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Dr. Jie ZouPHY 13611 Chapter 30 Sources of the Magnetic Field

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Dr. Jie ZouPHY 13612 Introduction Hans Christian Oersted (1777-1851) The first evidence of the close connection between electricity and magnetism was obtained accidentally by the Danish scientist Hans Christian Oersted in 1820. An electric current produces a magnetic field. Biot and Savart performed quantitative experiments on the force exerted by an electric current on a nearby magnet. Biot-Savart law: a mathematical expression to calculate the magnetic field produced at some point in space by a small current element.

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Dr. Jie ZouPHY 13613 Outline The Biot-Savart Law (30.1) Mathematical expression Applications in finding the total magnetic field produced by various current distributions Example 1: Thin, straight current-carrying wire Example 2: Circular current loop

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Dr. Jie ZouPHY 13614 The Biot-Savart Law The magnetic field dB at a point P associated with a length element ds of a wire carrying a steady current I is given by: 0 = 4 10 -7 T m/A, the permeability of free space. (1) The direction of dB is perpendicular to both ds and, and thus perpendicular the plane formed by ds and. (2) The magnitude of dB is proportional to I, ds, and sin , but inversely proportional to r 2.

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Dr. Jie ZouPHY 13615 Quick Quiz: where is the magnetic field the greatest? Consider the current in the length of wire shown in the figure below. Rank the points A, B, and C, in terms of magnitude of the magnetic field due to the current in the length element shown, from greatest to least.

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Dr. Jie ZouPHY 13616 Total magnetic field due to a current distribution Evaluate B by integration: (1) The above equation follows the principle of superposition. (2) The integral is taken over the entire current distribution. (3) The integrand is a cross product and therefore a vector quantity.

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Dr. Jie ZouPHY 13617 Example 1: Thin, straight current- carrying wire Consider a thin, straight wire carrying a constant current I and placed along the x axis as shown in the figure below. (1) Determine the magnitude and direction of the magnetic field at point P due to this current. Answer:, out of the page. (2) Find the magnetic field at P in the limit of an infinite long, straight wire. Answer: B = 0 I/(2 a), out of the page.

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Dr. Jie ZouPHY 13618 Magnetic field surrounding a long, straight current-carrying wire Direction of B: The magnetic field lines are circles concentric with the wire and lie in planes perpendicular to the wire. The right-hand rule: grasp the wire with the right hand, positioning the thumb along the direction of the current. The four fingers wrap in the direction of the magnetic field. Magnitude of B: B = 0 I/(2 a) B is constant on any circle of radius a.

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Dr. Jie ZouPHY 13619 Example 2: Curved wire segment Calculate the magnetic field at point O for the current-carrying wire segment shown. The wire consists of two straight portions and a circular arc of radius R, which subtends an angle . The arrowheads on the wire indicate the direction of the current. Answer: B = 0 I /(4 R), into the page at O. Can you also find the magnetic field at the center of a circular wire loop of radius R that carries a current I? Answer: B = 0 I/(2R),

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Dr. Jie ZouPHY 136110 Magnetic field lines surrounding a current loop (a-b) Magnetic field lines surrounding a current loop. (c) Magnetic field lines surrounding a bar magnet. Note the similarity between this line pattern and that of a current loop.

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