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30 - Sources of the Magnetic Field

# 30 - Sources of the Magnetic Field - Chapter 30 Sources of...

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Chapter 30 Sources of the Magnetic Field CHAPTE R OUTLI N E 30.1 The Biot–Savart Law 30.2 The Magnetic Force Between Two Parallel Conductors 30.3 Ampère’s Law 30.4 The Magnetic Field of a Solenoid 30.5 Magnetic Flux 30.6 Gauss’s Law in Magnetism 30.7 Displacement Current and the General Form of Ampère’s Law 30.8 Magnetism in Matter 30.9 The Magnetic Field of the Earth 926 A proposed method for launching future payloads into space is the use of rail guns, in which projectiles are accelerated by means of magnetic forces. This photo shows the firing of a projectile at a speed of over 3 km/s from an experimental rail gun at Sandia National Research Laboratories, Albuquerque, New Mexico. (Defense Threat Reduction Agency [DTRA])

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I n the preceding chapter, we discussed the magnetic force exerted on a charged particle moving in a magnetic field. To complete the description of the magnetic interaction, this chapter explores the origin of the magnetic field—moving charges. We begin by showing how to use the law of Biot and Savart to calculate the magnetic field produced at some point in space by a small current element. Using this formalism and the principle of superposition, we then calculate the total magnetic field due to various current distributions. Next, we show how to determine the force between two current-carrying conductors, which leads to the definition of the ampere. We also introduce Ampère’s law, which is useful in calculating the magnetic field of a highly symmetric configuration carrying a steady current. This chapter is also concerned with the complex processes that occur in magnetic materials. All magnetic effects in matter can be explained on the basis of atomic magnetic moments, which arise both from the orbital motion of electrons and from an intrinsic property of electrons known as spin. 30.1 The Biot–Savart Law Shortly after Oersted’s discovery in 1819 that a compass needle is deflected by a current-carrying conductor, Jean-Baptiste Biot (1774–1862) and Félix Savart (1791–1841) performed quantitative experiments on the force exerted by an electric current on a nearby magnet. From their experimental results, Biot and Savart arrived at a mathematical expression that gives the magnetic field at some point in space in terms of the current that produces the field. That expression is based on the following experimental observations for the magnetic field d B at a point P associated with a length element d s of a wire carrying a steady current I (Fig. 30.1): The vector d B is perpendicular both to d s (which points in the direction of the current) and to the unit vector r ˆ directed from d s toward P . The magnitude of d B is inversely proportional to r 2 , where r is the distance from d s to P . The magnitude of d B is proportional to the current and to the magnitude ds of the length element d s .
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