Question

A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter 15.0 cm, giving it a charge of -39.0 μC. What is the charge density on the surface of the sphere?

Answer 1

The charge density on the surface of the sphere is calculated by dividing the total charge by the surface area of the sphere. Using the formula for surface area of a sphere, we find the charge density to be approximately -549.3 μC/m².

Step-by-step explanation:

Charge Density on a Spherical Surface

The student is tasked with finding the charge density on the surface of a sphere. To calculate this, we need to use the formula for surface charge density, which is the charge divided by the surface area. Given the diameter of the sphere as 15.0 cm, we first find the radius (r = diameter / 2) which is 7.5 cm or 0.075 m. The surface area (A) of a sphere is given by the formula A = 4πr2. The charge density (σ) can be found using the formula σ = Q / A, where Q is the total charge.

Now, let's plug in the numbers:

1. Calculate the radius of the sphere: r = 7.5 cm = 0.075 m.
2. Calculate the surface area of the sphere: A = 4π(0.075 m)2.
3. Calculate the charge density: σ = -39.0 μC / A.

Substituting the values in, we get:

• A = 4π(0.075 m)2 = 4π(0.005625 m2) ≈ 0.071 m2.
• σ = -39.0 μC / 0.071 m2 ≈ -549.3 μC/m2.

Thus, the charge density on the surface of the sphere is approximately -549.3 μC/m2.