Question

A very long dielectric cylinder with radius a = 30mm is in vacuum and is charged with an electric charge whose volume density is 10⁻⁵C/m3. Required:

a) electric charge per unit length of the cylinder if the charge is uniformly distributed;
b) electric induction, electric field intensity, polarization vectors at a point located at a distance of 10 cm from the axis of the cylinder;
c) electric induction, electric field intensity, polarization vectors at a point located at a distance of 1cm from the axis of the cylinder;
d) the electric potential of the cylinder, if the reference potential is considered zero at a distance of 10m from the axis of the cylinder. The vacuum permittivity is 8.85*10⁻¹²F/m

Answer 1

a) The electric charge per unit length of the cylinder is λ = ρπa^2. b) At a distance of 10 cm from the axis of the cylinder, the electric induction is D = ελ/(2πR). c) At a distance of 1 cm from the axis of the cylinder, the electric field intensity and polarization vectors can be calculated using the same formulas as in part b. d) The electric potential of the cylinder can be found by integrating the electric field intensity equation.

Step-by-step explanation:

a) To find the electric charge per unit length of the cylinder, we need to calculate the total charge on the cylinder and divide it by the length of the cylinder. The volume of the cylinder is given by V = πa^2L, where a is the radius and L is the length. The total charge is given by Q = ρV, where ρ is the charge density. Therefore, the electric charge per unit length is λ = Q/L = ρπa^2.

b) At a distance of 10 cm from the axis of the cylinder, the electric induction is given by D = εE, where ε is the permittivity of the vacuum and E is the electric field intensity. The electric field intensity at a point outside a charged cylinder is given by E = λ/(2πεR), where R is the distance from the axis of the cylinder. The electric induction is then D = ελ/(2πR).

The electric field intensity and polarization vectors at this point can be found using the formula E = D - P, where P is the polarization vector. Since the cylinder is in vacuum, there is no polarization, so the electric field intensity and polarization vectors are the same.

c) At a distance of 1 cm from the axis of the cylinder, the electric field intensity and polarization vectors can be calculated using the same formulas as in part b.

d) To find the electric potential of the cylinder, we can use the formula V = -∫E·dl, where E is the electric field intensity and dl is an infinitesimal length element along the surface of the cylinder. The electric field intensity at a point outside a charged cylinder is given by E = λ/(2πεR), where R is the distance from the axis of the cylinder. The electric potential can then be found by integrating this equation.