A spherical capacitor contains a charge of $3.30$ nC when connected to a potential difference of $220$ V. If its plates are separated by vacuum and the inner radius of the outer shell is $4.00$ cm, calculate: (a) the capacitance; (b) the radius of the inner sphere; (c) the electric field just outside the surface of the inner sphere.

Verified step by step guidance

1. To find the capacitance of a spherical capacitor, use the formula for capacitance: $C=\frac{Q}{V}$, where $Q$ is the charge and $V$ is the potential difference. Substitute the given values: $Q=\; 3.30\; imes\; 10^\{-9\}\; \backslash (\backslash text\{\; C\}\backslash )$ and $V=\; 220\; \backslash (\backslash text\{\; V\}\backslash )$.
2. For part (b), use the formula for the capacitance of a spherical capacitor: , where $\mathrm{r\_1}$ is the radius of the inner sphere and $\mathrm{r\_2}$ is the radius of the outer shell. Rearrange the formula to solve for $\mathrm{r\_1}$.
3. Substitute the known values into the rearranged formula: $C$ from part (a), $\mathrm{r\_2}=\; 0.04\; \backslash (\backslash text\{\; m\}\backslash )$, and $\mathrm{\epsilon \_0}=\; 8.85\; imes\; 10^\{-12\}\; \backslash (\backslash text\{\; F/m\}\backslash )$.
4. For part (c), calculate the electric field just outside the surface of the inner sphere using the formula: . Substitute the values for $Q$, $\mathrm{\epsilon \_0}$, and $\mathrm{r\_1}$ from part (b).
5. Ensure all units are consistent, converting cm to m where necessary, and verify each calculation step to ensure accuracy in determining the capacitance, radius, and electric field.

Key Concepts

Capacitance of a Spherical Capacitor

Capacitance is a measure of a capacitor's ability to store charge per unit voltage. For a spherical capacitor, it depends on the radii of the inner and outer spheres and the permittivity of the medium between them. The formula for capacitance in a vacuum is C = 4πε₀(r₁r₂)/(r₂ - r₁), where r₁ and r₂ are the radii of the inner and outer spheres, respectively.

Electric Field in a Spherical Capacitor

The electric field in a spherical capacitor is determined by the charge on the inner sphere and the distance from the center. Just outside the surface of the inner sphere, the electric field E can be calculated using E = Q/(4πε₀r₁²), where Q is the charge and r₁ is the radius of the inner sphere. This field is radially outward and decreases with distance.

Potential Difference and Charge Relationship

The relationship between potential difference (voltage) and charge in a capacitor is given by Q = CV, where Q is the charge, C is the capacitance, and V is the voltage. This equation helps in determining the capacitance when the charge and voltage are known, and is fundamental in analyzing capacitors in circuits.