Chapter 12: Magnetic Fields

INTRODUCTION

  • Chapter 12: Sources of Magnetic Fields
      - 12.1 The Biot-Savart Law
      - 12.2 Magnetic Field Due to a Thin Straight Wire
      - 12.3 Magnetic Force between Two Parallel Currents
      - 12.4 Magnetic Field of a Current Loop
      - 12.5 Ampère’s Law
      - 12.6 Solenoids and Toroids
      - 12.7 Magnetism in Matter

  • In the previous chapter, the effect of a moving charged particle producing a magnetic field was discussed.

  • This interplay between electricity and magnetism is utilized in modern technology such as:
      - Computer hard drives
      - Telephones
      - Television
      - Computers
      - Internet

  • This chapter focuses on how magnetic fields are created by electric currents, using the Biot-Savart law as a fundamental tool.

  • Further discussions include the forces arising from current-carrying wires, magnetic loops, and principles underlying devices like solenoids and toroids.

  • Additionally, the chapter covers the behavior of magnetic materials under magnetic fields.

  • Figure 12.1 depicts an external hard drive, demonstrating the principle of magnetic encoding information for rapid storage and retrieval.

12.1 THE BIOT-SAVART LAW

Learning Objectives

  • Calculate the magnetic field from an arbitrary current in a line segment.

  • Apply the Biot-Savart law to specific geometries, such as a straight wire and a circular arc.

The Biot-Savart Law Overview

  • We acknowledge that mass creates a gravitational field, and charge creates an electric field.

  • Moving charge (current) also interacts with, and generates, a magnetic field.

  • Biot-Savart law is key for calculating the magnetic field produced by a current.

  • It is named after physicists Jean-Baptiste Biot and Félix Savart, who studied the interactions between currents and magnetic fields.

Biot-Savart Law Equation

  • The magnetic field B at a point P due to an element of wire carrying current I can be expressed as:
      extdextbfB=racextμ04extπracIextdextbflimesextbfrr3ext{d} extbf{B} = rac{ ext{μ}_0}{4 ext{π}} rac{I ext{d} extbf{l} imes extbf{r}}{r^3}
      - where:
        - μ0 = permeability of free space
        - d extbf{l} = infinitesimal section of the wire carrying current
        - r = distance vector from the wire to point P

Direction of Magnetic Field

  • The direction of dB can be found using the right-hand rule applied to the cross product.

  • The contribution from the wire element has no component parallel to r.

Example 12.1: Calculating Magnetic Fields of Short Current Segments

  • Consider a wire segment of 1.0 cm length carrying a current of 2.0 A vertically.

  • To find the magnetic field at point P, 1 meter from the wire in the x-direction, we apply the Biot-Savart law:

Strategy

  • The integral can be avoided given that the length of the segment is small compared to the distance x from point P. The distance remains constant across the wire segment.

Solution

  • Using the Biot-Savart law, compute dB and integrate over the length of the wire. The right-hand rule indicates the direction of B.

  • Significance: This approximation is valid if the segment length is small relative to distance P.

Example 12.2: Magnetic Field of a Circular Arc of Wire

  • For a wire carrying current I in a circular arc of radius R swept through an angle θ, the field at the center can be calculated using the Biot-Savart law.

Strategy

  • Since dl is perpendicular to the radius r, the contributions to the magnetic field sum directly.

Solution

  • The integral simplifies as contributions are in the same direction (out of the page). Directly use the integral of θ over the path to find B.

12.2 MAGNETIC FIELD DUE TO A THIN STRAIGHT WIRE

Learning Objectives

  • Use the Biot-Savart law to determine the magnetic field due to a thin, straight wire.

  • Establish the magnetic field’s dependence on distance and current flowing in the wire.

Exploring the Magnetic Field of a Thin Straight Wire

  • Surveyors notice that overhead lines create magnetic fields affecting compass readings.

  • Figure 12.5 shows a section of an infinitely long, straight wire carrying current I. What is the magnetic field at a distance R from the wire?

  • The magnetic field B at point P is derived as follows:

Contributions from Current Elements

  • The magnetic field at point P due to current element dl is evaluated using proper integration limits and geometry.

Integration and Evaluation

  • Substituting expressions for r and j into the Biot-Savart law, we evaluate the integral for the magnetic field from the wire.

Final Result

  • The magnetic field from an infinitely long, straight wire is given by:
      B=racextμ0I2extπRB = rac{ ext{μ}_0 I}{2 ext{π} R}

Magnetic Field Lines

  • Figure 12.6 illustrates that field lines are circular around the wire and identical in each perpendicular plane.

Example 12.3: Calculating Magnetic Field Due to Three Wires

  • A configuration of three wires carrying 2 A currents positioned at the corners of a square needs to determine the magnetic field at the fourth corner.

Solution Strategy

  1. Calculate contributions from each wire at the desired point.

  2. Resolve contributions into components before summing them appropriately.

12.3 MAGNETIC FORCE BETWEEN TWO PARALLEL CURRENTS

Learning Objectives

  • Explain how parallel wires can exert forces on each other.

  • Define the ampere and relate it to current-carrying wires.

Interplay Between Parallel Currents

  • Parallel wires carrying current generate forces due to their associated magnetic fields. Attractive or repulsive forces result depending on the direction of the currents.

Force Calculation

  • The force per unit length (F/l) between two long parallel conductors separated by distance r is given by:
      racFl=racextμ0I1I22extπrrac{F}{l} = rac{ ext{μ}_0 I_1 I_2}{2 ext{π} r}
      - F = magnetic force
      - j1 and j2 = currents in the wires' lengths

Practical Applications

  • This principle dictates behavior in high-power electrical systems and applications like circuit breakers.

Example 12.4: Forces on Wires

  • Two wires with currents of magnitude 5.0 mA, located accordingly, need to find magnetic forces between them.

Solution Strategy

  • Calculate the force per unit length using the proper distances and orientations of the currents.

12.4 MAGNETIC FIELD OF A CURRENT LOOP

Learning Objectives

  • Calculate the field due to a current in a loop of wire at specified points along the axis.

Magnetic Field from Current Loop

  • For a circular loop of radius R carrying current I, evaluate the field at point P along the axis of the loop.

Biot-Savart Law Application

  • Using the Biot-Savart law, determine the field contribution from the loop elements to point P.

Solution Derivation

  • Integrate to obtain the total magnetic field due to the complete loop at the specified point.

12.5 AMPÈRE’S LAW

Learning Objectives

  • Explain Ampère’s law relating magnetic fields to their sources.

Ampère’s Law Overview

  • Magnetic fields are not conservative, contrasting electric fields.

  • Ampère’s law relates the magnetic field produced by a current to the total current passing through any surface bounded by an integration path.

Formula Representation

  • ext{∮ B • dl = μ0 I{enclosed}

Essential Steps to Apply the Law

  1. Identify path symmetry.

  2. Choose an appropriate closed path for evaluating the line integral.

  3. Solve for the magnetic field.

12.6 SOLENOIDS AND TOROIDS

Learning Objectives

  • Relate the magnetic field of a solenoid/toroid to distance and current.

Solenoids

  • Long-wire coils in helix forms produce uniform magnetic fields.

Magnetic Field Calculation

  • B=μ0racNILB = μ_0 rac{N I}{L}

  • N: number of turns, L: length of the solenoid

Toroids

  • A donut-shaped coil containing wrapped wire with current generates magnetic fields.

Field Characteristics

  • In a tightly wound toroid, the magnetic field is primarily confined within the coil, with weaker external fields.

12.7 MAGNETISM IN MATTER

Learning Objectives

  • Classify magnetic materials (paramagnetic, diamagnetic, ferromagnetic) under external fields.

Types of Magnetic Materials

  • Paramagnetic materials align partially with fields; Diamagnetic materials create opposing fields; Ferromagnetic materials retain magnetization.

Microscopic Understanding

  • Atoms contribute magnetic dipoles subject to external fields, affecting their overall behavior.

Hysteresis and Susceptibility

  • Hysteresis describes how ferromagnetic materials behave when subjected to alternating magnetic fields, retaining magnetization post-field removal.

CHAPTER REVIEW

  • Includes key terms, equations, and summary statements.

  • Key Definitions: Ampère’s law, Biot-Savart law, magnetic domains, hysteresis, magnetic susceptibility, permeability of free space, solenoids, and toroids.

  • Key Equations:
    Bsolenoid=μ<em>0racnI2B_{solenoid} = μ<em>0 rac{n I}{2} B</em>toriod=racμ0NI2extπrB</em>{toriod} = rac{μ_0 N I}{2 ext{π} r}

  • Conceptual Insights: Behavior and definitions of magnetic materials related to external fields and current configurations.