To find the electric field at the center of the disc, we can use the principle of superposition. The electric field at the center of the disc can be thought of as the sum of the electric fields due to all the infinitesimally small charge elements on the disc. Each of these charge elements contributes to the electric field, and we can integrate their contributions to find the total electric field.
Let's consider an infinitesimally small charge element, dQ, on the disc. The electric field due to this element at the center of the disc can be calculated using Coulomb's law:
dE = (1 / (4πε₀)) * (dQ / r²)
Where:
dE is the electric field due to the infinitesimal charge element dQ.
ε₀ is the permittivity of free space.
r is the distance from dQ to the center of the disc.
Since we're interested in the electric field at the center of the disc, r is simply the radius of the disc, R.
Now, we need to find the total electric field by integrating over all the charge elements on the disc:
E_total = ∫(dE) = (1 / (4πε₀)) * ∫(dQ / R²)
The surface charge density σ is defined as the charge per unit area:
σ = Q / (πR²)
Where Q is the total charge on the disc. We can rearrange this equation to solve for Q:
Q = σ * πR²
Now, substitute this expression for Q into the integral for E_total:
E_total = (1 / (4πε₀)) * ∫((σ * πR²) / R²) dA
Here, dA represents the infinitesimal area element on the disc. We're integrating over the entire disc, so we don't need to worry about the angle or direction. We can pull out the constants:
E_total = (σ / (4ε₀)) * πR² * ∫(dA / R²)
Now, the integral of dA over the entire disc is just the total area of the disc, which is πR²:
E_total = (σ / (4ε₀)) * πR² * (πR² / R²)
E_total = (σ / (4ε₀)) * πR² * π
E_total = (σπ² / (4ε₀))
Now, we have the electric field at the center of the disc, E_total:
E_total = (σπ² / (4ε₀))
Now, let's find the electric field along the axis at a distance R from the center of the disc. This is a standard result and can be found using the formula for the electric field due to a charged ring:
E_axis = (1 / (4πε₀)) * (2πσR) / (R² + z²)^(3/2)
Here, z is the distance along the axis from the center of the disc.
In this case, z = R, so:
E_axis = (1 / (4πε₀)) * (2πσR) / (R² + R²)^(3/2)
E_axis = (σ / (2ε₀)) * (1 / (2R))
E_axis = (σ / (4ε₀R))
Now, let's compare the electric field at the center of the disc (E_total) with the electric field along the axis at a distance R from the center of the disc (E_axis):
(E_total / E_axis) = ((σπ² / (4ε₀)) / (σ / (4ε₀R)))
(E_total / E_axis) = (σπ² / (4ε₀)) * ((4ε₀R) / σ)
(E_total / E_axis) = π²R
Now, we can calculate the percentage reduction:
Percentage reduction = [(E_total - E_axis) / E_total] * 100
Percentage reduction = [(π²R - σ / (4ε₀R)) / (σπ² / (4ε₀))] * 100
Percentage reduction = [(π²R - σ / (4ε₀R)) / (σπ² / (4ε₀))] * 100
Percentage reduction = [(π²R - 1 / (4R)) / (π²)] * 100
Now, plug in the value of π² ≈ 9.87:
Percentage reduction = [(9.87R - 1 / (4R)) / 9.87] * 100
Now, let's calculate the percentage reduction for R = 1 (since we have no specific value for R):
Percentage reduction ≈ [(9.87 * 1 - 1 / (4 * 1)) / 9.87] * 100
Percentage reduction ≈ [(9.87 - 0.25) / 9.87] * 100
Percentage reduction ≈ (9.62 / 9.87) * 100
Percentage reduction ≈ 97.47%
So, the electric field along the axis at a distance R from the center of the disc reduces by approximately 97.47% compared to the electric field at the center of the disc.
Therefore, the correct option is:
A. Reduces by 70.7%.
