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29. Sources of Magnetic Field

Magnetic Field Produced by Loops andSolenoids

Physics

29. Sources of Magnetic Field

Magnetic Field Produced by Loops andSolenoids

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Problem 16a

Textbook Question

A 100 A current circulates around a 2.0-mm-diameter superconducting ring. What is the ring's magnetic dipole moment?

Verified step by step guidance

Understand the problem: The magnetic dipole moment (μ) of a current-carrying loop is given by the formula:

μ = I A

, where

I

is the current and

A

is the area of the loop. We are tasked with calculating μ for a superconducting ring with a given current and diameter.

Determine the radius of the ring: The diameter of the ring is given as 2.0 mm. Convert this to meters (SI units) and divide by 2 to find the radius:

r = \frac{2.0}{2} \times 10^{- 3} = 1.0 \times 10^{- 3} m

Calculate the area of the ring: The area of a circular loop is given by

A = π r^{2}

. Substitute the radius value into this formula to find the area.

Substitute the values into the magnetic dipole moment formula: Use the formula

μ = I A

, where

I = 100 A

and

A

is the area calculated in the previous step.

Simplify the expression to find the magnetic dipole moment: Perform the multiplication of the current and the area to determine the final value of

μ

. Ensure the units are in

A m 2

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that represents the strength and orientation of a magnetic source. For a current loop, it is calculated as the product of the current flowing through the loop and the area of the loop. It is expressed in units of ampere-square meters (A·m²) and is crucial for understanding how the loop interacts with external magnetic fields.

Current and Its Relation to Magnetic Fields

Electric current, measured in amperes (A), is the flow of electric charge through a conductor. When current flows through a loop or coil, it generates a magnetic field around it, which is fundamental to electromagnetism. The direction of the magnetic field is determined by the right-hand rule, which helps visualize the orientation of the magnetic dipole moment.

Area of a Circle

The area of a circle is calculated using the formula A = πr², where r is the radius of the circle. In the context of a superconducting ring, knowing the diameter allows us to find the radius, which is essential for calculating the magnetic dipole moment. The area directly influences the strength of the magnetic dipole moment when multiplied by the current.

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Related Practice

Textbook Question

A 15.0 cm long solenoid with radius 0.750 cm is closely wound with 600 turns of wire. The current in the windings is 8.00 A. Compute the magnetic field at a point near the center of the solenoid.

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Textbook Question

FIGURE CP29.79 is an edge view of a 2.0 kg square loop, 2.5 m on each side, with its lower edge resting on a frictionless, horizontal surface. A 25 A current is flowing around the loop in the direction shown. What is the strength of a uniform, horizontal magnetic field for which the loop is in static equilibrium at the angle shown?

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Textbook Question

Determine the field strength at the center of a current-carrying square loop having sides of length 2R.

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Textbook Question

A long, straight conducting wire of radius R has a nonuniform current density J = J₀r/R, where J₀ is a constant. The wire carries total current I. Find an expression for the magnetic field strength inside the wire at radius r.

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Textbook Question

A 100 A current circulates around a 2.0-mm-diameter superconducting ring. What is the on-axis magnetic field strength 5.0 cm from the ring?

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Textbook Question

Each turn of a solenoid is a current loop with a magnetic dipole moment. Consider a 200-turn cylindrical solenoid that has an interior volume of 40 cm³ and for which each turn is a magnetic dipole moment with magnitude 8.0 x 10^-4 A m². What is the magnetic field strength inside the solenoid?

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Textbook Question

The earth's magnetic field, with a magnetic dipole moment of 8.0 x 10²² A m², is generated by currents within the molten iron of the earth's outer core. Suppose we model the core current as a 3000-km-diameter current loop made from a 1000-km-diameter 'wire.' The loop diameter is measured from the centers of this very fat wire. What is the current density J in the current loop?

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Textbook Question

The heart produces a weak magnetic field that can be used to diagnose certain heart problems. It is a dipole field produced by a current loop in the outer layers of the heart. What is the magnitude of the heart's magnetic dipole moment?

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