To find the force \( F \) required to hold together the two hemispherical shells of a uniformly charged thin spherical shell, we need to consider the electric field produced by the charged shell and how it interacts with the charges on the hemispheres. Let's break this down step by step.

Understanding the Electric Field of a Charged Shell

A uniformly charged thin spherical shell creates an electric field outside the shell, but interestingly, inside the shell, the electric field is zero. This property is crucial for our calculations. The total charge \( Q \) on the spherical shell can be expressed as:

• \( Q = \sigma \cdot A \)
• where \( A \) is the surface area of the sphere, given by \( A = 4\pi R^2 \).

Thus, the total charge becomes:

\( Q = \sigma \cdot 4\pi R^2 \)

Electric Field Outside the Shell

For points outside the shell, the electric field \( E \) at a distance \( r \) from the center (where \( r > R \)) is given by:

\( E = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q}{r^2} \)

Substituting for \( Q \), we have:

\( E = \frac{1}{4\pi \epsilon_0} \cdot \frac{\sigma \cdot 4\pi R^2}{r^2} = \frac{\sigma R^2}{\epsilon_0 r^2} \)

Force on Each Hemispherical Shell

Now, consider the force acting on one of the hemispherical shells due to the electric field created by the other half. The electric field \( E \) at the surface of one hemisphere (at \( r = R \)) is:

\( E = \frac{\sigma R^2}{\epsilon_0 R^2} = \frac{\sigma}{\epsilon_0} \)

The force \( F \) acting on one hemisphere due to the electric field of the other hemisphere can be calculated using the formula:

\( F = E \cdot Q_{\text{hemisphere}} \)

Here, \( Q_{\text{hemisphere}} \) is the charge on one hemisphere, which is half of the total charge:

\( Q_{\text{hemisphere}} = \frac{Q}{2} = \frac{\sigma \cdot 4\pi R^2}{2} = 2\pi \sigma R^2 \)

Substituting this into the force equation gives:

\( F = \left(\frac{\sigma}{\epsilon_0}\right) \cdot \left(2\pi \sigma R^2\right) \)

Thus, simplifying this expression, we find:

\( F = \frac{2\pi \sigma^2 R^2}{\epsilon_0} \)

Final Result

In conclusion, the force \( F \) required to hold the two hemispherical shells together is:

\( F = \frac{2\pi \sigma^2 R^2}{\epsilon_0} \)

This equation shows how the force depends on the surface charge density \( \sigma \), the radius \( R \) of the shells, and the permittivity of free space \( \epsilon_0 \). Understanding this relationship is essential in electrostatics, particularly when dealing with charged objects and their interactions.