Spherical Shell Electric Field: Problem Overview

This problem involves determining the radius of a thin spherical shell when its surface charge density, the electric field intensity at a certain distance from its center, and the medium (air) are known. We will use the fundamental principles of electrostatics, specifically the formula for the electric field produced by a uniformly charged spherical shell.

Given Data for Spherical Shell Calculation

Let's list the given parameters for the thin spherical shell:

• Surface charge density ($\sigma$) = \(88.54 \, \text{C/m}^2\)
• Electric field intensity (E) = \(5.625 \times 10^{12} \, \text{N/C}\)
• Distance from the center of the shell (r) = \(12 \, \text{mm}\)

It's crucial to work with consistent units, so we convert the distance from millimeters (mm) to meters (m):

• \(r = 12 \, \text{mm} = 12 \times 10^{-3} \, \text{m}\)

We also need the permittivity of free space (air medium), which is a standard physical constant:

• Permittivity of free space (\(\epsilon_0\)) = \(8.854 \times 10^{-12} \, \text{F/m}\)

Electric Field Formula for a Charged Sphere

For a thin spherical shell with uniform surface charge density \(\sigma\) and radius \(R\), the electric field intensity \(E\) at a point outside the shell at a distance \(r\) from its center (\(r > R\)) is given by the formula derived from Gauss's Law:

$$E = \frac{\sigma R^2}{\epsilon_0 r^2}$$

In this formula:

• \(E\) is the electric field intensity.
• \(\sigma\) is the surface charge density of the shell.
• \(R\) is the radius of the spherical shell (what we need to find).
• \(\epsilon_0\) is the permittivity of free space.
• \(r\) is the distance from the center of the shell to the point where the electric field is measured.

From this formula, we can rearrange it to solve for the radius \(R\):

$$R^2 = \frac{E \epsilon_0 r^2}{\sigma}$$

So, the radius \(R\) will be:

$$R = \sqrt{\frac{E \epsilon_0 r^2}{\sigma}}$$

Radius Calculation for Thin Spherical Shell

Now, let's substitute the given values into the derived formula for \(R\):

$$R = \sqrt{\frac{(5.625 \times 10^{12} \, \text{N/C}) \times (8.854 \times 10^{-12} \, \text{F/m}) \times (12 \times 10^{-3} \, \text{m})^2}{88.54 \, \text{C/m}^2}}$$

Let's perform the calculation step-by-step:

First, simplify the terms involving \(\epsilon_0\) and \(\sigma\):

$$\frac{8.854 \times 10^{-12}}{88.54} = \frac{1}{10} \times 10^{-12} = 0.1 \times 10^{-12} = 1 \times 10^{-13}$$

Next, calculate the square of the distance \(r\):

$$(12 \times 10^{-3})^2 = 144 \times (10^{-3})^2 = 144 \times 10^{-6} \, \text{m}^2$$

Now, substitute these simplified values back into the equation for \(R\):

$$R = \sqrt{(5.625 \times 10^{12}) \times (1 \times 10^{-13}) \times (144 \times 10^{-6})}$$

Combine the powers of 10:

$$10^{12} \times 10^{-13} \times 10^{-6} = 10^{(12 - 13 - 6)} = 10^{-7}$$

Multiply the numerical parts:

$$5.625 \times 1 \times 144 = 810$$

So, we have:

$$R = \sqrt{810 \times 10^{-7}}$$

To make the number under the square root easier to handle, let's adjust the exponent:

$$R = \sqrt{81 \times 10 \times 10^{-7}} = \sqrt{81 \times 10^{-6}}$$

Finally, take the square root:

$$R = \sqrt{81} \times \sqrt{10^{-6}}$$

$$R = 9 \times 10^{-3} \, \text{m}$$

Convert the radius back to millimeters:

$$R = 9 \times 10^{-3} \, \text{m} = 9 \, \text{mm}$$

Final Radius of the Spherical Shell

Based on our calculations, the radius of the thin spherical shell is \(9.0 \, \text{mm}\).

This result aligns with one of the provided options, confirming the accuracy of our calculation process for the spherical shell's radius.