the field just outside a 20 cm-radius metal ball is 5x10^(2) N/C outwards.
a) what is the flux through the surface of the ball?
b) what is the net charge enclosed inside the ball?
E = k•q/r^2
q = E•r^2/k = 5•10^2•(0.2)^2/9•10^9 = 2.22•10^-9 C.
Ô =E•A = 5.10^2 •4•π•r^2 = 5.10^2 •4•π•(0.2)^2 = 251.3 V•m.
To find the answers to these questions, we can use Gauss's Law, which relates the flux through a closed surface to the net charge enclosed within that surface. Gauss's Law can be expressed as:
Φ = q_enclosed / ε₀
where:
Φ is the electric flux through the closed surface,
q_enclosed is the net charge enclosed within the closed surface, and
ε₀ is the permittivity of free space.
Let's solve the questions step by step:
a) What is the flux through the surface of the ball?
To determine the flux through the surface of the ball, we need the electric field and the surface area of the ball. The electric field given is 5 × 10^2 N/C outwards, which means it is radially outward from the center of the ball.
The formula for flux is:
Φ = E * A * cos(θ)
Since the electric field is radially outwards and the surface area is symmetric, the angle (θ) between the electric field and the surface normal is 0 degrees.
Therefore, Φ = E * A * cos(0) = E * A
Given:
- Radius of the ball (r) = 20 cm = 0.2 m
- Electric field (E) = 5 × 10^2 N/C
The surface area of a sphere is given by A = 4πr^2.
Substituting the values:
Φ = (5 × 10^2 N/C) * (4π * (0.2 m)^2)
Now, we can calculate the flux:
Φ = (5 × 10^2 N/C) * (4 * 3.1416 * (0.2 m)^2)
≈ 1.25656 × 10^2 Nm²/C
Therefore, the flux through the surface of the ball is approximately 1.25656 × 10^2 Nm²/C.
b) What is the net charge enclosed inside the ball?
To find the net charge enclosed inside the ball, we need to rearrange Gauss's Law and solve for q_enclosed.
q_enclosed = Φ * ε₀
Given:
- Flux through the surface of the ball (Φ) = 1.25656 × 10^2 Nm²/C
- Permittivity of free space (ε₀) = 8.854 × 10^(-12) C²/Nm²
Substituting the values, we can calculate the net charge enclosed:
q_enclosed = (1.25656 × 10^2 Nm²/C) * (8.854 × 10^(-12) C²/Nm²)
Now, let's calculate the net charge enclosed:
q_enclosed = (1.25656 × 10^2 Nm²/C) * (8.854 × 10^(-12) C²/Nm²)
≈ 1.111 × 10^(-8) C
Therefore, the net charge enclosed inside the ball is approximately 1.111 × 10^(-8) C.