A disc has charge distribution sigma and radius R . What is the Electric Field in N/C at a point P , located a distance x directly above the center of the disc?Write your answer in terms of sigma, R ,x , k, and pi (type "pi"). Indicate multiplication with a "*" and division with a "/".
The charge between radii r and r + dr on the disk is:
sigma 2 pi r dr
The contribution to the potential from this charge is:
sigma/(4 pi epsilon) 2 pi r dr/sqrt(r^2 + x^2)
Integrating over r from 0 to R gives:
V(x) = sigma/(2 epsilon) [sqrt(R^2 + x^2) - x]
By symmetry, the electric field only has a component in the x-direction, this is minus the derivative of the potential w.r.t. x:
E(x) = sigma/(2 epsilon) [1 - x/sqrt(R^2 + x^2)]
To find the electric field at point P above the center of the disc, we can use Gauss's law. Gauss's law tells us that the electric field at a point above the center of a disc with charge distribution sigma is given by:
E = (sigma / (2 * epsilon0)) * (1 - (x / sqrt(x^2 + R^2)))
where epsilon0 is the electric constant, also known as the permittivity of free space.
In terms of sigma, R, x, k (Coulomb's constant), and pi, the equation becomes:
E = (sigma / (2 * k * pi)) * (1 - (x / sqrt(x^2 + R^2)))
Remember to substitute the value of k as k = 1 / (4 * pi * epsilon0) when calculating the electric field.
Therefore, the electric field at point P is given by:
E = (sigma / (2 * k * pi)) * (1 - (x / sqrt(x^2 + R^2)))
To find the electric field at a point P located a distance x directly above the center of the disc with charge distribution sigma and radius R, we can use the electric field formula for a charged disk.
The electric field formula for a charged disk at a point on the axis perpendicular to the disk's plane is given by:
E = (sigma / (2 * epsilon0)) * (1 - (x / sqrt(x^2 + R^2))),
where
- E is the electric field in N/C,
- sigma is the charge distribution on the disk in Coulombs per square meter (C/m^2),
- epsilon0 is the electric constant (~ 8.85 x 10^-12 C^2/(N * m^2)),
- x is the distance from the center of the disk to point P directly above it in meters, and
- R is the radius of the disk in meters.
However, in the formula above, we need to replace epsilon0 with the Coulomb constant, k, which is given by k = 1 / (4 * pi * epsilon0).
Therefore, the electric field, E, at point P is:
E = (sigma / (2 * k)) * (1 - (x / sqrt(x^2 + R^2))).
Note that the value of pi in this formula is simply "pi".