A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter 19.0{\rm cm} , giving it a charge of -13.0\mu {\rm C}
Find the electric field just outside the paint layer.
Find the electric field 8.50{\rm cm} outside the surface of the paint layer.
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To find the electric field just outside the paint layer, you can use Gauss's law. Gauss's law states that the electric flux through any closed surface is equal to the total charge enclosed divided by the electric constant (ε₀).
1. Calculate the total charge enclosed:
The charge given to the plastic sphere is -13.0 μC. Since the paint is spread uniformly over the surface, the entire charge is enclosed by the paint layer.
2. Calculate the surface area of the paint layer:
The surface area of a sphere is given by the formula A = 4πr², where r is the radius of the sphere. Since the diameter of the sphere is given as 19.0 cm, the radius can be calculated as r = 19.0 cm / 2 = 9.50 cm.
3. Convert the radius to meters:
Since the electric constant (ε₀) is given in SI units, it is important to convert the radius to meters. The conversion factor from cm to m is 1 cm = 0.01 m.
4. Calculate the electric field just outside the paint layer using Gauss's law:
The electric flux (Φ) through a closed surface is given by Φ = E * A, where E is the electric field and A is the surface area of the closed surface. The electric flux is also equal to Q / ε₀, where Q is the total charge enclosed and ε₀ is the electric constant.
Setting these two equations equal to each other, we get:
E * A = Q / ε₀
Solving for E, we have:
E = (Q / ε₀) / A
Now we can substitute the values:
Q = -13.0 μC = -13.0 * 10^(-6) C
ε₀ = 8.854 * 10^(-12) C² / N·m²
A = 4πr² = 4π(9.50 cm * 0.01 m / cm)²
Calculating these values, we can find the electric field just outside the paint layer.
To find the electric field 8.50 cm outside the surface of the paint layer, we can use the principle of superposition. The electric field outside the paint layer can be calculated by adding the contributions from the charges in the paint layer and the remaining charge at the center of the sphere.
Since the paint is spread uniformly, the charge density on the surface is given by:
σ = Q / A
where σ is the charge density, Q is the total charge, and A is the surface area.
1. Calculate the charge density:
The total charge Q is given as -13.0 μC = -13.0 * 10^(-6) C.
The surface area of the paint layer A is the same as calculated above.
2. Calculate the electric field due to the charge in the paint layer:
The electric field due to a charged thin layer is given by the formula:
E = σ / (2ε₀)
where E is the electric field, σ is the charge density, and ε₀ is the electric constant.
3. Calculate the electric field due to the remaining charge at the center of the sphere:
The electric field due to a point charge at a distance r from the point charge is given by the formula:
E = kQ / r²
where E is the electric field, k is the Coulomb's constant (k = 1 / (4πε₀)), Q is the charge, and r is the distance from the point charge.
Setting up the equation for the electric field at a distance 8.50 cm from the center of the sphere and solving for E, we can find the electric field 8.50 cm outside the surface of the paint layer.
To find the electric field just outside the paint layer, we can use Gauss's Law. Here's how we can solve this:
1. Identify the Gaussian surface: In this case, we want to find the electric field just outside the paint layer. Since the paint is spread uniformly over the surface of the sphere, we can consider a Gaussian sphere with the same radius as the plastic sphere.
2. Apply Gauss's Law: Gauss's Law states that the electric flux through a closed surface equals the charge enclosed by that surface divided by the permittivity of the medium. Mathematically, it can be written as Φ = Q_enclosed / ε₀, where Φ is the electric flux, Q_enclosed is the charge enclosed, and ε₀ is the permittivity of free space.
3. Calculate the charge enclosed: The charge enclosed by our Gaussian sphere is the charge of the paint layer, which is given as -13.0 μC.
4. Calculate the electric field just outside the paint layer: Since the Gaussian sphere has the same radius as the plastic sphere, the electric field just outside the paint layer will be the same as the electric field due to the entire plastic sphere. The electric field due to a uniformly charged sphere at a point just outside its surface can be found using Coulomb's Law.
The electric field due to a uniformly charged sphere at a point just outside its surface is given by the equation:
E = k * Q / r², where E is the electric field, k is the electrostatic constant (k = 8.99 × 10^9 Nm²/C²), Q is the charge of the sphere, and r is the radius of the sphere.
5. Apply the formula to find the electric field just outside the paint layer:
E = (8.99 × 10^9 Nm²/C²) * (-13.0 μC) / (0.19 m)²
Now, to find the electric field 8.50 cm outside the surface of the paint layer, we'll use the same formula but with a larger radius.
1. Calculate the electric field 8.50 cm outside the surface of the paint layer:
E = (8.99 × 10^9 Nm²/C²) * (-13.0 μC) / (0.195 m)²
By substituting the values into the equations, you can calculate both electric fields.
total q=13microC
E=k*totalcharge/radius