Chapter 28 Physics

1. Overview of Magnetic Fields and Current

• In previous studies, compass needles were used to map magnetic fields, specifically those of permanent magnets.

• Electric current flowing through a wire generates a magnetic field, similar to how a compass reveals magnetic influence.

1. Biot-Savart Law

2.1 Basic Understanding

• In Chapter 27, it was established that magnetic fields have an effect on moving charges.

• This chapter focuses on how moving charges contribute to the generation of magnetic fields.

2.2 Formulation of Magnetic Fields

• The magnetic field produced by a differential current element (ids) can be described by: $dB = \frac{\mu_0}{4\pi}\cdot \frac{ids\sin\theta}{r^2}$

  • Here, $\mu_0$ represents the magnetic permeability of free space.

  • The current element (ids) creates a magnetic field whose direction must be taken into account, unlike in electric field calculations.

2.3 Biot-Savart Law Expression

• The Biot-Savart Law states that the differential magnetic field created by a current element is given by:

  $dB = \frac{\mu_0}{4\pi} \cdot \frac{ids \sin\theta}{r^2}$

• Numerical value of magnetic permeability, $\mu_0 = 4\pi \times 10^{-7} \text{ Tm/A}$.

1. Magnetic Field from a Long, Straight Wire

3.1 Calculation Methodology

• The magnetic field at any point due to the entire length of a straight wire can be determined by summing the contributions from all differential elements using the Biot-Savart Law.

• A simplification involves calculating the field due to just the right half of the wire and then multiplying it by two.

3.2 Expression for Magnetic Field (Magnitude)

• The equation for the magnitude of the magnetic field from a straight wire is: $B = \frac{\mu_0 i}{2\pi d}$

• Substituting for variables, the integration gives rise to:

  $B = \mu<em>0 i \int</em>0^{\infty} \frac{ds \sin \theta}{r^2}$

3.3 Integration Process

• To compute $B$, necessary relations involve:

  • The trigonometric definition of angles and lengths related to the geometry of the wire.

3.4 Direction of Magnetic Field

• The direction of the magnetic field can be determined with the right-hand rule, which states that if you point the thumb of your right hand in the direction of current, the curled fingers will point in the direction of the magnetic field.

1. Two Parallel Wires

4.1 Interaction of Current-Carrying Wires

• When two parallel wires carry currents, each wire experiences a force due to the magnetic field created by the other wire.

• The equation for the magnetic field produced by a wire at a distance (d) is:

  $B = \frac{\mu_0 i}{2\pi d}$

4.2 Magnitude of Magnetic Force

• The force $F$ on a length $L$ of wire 2 due to wire 1 is expressed as:

  $F = i<em>2 L B</em>1 = i<em>2 L \left(\frac{\mu</em>0 i_1}{2\pi d}\right)$

4.3 Newton’s Third Law

• Forces exerted between the wires are equal in magnitude and opposite in direction as stated by Newton’s Third Law.

1. Definition of the Ampere

5.1 SI Definition

• The ampere is defined as the constant current that maintains a force of $2 \times 10^{-7} \text{ N}$ per meter of length between two parallel conductors.

  • When $i<em>1 = i</em>2 = 1\text{ A}$ and the distance $d = 1\text{ m}$, this results in the defined force.

5.2 Formulation of Magnetic Permeability

• To solve for $\mu<em>0$, one utilizes the fundamental constants leading to: $\mu</em>0 = 4\pi \times 10^{-7} \text{ Tm/A}$.

1. Electromagnetic Rail Accelerator

6.1 Use Cases

• These accelerators are under study primarily for launching projectiles at high speeds and accelerating fusion fuel pellets.

• Example specifications:

  • Parallel rails of radius $r = 5.00 \text{ cm}$

  • Separation distance $d = 25.0 \text{ cm}$

  • Rail length $L = 5.00 \text{ m}$

  • Kinetic energy $K = 32.0 \text{ MJ}$

6.2 Problem Statement

• Determine the current required to accelerate a projectile with the outlined properties.

6.3 Calculations and Forces

• The total force from the magnetic fields acting on the conductor must integrate throughout the length between the rails.

1. Magnetic Fields due to Wire Loop

7.1 Magnetic Field at the Center

• For a circular loop with current, the magnitude of the magnetic field at the center is given by:

  $B = \frac{\mu_0 i}{2R}$

7.2 Magnetic Field along the Axis

• A different calculation applies if the magnetic field is being determined at points along the axis. Integration around the loop provides more complex relationships involving distances squared.

1. Ampere's Law

8.1 Overview

• Analogous to Gauss’s Law but for current-carrying distributions.

• Ampere's law states:

  $\oint B \cdot ds = \mu<em>0 i</em>{\text{enc}},$

  allowing simplifications with symmetrical current distributions.

8.2 Example: Long Straight Wire

• For a straight wire with uniform current, the magnetic field calculations yield:

  $B<em>{\text{inside}} = \frac{\mu</em>0 i}{2\pi R}$ for the field within the wire.

1. Magnetic Fields of Solenoids and Toroids

9.1 Solenoids

• Solenoids consist of many turns of wire, and the field within is uniform, as described by:

  $B = \mu_0 n i$

  where $n$ is the number of turns per unit length.

9.2 Toroids

• A toroidal structure provides a confined magnetic field inside while keeping the exterior zero. Employing Ampere's Law gives:

  $B = \frac{\mu_0 N i}{2\pi r}$.

1. Atoms and Magnetism

10.1 Atomic Structure

• Electrons in atoms create current loops contributing to magnetic fields, with orientation affecting overall magnetism.

10.2 Magnetic Properties of Matter

10.2.1 Diamagnetism

• Materials with negative magnetic susceptibility \chi_m < 0 oppose external magnetic fields.

10.2.2 Paramagnetism

• Positive magnetic susceptibility \chi_m > 0 indicates alignment under external fields.

10.2.3 Ferromagnetism

• Ferromagnetic materials maintain alignment after external fields are removed.

10.3 Temperature Dependence

• Heating can disrupt ferromagnetic alignment, with a critical temperature known as the Curie temperature.