Ch 29: The Magnetic Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 29: The Magnetic Field

Problem 16a

Chapter 29, Problem 16a

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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