A thin copper plate of diameter 11.0 cm is charged to 9.00 nC. What is the strength of the electric field 0.1 mm above the center of the top surface plate?
To find the strength of the electric field above the center of the top surface of the copper plate, we need to use the formula for the electric field due to a charged disk.
The formula for the electric field due to a uniform charged disk at a point above its center is given by:
E = (σ / (2 * ε₀)) * (1 - (z / √(z² + R²)))
Where:
E is the electric field strength,
σ is the surface charge density,
ε₀ is the permittivity of free space,
z is the vertical distance from the center of the disk, and
R is the radius of the disk.
Given:
Diameter of the plate = 11.0 cm.
So, the radius (R) = 11.0 cm / 2 = 5.5 cm = 0.055 m.
Charge on the plate = 9.00 nC = 9.00 * 10⁻⁹ C.
We need to find the electric field strength at a distance of 0.1 mm above the center of the top surface of the plate.
So, z = 0.1 mm = 0.1 * 10⁻³ m.
Let's substitute the values into the formula and calculate the electric field strength.
E = ((9.00 * 10⁻⁹ C) / (2 * (8.85 * 10⁻¹² F/m))) * (1 - (0.1 * 10⁻³ m / √((0.1 * 10⁻³ m)² + (0.055 m)²)))
Now we can solve the equation to find the electric field strength.
To find the electric field strength 0.1 mm above the center of the top surface of the charged copper plate, we can use the formula for the electric field of a uniformly charged circular plate.
The formula for the electric field strength above the center of a charged circular plate is given by:
E = (σ / (2ε₀))(1 - (d / √(R² + d²)))
where:
E is the electric field strength,
σ is the surface charge density,
ε₀ is the permittivity of free space,
d is the distance from the center of the plate to the point above it, and
R is the radius of the plate.
Given:
σ = 9.00 nC = 9.00 × 10⁻⁹ C (surface charge density)
d = 0.1 mm = 0.1 × 10⁻³ m (distance above the center of the plate)
R = diameter / 2 = 11.0 cm / 2 = 5.5 cm = 5.5 × 10⁻² m (radius)
Now, we can substitute the given values into the formula to find the electric field strength:
E = (9.00 × 10⁻⁹ C / (2 × 8.854 × 10⁻¹² C²/N·m²))(1 - (0.1 × 10⁻³ m / √((5.5 × 10⁻² m)² + (0.1 × 10⁻³ m)²)))
Simplifying the expression inside the square root:
E = (9.00 × 10⁻⁹ C / (2 × 8.854 × 10⁻¹² C²/N·m²))(1 - (0.1 × 10⁻³ m / √(3.025 × 10⁻⁴ m² + 1.00 × 10⁻⁶ m²)))
E = (9.00 × 10⁻⁹ C / (2 × 8.854 × 10⁻¹² C²/N·m²))(1 - (0.1 × 10⁻³ m / √(3.0251 × 10⁻⁴ m²)))
Calculating the value inside the square root:
E = (9.00 × 10⁻⁹ C / (2 × 8.854 × 10⁻¹² C²/N·m²))(1 - (0.1 × 10⁻³ m / 0.017373988 m))
E = (9.00 × 10⁻⁹ C / (2 × 8.854 × 10⁻¹² C²/N·m²))(1 - 0.005751903)
Evaluating the expression inside the parentheses:
E = (9.00 × 10⁻⁹ C / (2 × 8.854 × 10⁻¹² C²/N·m²))(0.994248097)
Evaluating the expression inside the brackets:
E = 9.00 × 10⁻⁹ C / (2 × 8.854 × 10⁻¹² C²/N·m²) × 0.994248097
Calculating the electric field strength:
E ≈ 5.09 × 10⁴ N/C
Therefore, the strength of the electric field 0.1 mm above the center of the top surface of the copper plate is approximately 5.09 × 10⁴ N/C.
Well, the charge distributes on both sides of the plate, so
E=Q/(area*2*epsilion)
area=PI(.11)^2, Q=9nC