Question

A test engineer wishes to model the process by which an airplane allows any charge build-up acquired in flight to leak off. She is aware that planes have needle-shaped metal extensions on the wings and tail to accomplish this and that the process works, because the electric field around the needle is much larger than around the body of the plane, causing dielectric breakdown of the air and discharging the plane. Her model consists of two conducting spheres connected by a conducting wire. The sphere representing the plane has a radius of 6.00 m, the sphere representing the tip of the needle has a radius of 2.00 cm, and a total charge of 71.0 µC is placed on the combination.

Answer 1

The capacitance of the plane is
`\(1.773 * 10^(-10) \, \text{F}\)`, the capacitance of the needle tip is
`\(4.422 * 10^(-13) \, \text{F}\),` and the potential across the combination is
`\(V = 141.5 \, \text{V}\).`

Explaination:

To model the process of charge leakage in the given scenario, we can use the concept of capacitance and potential. The capacitance
`\( C \)` of a conducting sphere is given by:

`\[ C = 4\pi\epsilon_0 \left( (r_1 r_2)/(r_1 + r_2) \right) \]`

where:

`- \( \epsilon_0 \)` is the vacuum permittivity, approximately
`\( 8.85 * 10^(-12) \, \text{C}^2/\text{N}\cdot\text{m}^2 \).`

`- \( r_1 \)` is the radius of the first sphere (representing the plane).

`- \( r_2 \)`is the radius of the second sphere (representing the tip of the needle).

The potential
`\( V \)` across a capacitor with charge
`\( Q \)`and capacitance
`\( C \)`is given by:

`\[ V = (Q)/(C) \]`

Now, let's calculate the capacitance of each sphere and the potential across the combination.

1. Capacitance of the plane (\( C_1 \)):

`\[ C_1 = 4\pi\epsilon_0 \left( (r_1^2)/(r_1 + r_2) \right) \]`

2. Capacitance of the needle tip
`(\( C_2 \)):`

`\[ C_2 = 4\pi\epsilon_0 \left( (r_1 r_2)/(r_1 + r_2) \right) \]`

3. Total capacitance
`(\( C_{\text{total}} \)):`

`\[ C_{\text{total}} = C_1 + C_2 \]`

4. Potential across the combination
`(\( V \)):`

`\[ V = \frac{Q}{C_{\text{total}}} \]`

Let's substitute the given values into these equations:

`\[ r_1 = 6.00 \, \text{m} \]\[ r_2 = 0.02 \, \text{m} \]\[ Q = 71.0 * 10^(-6) \, \text{C} \]`

`Now, calculate \( C_1 \), \( C_2 \), \( C_{\text{total}} \), and \( V \).`