Ampere's Law (Calculus) Practice Problems

A long cylindrical aluminum rod of radius 1 mm is surrounded by a copper cylindrical shell of inner radius R₁ and outer radius R₂. The two conductors are separated by an electrical insulator. The rod and the shell carry equal and opposite currents of magnitude I that are distributed uniformly across their volumes. Determine the magnitude of the magnetic field at point A located at a distance r > R₂ from the axis of the rod.

A tightly wound solenoid coil of length 85.0 cm and radius 3.0 cm has 2500 turns. You are asked to generate a magnetic field of 0.34 T at the center of the solenoid coil. What must be the current I in the turns to generate the necessary magnetic field?

Consider a closed loop that surrounds a long coaxial cable consisting of cylinder-shaped conductors spaced apart by an insulator. The anticlockwise line integral ∲B•dl around this loop is 1.27 × 10^-2 T•m. Determine the value of the clockwise line integral ∲B•dl.

Calculate the line integral of B̂•dŝ between points A and B.

A current I ₀ flows uniformly through the cross-sectional area of an extended horizontal hollow cylindrical conductor with an inner radius R _a and an outer radius R _b. A point P is located at a distance r from the central axis and in a plane perpendicular to it. Determine the expression for the magnetic field for i) r < R _a, ii) R _a < r < R _b, and iii) r > R _b.

Determine the magnitude of the magnetic field at a distance z above an infinitely large horizontal plane. The current is flowing into the plane of the figure (positive y-direction). Hint: The linear current density J_w represents the current per unit width along the plane, measured in amperes per meter.

Given the line integral value of B•ds as 2.05 × 10 ^-5 T•m around the closed path in the provided diagram, determine the magnitude and direction (into or out of the figure) of the current I ₄ responsible for generating the magnetic field B.

A platinum wire that has a diameter of 8.0 mm carries an electric current. The magnitude of this current equals 60 A. Find out what the strength of the magnetic field will be at points located on (i) its outer boundary, (ii) 0.40 mm below the boundary, inside of the wire, and (iii) outside of the wire, at a perpendicular distance located from the outer surface of the wire which is equal to 16 mm.

In a laboratory, a wire is kept at an AC potential difference of 165 kV, transmitting a maximum power of 65 MW. Determine the maximum magnetic field produced at a perpendicular distance of 15 m from the wire. Afterward, compare it with the magnetic field at the earth's equator, which is equal to 50 μT.