, \text{m} \]\[ Q = 71.0 * 10^(-6) \, \text{C} \]](https://img.qammunity.org/2024/formulas/physics/high-school/vjtyqsudx2t9a7fnr2ued2fplb3nwdpbll.png)
