An infinite sheet with a negative surface charge density runs parallel to the x axis. Another infinite sheet with an equal but opposite sign surface charge runs parallel to the y axis. The difference in electric potential from the point x = +5.2 m, y = +3.1 m to the point x = +3.1 m, y = +5.2 m is 8.6 x 106 V. What is the surface charge density in micro-coulomb/m2 for the sheet on the y axis?
To determine the surface charge density on the sheet along the y-axis, we can use the formula for electric potential difference (ΔV) between two points in an electric field:
ΔV = -Ed
Where:
- ΔV is the electric potential difference
- E is the electric field strength
- d is the distance between the two points
In this case, we are given that the electric potential difference is 8.6 x 10^6 V and the distance (d) between the two points is given by the Pythagorean theorem:
d = sqrt((x1 - x2)^2 + (y1 - y2)^2)
Using the given values:
x1 = +5.2 m
x2 = +3.1 m
y1 = +3.1 m
y2 = +5.2 m
Calculating the distance d:
d = sqrt((5.2 - 3.1)^2 + (3.1 - 5.2)^2)
d = sqrt(2.1^2 + (-2.1)^2)
d = sqrt(4.41 + 4.41)
d = sqrt(8.82)
d ≈ 2.97 m
Now we have the values for ΔV (8.6 x 10^6 V) and d (2.97 m). We can rearrange the formula to solve for the electric field strength (E):
E = -ΔV / d
E = -(8.6 x 10^6) / (2.97)
E ≈ -2.89 x 10^6 V/m
Given that the sheet along the y-axis has an equal but opposite sign surface charge density compared to the sheet along the x-axis, the electric field strength due to the sheet along the y-axis is equal in magnitude but opposite in direction to the electric field strength along the x-axis.
Now, we can relate the electric field strength to the surface charge density (σ) using the formula:
E = σ / (2ε₀)
Where:
- E is the electric field strength
- σ is the surface charge density
- ε₀ is the permittivity of free space (ε₀ ≈ 8.85 x 10^-12 C^2/N*m^2)
Rearranging the formula to solve for σ:
σ = E * (2ε₀)
Substituting the value for E (-2.89 x 10^6 V/m) and the value for ε₀ (8.85 x 10^-12 C^2/N*m^2):
σ = (-2.89 x 10^6) * (2 * 8.85 x 10^-12)
σ ≈ -51 x 10^-6 C/m^2
However, since we are asked for the surface charge density of the sheet on the y-axis and it is given that it has an equal but opposite sign to the sheet on the x-axis, the surface charge density of the y-axis sheet would be:
σ = 51 x 10^-6 C/m^2
Therefore, the surface charge density of the sheet on the y-axis is 51 micro-coulombs per square meter (μC/m²).