To determine the minimum force required to hold two identical non-conducting thin hemispherical shells together when they are charged, we need to analyze the electrostatic forces acting between them. The charge \( Q \) is uniformly distributed over the surfaces of the hemispheres, and we can use Coulomb's law to find the force between them.
Understanding the Setup
Imagine two hemispherical shells that, when placed together, form a complete sphere. Each shell has a radius \( R \) and carries a total charge \( Q \). Since the shells are non-conducting, the charge remains fixed on their surfaces. When they come into contact, they exert electrostatic forces on each other due to the charges.
Electrostatic Force Calculation
To find the force \( F \) required to hold the shells together, we can consider the interaction between the charges on the two hemispheres. The force between two point charges is given by Coulomb's law:
F = k \frac{Q_1 Q_2}{r^2}
In our case, since both hemispheres have the same charge \( Q \), we can simplify this to:
F = k \frac{Q^2}{r^2}
However, the distance \( r \) in this scenario is not simply the radius \( R \) of the hemispheres. Instead, we need to consider the effective distance between the charges when they are in contact. The effective distance for the force calculation is actually \( 2R \) because we are considering the interaction across the diameter of the sphere formed by the two hemispheres.
Adjusting for Geometry
When the two hemispheres are brought together, the effective distance between the charges can be approximated as \( R \) for the purpose of calculating the force acting on each hemisphere due to the other. Thus, we can express the force as:
F = k \frac{Q^2}{(2R)^2} = k \frac{Q^2}{4R^2}
Minimum Force Required
Now, this force \( F \) represents the electrostatic repulsion between the two hemispheres. To hold them together against this repulsive force, we need to apply an equal and opposite force. Therefore, the minimum force required to keep the hemispheres in contact is:
F = k \frac{Q^2}{4R^2}
Conclusion on the Options
From the options provided:
- F = kQ² / 4R²
- F = kQ² / 8R²
- F = kQ² / 16R²
- kQ² / 8R² > F > kQ² / 16R²
The correct answer is:
F = kQ² / 4R²
This indicates that the minimum force required to hold the two hemispherical shells together, given their uniform charge distribution, is directly proportional to the square of the charge and inversely proportional to the square of the radius of the shells.
