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?

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Magnetic Field Produced by Moving Charges | 11 mins | 0 completed | Learn |

Magnetic Field Produced by Straight Currents | 29 mins | 0 completed | Learn Summary |

Magnetic Force Between Parallel Currents | 13 mins | 0 completed | Learn |

Magnetic Force Between Two Moving Charges | 9 mins | 0 completed | Learn |

Magnetic Field Produced by Loops and Solenoids | 43 mins | 0 completed | Learn Summary |

Toroidal Solenoids aka Toroids | 12 mins | 0 completed | Learn |

Biot-Savart Law with Calculus | 16 mins | 0 completed | Learn |

Ampere's Law with Calculus | 17 mins | 0 completed | Learn |

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Ampere's Law |

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?

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Concept #1: Ampere's Law with Calculus

Example #1: Magnetic Field Inside a Solenoid

Practice #1: Magnetic Field Inside Solid Cylindrical Conductor

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

To understand Ampère's law and its application. Ampère's law is often written Part A The integral on the left is a. the integral throughout the chosen volume b. the surface integral over the open surface. c. the surface integral over the closed surface bounded by the loop. d. the line integral along the closed lop e. the line integral tom stat to finishPart B The circle on the integral means that B(r) must be integrated a. over a circle or a sphere b. along any closed path that you choose. c. along the path of a closed physical conductor d. over the surface bounded by the current-carrying wire.

Ampere's Law involves integrating the magnetic field in a closed volume, on a closed area, or around a closed path?

Learning goal: To understand Ampere's law and its application.Ampere's law is often writtenPart (a) The integral on the left is:A) the integral throught the chosen volumeB) the surface integral over the open surfaceC) the surface integral over the closed surface bounded by the loop.D) the line integral along the closed loop.E) the line integral from strat to finish. Part (b) The circle on the integral means the B(r) must be integratedA) Over a circle or a sphereB) along any closed path you choose.C) along the path of a closed physical conductor.D) over the surface bounded by the current-carrying wire.

Learning Goal: To apply Ampère's law to find the magnetic field inside an infinite solenoid. In this problem we will apply Ampère's law, written to calculate the magnetic field inside a very long solenoid (only a relatively short segment of the solenoid is shown in the pictures). The segment of the solenoid shown in (Figure 1) has length L, diameter D, and n turns per unit length with each carrying current I. It is usual to assume that the component of the current along the z axis is negligible. (This may be assured by winding two layers of closely spaced wires that spiral in opposite directions.) From symmetry considerations it is possible to show that far from the ends of the solenoid, the magnetic field is axial.a) What is Iencl, the current passing through the chosen loop? Express your answer in terms of L (the length of the Amperean loop along the axis of the solenoid) and other variables given in the introduction.b) Find Bin, the z component of the magnetic field inside the solenoid where Ampere's law applies. Express your answer in terms of L, D, n, I, and physical constants such as μ0.

What quantity is represented by the symbol J? a. Resistivity b. Conductivity c. Current density d. Complex impedance e. Johnston’s constant

Magnetic Field inside a Very Long Solenoid Learning Goal: To apply Ampère's law to find the magnetic field inside an infinite solenoid. In this problem we will apply Ampère's law to calculate the magnetic field inside a very long solenoid (only a relatively short segment of the solenoid is shown in the pictures). The segment of the solenoid shown in (Figure 1) has length L, diameter D, and n turns per unit length with each carrying current I. It is usual to assume that the component of the current along the z axis is negligible. (This may be assured by winding two layers of closely spaced wires that spiral in opposite directions.) From symmetry considerations it is possible to show that far from the ends of the solenoid, the magnetic field is axial.Part DWhat is Iencl, the current passing through the chosen loop?Express your answer in terms of L (the length of the Ampèrean loop along the axis of the solenoid) and other variables given in the introduction.Part EFind Bin, the z component of the magnetic field inside the solenoid where Ampère's law applies.Express your answer in terms of L, D, n, I, and physical constants such as ?0.

An electric current is flowing through a long cylindrical conductor with radius a = 0.15 m. The current density J= 5.5 A/m2 uniform in the cylinder. In this problem, we consider an imaginary cylinder with radius r around the axis AB (a) When r is less than a, express the current inside the imaginary cylinder ir in terms of r and J.(b) Express the magnitude of the magnetic field B at r in terms of the current through the imaginary cylinder ir and its radius.

An electric current is flowing through a long cylindrical conductor with radius a = 0.15 m. The current density J= 5.5 A/m2 uniform in the cylinder. In this problem, we consider an imaginary cylinder with radius r around the axis AB (a) When r is greater than a, express the current inside the imaginary cylinder in terms of r, a, and J.(b) Express the magnitude of the magnetic field B at r > a in terms i and r.

Inside the toroid, in which direction does the magnetic field point?

An electric current is flowing through a long cylindrical conductor with radius a = 0.15 m. The current density J= 5.5 A/m2 is uniform in the cylinder. In this problem we consider an imaginary cylinder with radius r around the axis AB (a) Express B in terms of J, a, and r.(b) For r = 2a, calculate the numerical value of B in Tesla

An electric current is flowing through a long cylindrical conductor with radius a = 0.15 m. The current density J= 5.5 A/m2 uniform in the cylinder. In this problem, we consider an imaginary cylinder with radius r around the axis AB (a) Express B in terms of J and r.(b) For r = 0.5a, calculate the numerical value of B in Tesla

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