A ring of radius a carries a charge whose density varies directly proportional to the angular position between the points zero and 2π there is a very thin membrane of an insulator material separating these two points. Calculate the electric field vector at a point P on the symmetry axis perpendicular to the plane of the ring at a distance z from its center.

Solution

A square loop of side a carries a uniformly distributed charge Q. Determine the electric field vector at points on the line perpendicular to the plane of the loop at a distance z from its center.

Solution

A disk of radius a carries a uniformly distributed charge Q. Calculate the electric field vector:
a) At a point P on the axis of symmetry perpendicular to the plane of the disk at a distance z from its center;
b) In the case that the radius a of the disk is much greater than the distance from the point P to the disk, a>>z.

Solution with element of area obtained by geometry

Solution with element of area obtained by Jacobian

A hemispherical bowl of radius a carries a uniformly distributed charged Q. Calculate the electric field vector at a point P in the center of the hemisphere base.

Solution with element of area obtained by geometry

Solution with element of area obtained by Jacobian

A disk of radius a has at the center a hole of radius b and carries a uniformly distributed charge Q. Calculate the electric field vector at a point P on the symmetry axis perpendicular to the plane of the disk at a distance z from its center.

Solution with element of area obtained by geometry

Solution with element of area obtained by Jacobian

A semicircular plate has an outer radius of a and an inner radius of b. The plate carries a total charge Q distributed non uniformly, the charge is directly proportional to the central angle θ that a semicircle \( 0 \leq \theta \leq \pi \). Calculate the electric field vector at a point P on the axis of the semicircle perpendicular to the plane passing through the center of curvature at a distance z from its center.

Solution with element of area obtained by geometry

Solution with element of area obtained by Jacobian

A disk of radius a carries a charge whose density varies directly proportional to the radial position. Calculate the electric field vector at a point P on the symmetry axis perpendicular to the plane of the disk at a distance z of its center.

Solution with element of area obtained by geometry

Solution with element of area obtained by Jacobian

A wire is folded in a semicircular form of radius a, this wire carries a uniformly distributed charge Q. At point P, in the center of the semicircle, this distribution of charges produces an electric field of magnitude E₁.
If the wire is substituted by a point charge of the same value Q and at a distance a, equal to the radius of the semicircle, from the point P it produces at this point an electric field of magnitude E₂.
Calculate the ratio E₁/E₂, between the magnitudes of the electric fields produced by semicircular charges and by the point charge.

Solution