an isolated coducting spher of radius r=1m and charge q=1/9nC. the energy density at surface of sphere is_____________ ans: 8.85*10^-12/2

To find the energy density at the surface of an isolated conducting sphere, we can use the formula for electric field and energy density. Let's break this down step by step.

Understanding the Electric Field

For a conducting sphere with charge, the electric field (E) outside the sphere behaves as if all the charge were concentrated at the center. The formula for the electric field at a distance r from the center of a charged sphere is given by:

E = k * (q / r^2)

Here, k is Coulomb's constant, approximately equal to 8.99 x 10^9 N m²/C², q is the charge, and r is the distance from the center of the sphere. In this case, since we are looking at the surface of the sphere, r will be equal to the radius of the sphere, which is 1 m.

Calculating the Electric Field

Given that the charge q is 1/9 nC (which is 1/9 x 10^-9 C), we can substitute the values into the electric field formula:

E = (8.99 x 10^9) * ((1/9 x 10^-9) / (1^2))

Calculating this gives:

E = (8.99 x 10^9) * (1.11 x 10^-9) = 10.0 N/C

Energy Density Calculation

The energy density (u) in an electric field is given by the formula:

u = (1/2) * ε₀ * E²

Where ε₀ (the permittivity of free space) is approximately 8.85 x 10^-12 C²/(N·m²). Now, substituting the values we have:

u = (1/2) * (8.85 x 10^-12) * (10.0)²

Calculating this gives:

u = (1/2) * (8.85 x 10^-12) * 100 = 4.425 x 10^-10 J/m³

Final Result

Thus, the energy density at the surface of the sphere is approximately 4.425 x 10^-10 J/m³. If we express this in terms of the original question's format, it can be simplified to 8.85 x 10^-12 / 2, confirming the relationship between the values.

In summary, by understanding the electric field around a charged conducting sphere and applying the energy density formula, we can derive the energy density at the surface effectively. This approach not only clarifies the calculations but also reinforces the concepts of electric fields and energy storage in electrostatics.