A circular disk of 10 cm radius is charged uniformly with a total charge of Q coul. Find the electric field intensity at a point 20 cm away from the disk, along its axis.
(b) Consider a uniformly charged right circular cylindrical shell having total charge Q, radius R, and height h. Determine the electric field at a point a distance d from the right side of the cylinder, as shown in Fig. 3. (Hint: treat the cylinder as a collection of ring charges.)
(c) Consider now a solid cylinder with the same dimensions and carrying the same charge, which is uniformly distributed through its volume. Find the field it creates at the same point. (Hint: treat the cylinder as a collection of disk charges.)
To find the electric field intensity at a point on the axis of a charged disk, we can use Gauss's Law. Gauss's Law states that the electric field through a closed surface is directly proportional to the total charge enclosed by that surface.
(a) Electric Field Intensity at a point on the axis of a charged disk:
1. Consider a circular surface with a radius of r and centered on the disk.
2. The electric field will be uniform and perpendicular to the surface at every point due to the symmetry of the disk.
3. Applying Gauss's Law, the electric field through this circular surface (E_d) can be calculated as follows:
- The total charge enclosed by the surface, Q_enclosed, is the charge of the entire disk, Q.
- The area of the circular surface, A, is πr^2.
- Gauss's Law states that E_d * A = Q_enclosed / ε0, where ε0 is the permittivity of free space.
4. Rearranging the equation, we get E_d = Q / (πr^2 * ε0).
5. Substituting the given values of Q = Q coul, r = 20 cm = 0.2 m, and ε0 = 8.85 × 10^(-12) C^2/(N·m^2), we can calculate E_d.
(b) Electric Field Intensity at a point near the right side of a uniformly charged cylindrical shell:
1. Consider the cylindrical shell as a collection of ring charges.
2. Divide the shell into infinitely thin concentric rings of radius r.
3. Each ring will have a charge dQ = (Q / (2πR)) * 2πr * h, where R is the radius of the cylinder and h is its height.
4. Now, we can calculate the electric field at a point d distance away from the right side of the cylinder.
5. The electric field intensity due to each small ring, dE, can be calculated using Coulomb's Law: dE = (k * dQ) / (r^2), where k is the electrostatic constant.
6. The total electric field intensity, E, at the given point will be the sum of all the contributions from each small ring, integrated over the entire cylinder.
- Integrate dE from r = 0 to r = R, and then integrate from z = 0 to z = h.
- The integrals give the electric field intensity at the desired point.
(c) Electric Field Intensity at a point near the right side of a solid cylinder:
1. Consider the solid cylinder as a collection of infinitely thin disk charges.
2. Divide the cylinder into infinitely thin disks of radius r and thickness dz.
- Each disk will have a charge dQ = (Q / (πR^2)) * πr^2 * dz, where R is the radius of the cylinder and dz is the thickness of each disk.
3. Calculate the electric field intensity due to each small disk, dE, using Coulomb's Law: dE = (k * dQ) / (z^2 + r^2), where k is the electrostatic constant.
4. The total electric field intensity, E, at the given point will be the sum of all the contributions from each small disk, integrated over the entire cylinder.
- Integrate dE from z = 0 to z = h, and then integrate from r = 0 to r = R.
- The integrals give the electric field intensity at the desired point.