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 39a

Chapter 29, Problem 39a

What is the magnitude of the torque on the current loop in FIGURE EX29.39?

Verified step by step guidance

Step 1: Identify the formula for torque on a current loop. The torque (τ) on a current loop in a magnetic field is given by τ = μ × B, where μ is the magnetic moment of the loop and B is the magnetic field.

Step 2: Calculate the magnetic moment (μ) of the loop. The magnetic moment is given by μ = I × A, where I is the current in the loop and A is the area of the loop. Use the dimensions provided in the figure to calculate the area (A = length × width).

Step 3: Determine the magnetic field (B) at the location of the loop. The magnetic field due to a long straight wire carrying current (I) is given by B = (μ₀ × I) / (2π × r), where μ₀ is the permeability of free space, I is the current in the wire, and r is the distance from the wire to the loop.

Step 4: Compute the torque (τ) using the values of μ and B. Ensure that the direction of the magnetic moment and the magnetic field are considered to calculate the cross product μ × B.

Step 5: Verify the units and ensure consistency in calculations. The torque should be expressed in Newton-meters (N·m).

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Key Concepts

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

Torque

Torque is a measure of the rotational force applied to an object, calculated as the product of the force and the distance from the pivot point (lever arm). It is a vector quantity, meaning it has both magnitude and direction, and is crucial in understanding how forces cause objects to rotate. The formula for torque (τ) is τ = r × F, where r is the distance from the pivot to the point of force application, and F is the force applied.

Magnetic Field

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. It is represented by magnetic field lines that indicate the direction and strength of the field. In the context of current-carrying loops, the magnetic field interacts with the current to produce torque, influencing the loop's orientation in the field.

Current Loop

A current loop is a closed loop of wire through which electric current flows. When placed in a magnetic field, the interaction between the magnetic field and the current generates a torque that can cause the loop to rotate. The magnitude of the torque depends on the current flowing through the loop, the area of the loop, and the strength of the magnetic field, making it essential for applications in electric motors and generators.

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Force and Torque on Current Loops

Related Practice

Textbook Question

A small bar magnet experiences a 0.020 N m torque when the axis of the magnet is at 45° to a 0.10 T magnetic field. What is the magnitude of its magnetic dipole moment?

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Textbook Question

What is the loop's equilibrium orientation?

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Textbook Question

A long wire carrying a 5.0 A current perpendicular to the xy-plane intersects the x-axis at x = -2.0 cm. A second, parallel wire carrying a 3.0 A current intersects the x-axis at x = +2.0 cm. At what point or points on the x-axis is the magnetic field zero if (a) the two currents are in the same direction and (b) the two currents are in opposite directions?

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Textbook Question

The two 10-cm-long parallel wires in FIGURE EX29.33 are separated by 5.0 mm. For what value of the resistor R will the force between the two wires be 5.4 x 10⁻⁵ N?

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Textbook Question

FIGURE EX29.37 is a cross section through three long wires with linear mass density 50 g/m. They each carry equal currents in the directions shown. The lower two wires are 4.0 cm apart and are attached to a table. What current I will allow the upper wire to 'float' so as to form an equilateral triangle with the lower wires?

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Textbook Question

The Hall voltage across a conductor in a 55 mT magnetic field is 1.9 μV. When used with the same current in a different magnetic field, the voltage across the conductor is 2.8 μV. What is the strength of the second field?

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