Concept #1: Ampere's Law with Calculus

Example #1: Magnetic Field Inside a Solenoid

Practice: A solid, cylindrical conductor carries a uniform current density, J. If the radius of the cylindrical conductor is R, what is the magnetic field at a distance 𝒓 from the center of the conductor when r < R? What about when r > R?

A hollow cylindrical conductor (inner radius = a, outer radius = b) carries a current I uniformly spread over its cross section. Which graph below correctly gives the magnetic field as a function of the distance r from the center of the cylinder?

A coaxial cable has two components: a thin wire at its center, carrying a current "forward", and a thin cylindrical shell surrounding the wire, carrying a current "backwards", with each current being the same magnitude. If the magnitude of the current was 1 A, and the cylindrical shell had a radius of 2 cm, what is the magnitude of the magnetic field at (a) some distance r < 2 cm from the inner wire and (b) some distance r > 2 cm from the inner wire?

A solid, cylinderical conductor of radius 1.5 cm carries a current density of 14 A/m 2. What is the magnetic field of this conductor at its surface?

Consider the toroid shown in the figure (inner radius 5cm, outer radius 10cm). What is the magnitude of the magnetic field inside a toroid of 1200 turns carrying a current 0.8 A at a distance 7cm from the center of the toroid? (Hint: Use Ampere's Law applied to a closed loop in the toroid mid-plane, as shown by a dash line)
A) 2.74 mT
B) 1.67 mT
C) 3.33 mT
D) 4.92 mT
E) 0.83 mT

A solid, cylinderical conductor of radius 1.5 cm carries a current density of 14 A/m 2. What is the magnetic field of this conductor at its surface?

Consider N parallel wires, each carrying a current i. If an Amperian loop was chosen that enclosed some of the wires, which of the following statements is true:
(a) The line integral ∫ Bdl would be independent of the number of wires enclosed
(b) The line integral ∫ Bdl would be independent of the radius of the Amperian loop (for a given number of wires enclosed)
(c) Halving the number of wires enclosed would reduce the line integral ∫ Bdl by 1/4
(d) The line integral ∫ Bdl is independent of the current in each wire i, only depending on the number of wires enclosed