91Ó°ÊÓ

Surface charge density is a measure of how much electric charge is accumulated per unit area on the surface of a conductor. This concept is crucial in understanding how charges distribute themselves on surfaces and their effect on electric fields.To calculate surface charge density, represented by \( \sigma \), you need to know the total charge \( Q \) on the surface and the area \( A \) over which it is spread. The formula is:\[\sigma = \frac{Q}{A}\]The area \( A \) for a sphere is given by \( A = 4 \pi r^2 \), where \( r \) is the radius.In our exercise, after determining the charge on the sphere using the electric field information given, we use this formula to find out how much charge is spread per square meter of the sphere’s surface. This is essential to understand the amount of charge concentrated in a specific area.

An electric field is a region around a charged object where forces are exerted on other charges. The strength and direction of this field depend on the magnitude of the charge and the distance from the charge.The electric field \( E \) at a distance \( r \) from the center of a uniformly charged sphere is given by the formula:\[ E = \frac{Q}{4\pi\varepsilon_0 r^2} \]where:

• \( Q \) is the charge enclosed by the Gaussian surface,
• \( \varepsilon_0 \) is the permittivity of free space, approximately \(8.85 \times 10^{-12} \text{ C}^2/\text{Nm}^2\),
• \( r \) is the radial distance from the sphere's center.

Using this equation, the charge causing the given electric field strength can be determined. Knowing the electric field helps us understand the influence of the sphere's charge over a certain distance.

Capacitance refers to the ability of a system to store electric charge. It is a key concept in electrostatics, particularly when discussing conductors and insulators.The capacitance \( C \) of an isolated spherical conductor is given by:\[ C = 4 \pi \varepsilon_0 a \]where:

• \( a \) is the radius of the sphere
• \( \varepsilon_0 \) is the permittivity of free space.

This formula shows that the capacitance of a sphere depends only on its physical size and the properties of the surrounding space and not on the charge or potential.Finding the capacitance is crucial because it tells us the sphere’s potential to store charge. It plays a significant role in applications where charge storage and management are needed. In our exercise, we calculate the sphere's capacitance using the given radius.