An infinitely long cylinder of radius 4.0 cm carries a uniform volume charge density p = 200 nC/m³. What is the electric field at r = 3.9 cm?

Question

Quizplus · Accepted Answer

Gauss's law states that the electric field at a point outside a charged cylinder is the same as if all the charge were concentrated on the axis of the cylinder. In this case, the electric field at r = 3.9 cm is the same as if all the charge were concentrated on the axis of the cylinder, which is a distance of 4.0 cm from the point. The electric field due to a point charge is given by E = kq/r^2, where k is Coulomb's constant, q is the charge, and r is the distance from the charge. In this case, the charge is Q = pV, where p is the volume charge density and V is the volume of the cylinder. The volume of the cylinder is V = πr^2h, where h is the height of the cylinder. In this case, the height of the cylinder is infinite, so the volume of the cylinder is infinite. Therefore, the charge on the cylinder is infinite. The electric field due to an infinite line charge is given by E = 2kλ/r, where λ is the linear charge density. In this case, the linear charge density is λ = pπr^2. Therefore, the electric field at r = 3.9 cm is E = 2kλ/r = 2k(pπr^2)/r = 2kρπr. Plugging in the values for k, ρ, and r, we get E = 2(9 × 10^9 N m^2/C^2)(200 × 10^-9 C/m^3)(π)(0.039 m) = 0.44 kN/C.

An Infinitely Long Cylinder of Radius 4