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When dealing with spherical capacitors, it is essential to understand how capacitance is determined by the geometry of the capacitor and the properties of the material between its plates. The capacitance of a spherical capacitor can be calculated if the radii of the inner and outer spheres and the permittivity of the material between them are known.
For a spherical capacitor, the formula for capacitance is:

• \( C = \frac{4\pi\varepsilon_0 r_1r_2}{r_2-r_1} \)

where \( r_1 \) is the radius of the inner sphere, \( r_2 \) is the radius of the outer sphere, and \( \varepsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12}\, \mathrm{F/m} \).

In practice, once you know the charge \( Q \) on the capacitor and the potential difference \( V \) across it, you can compute the capacitance using \( C = \frac{Q}{V} \). This relation allows us to link the electrical properties directly to the measurable physical characteristics of the system. In our particular problem, with the given charge of 3.30 nC and a voltage of 220 V, the capacitance calculation yields \( 1.50 \times 10^{-11} \mathrm{F} \).

The electric field in a spherical capacitor is crucial because it helps to understand the distribution of force around the charged bodies. After we have calculated the capacitance, the next step is to calculate the electric field just outside the surface of the inner sphere.

The electric field \( E \) at a distance from a point charge is given by:

• \( E = \frac{Q}{4\pi\varepsilon_0 r_1^2} \)

where \( Q \) is the charge and \( r_1 \) is the radius of the inner sphere. This formula is derived from Gauss's law, which addresses how an electric field emanates from a charged body.
For our specific problem, with a charge of 3.30 nC and an inner radius of 0.035 m, the electric field is determined to be approximately \( 2.61 \times 10^4 \mathrm{N/C} \). This calculation provides insight into the force experienced by a hypothetical 1 C charge placed just outside the inner sphere.

Knowing the radii of the inner and outer parts of a spherical capacitor is essential for calculating both the capacitance and understanding the capacitor's spatial configuration.

To find the inner radius \( r_1 \), we manipulate the capacitance formula:

• \( r_1 = \frac{Cr_2}{4\pi\varepsilon_0 r_2 + Cr_2/C} \)

This rearrangement allows calculation of the inner radius if the outer radius and capacitance are known.
In the given exercise, the outer radius \( r_2 \) is specified as 4.00 cm \((0.04 \mathrm{m})\). Substituting the values, we derive the inner radius \( r_1 \) to be about 3.5 cm. The accurate determination of these radii assists in applying the theoretical formulae accurately, ensuring a correct physical understanding of a spherical capacitor's behavior in real situations.