A non-conducting disk of radius 10cm lies in the x = 0 plane with its center at the origin. The disk is
uniformly charged and has a total charge +30ìC. Find Ex on the x-axis at (a) x = 2 cm, (b) x = 5cm, (c) x
= 10 cm, and (d) x = 20 cm.
To find the electric field (Ex) on the x-axis at different points, we can use the concept of electric field due to a uniformly charged disk. The electric field due to a uniformly charged disk on the axial line (x-axis) is given by the formula:
Ex = (σ / 2ε₀) * [1 - (z / √(z² + R²))]
Where:
Ex is the electric field on the x-axis,
σ is the charge density of the disk (charge per unit area),
ε₀ is the permittivity of free space,
z is the distance from the center of the disk to the point on x-axis, and
R is the radius of the disk.
In this case, the disk is uniformly charged and has a total charge of +30ìC. To find the charge density (σ), we divide the total charge by the area of the disk:
σ = Total charge / Area
= +30ìC / (π * R²)
= +30ìC / (π * (0.1m)²)
= +30ìC / (0.01π m²)
= +3000ìC / π m²
Now we can substitute the values of σ, ε₀, R, and z into the formula to find Ex at different x-axis points:
(a) x = 2 cm, z = 0.02 m
Ex = (+3000ìC / π m²) / (2 * ε₀) * [1 - (0.02 / √(0.02² + 0.1²))]
(b) x = 5 cm, z = 0.05 m
Ex = (+3000ìC / π m²) / (2 * ε₀) * [1 - (0.05 / √(0.05² + 0.1²))]
(c) x = 10 cm, z = 0.10 m
Ex = (+3000ìC / π m²) / (2 * ε₀) * [1 - (0.10 / √(0.10² + 0.1²))]
(d) x = 20 cm, z = 0.20 m
Ex = (+3000ìC / π m²) / (2 * ε₀) * [1 - (0.20 / √(0.20² + 0.1²))]
Note: Make sure to convert all units to meters for accurate calculations.