Consider a thin spherical shell of radius 16.0 cm with a total charge of +28.0 µC distributed uniformly on its surface. (Take radially outward as the positive direction.)
Find the electric field 10.0 cm from the center of the charge distribution.
jko
To find the electric field at a point 10.0 cm from the center of the charge distribution, we can use Gauss's law. Gauss's law states that the electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space (ε0).
1. Determine the total charge enclosed by a Gaussian surface of radius 10.0 cm. Since the charge distribution is uniform and the shell is thin, the entire charge (+28.0 µC) is enclosed.
2. Calculate the electric flux through the Gaussian surface. The electric flux through a Gaussian surface of radius r is given by Φ = qenclosed/ε0, where qenclosed is the charge enclosed and ε0 is the permittivity of free space (8.85 x 10^-12 C^2/N*m^2).
Φ = (28.0 x 10^-6 C) / (8.85 x 10^-12 C^2/N*m^2)
= 3.16 x 10^5 N*m^2/C
3. Calculate the area of the Gaussian surface. The area of a spherical Gaussian surface is given by A = 4πr^2, where r is the radius of the Gaussian surface.
A = 4π(0.1 m)^2
= 0.040 π m^2
4. Calculate the electric field. The electric field is given by E = Φ / A.
E = (3.16 x 10^5 N*m^2/C) / (0.040 π m^2)
≈ 2.52 x 10^6 N/C
Therefore, the electric field at a point 10.0 cm from the center of the charge distribution is approximately 2.52 x 10^6 N/C, directed radially outward.
To find the electric field at a point 10.0 cm from the center of the charge distribution, we can use the principle of superposition.
The electric field due to a uniformly charged spherical shell at a point outside the shell is the same as if all the charge of the shell were concentrated at its center.
Here's how we can calculate it step by step:
1. First, we calculate the electric field due to a point charge. The electric field due to a point charge Q at a distance r from it is given by the equation:
E = k * Q / r^2
Where E is the electric field magnitude, Q is the charge, r is the distance from the charge, and k is the electrostatic constant (k ≈ 8.988 × 10^9 N m^2/C^2).
2. In our case, we have a uniformly charged spherical shell. The total charge of the shell is +28.0 µC.
To find the electric field at a point 10.0 cm from the center, we consider that all the charge is concentrated at the center.
3. We convert the radius of the shell from centimeters to meters:
r = 16.0 cm = 0.16 m
4. Now we can calculate the electric field due to the point charge at the center. Since all the charge is concentrated at the center, we can use the equation:
E(total) = k * Q / r^2
Plugging in the values:
E(total) = (8.988 × 10^9 N m^2/C^2) * (28.0 × 10^(-6) C) / (0.16 m)^2
5. Simplifying the equation:
E(total) = (8.988 × 10^9 N m^2/C^2) * (28.0 × 10^(-6) C) / (0.16)^2 m^2
E(total) = 3.138 × 10^4 N/C
6. This gives us the electric field due to the uniformly charged spherical shell at its center. But we are interested in finding the electric field 10.0 cm from the center.
7. Since the electric field due to a uniformly charged spherical shell is the same as if all the charge were concentrated at the center, the electric field 10.0 cm from the center will be the same as the electric field at the center.
Therefore, the electric field 10.0 cm from the center is:
E = 3.138 × 10^4 N/C.
So, the electric field 10.0 cm from the center of the charge distribution is approximately 3.138 × 10^4 N/C, radially outward from the center.