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Chapter 30 Sources of the Magnetic Field Dr. Jie Zou PHY 1361 1 Introduction The first evidence of the close connection between electricity and magnetism was obtained accidentally by the Danish scientist Hans Christian Oersted in 1820. Hans Christian Oersted (1777-1851) Dr. Jie Zou 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. PHY 1361 2 Outline The Biot-Savart Law (30.1) Mathematical expression Applications in finding the total magnetic field produced by various current distributions Dr. Jie Zou Example 1: Thin, straight current-carrying wire Example 2: Circular current loop PHY 1361 3 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 Ids rˆ dB 4 r 2 Dr. Jie Zou 0 = 4 10-7 Tm/A, the permeability of free space. (1) The direction of dB is perpendicular to both ds and r̂ , and thus perpendicular the plane formed by ds and r̂ . (2) The magnitude of dB is proportional to I, ds, and sin , but inversely proportional to r2. PHY 1361 4 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. Dr. Jie Zou PHY 1361 5 Total magnetic field due to a current distribution Evaluate B by integration: 0 I ds rˆ B 4 r 2 (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. Dr. Jie Zou PHY 1361 6 Example 1: Thin, straight currentcarrying wire Consider a thin, straight wire carrying a constant current I and placed along the x axis as shown in the figure below. Dr. Jie Zou (1) Determine the magnitude and direction of the magnetic field at point P due to this current. Answer: B 0 I cos 1 cos 2 , out of the page. 4a (2) Find the magnetic field at P in the limit of an infinite long, straight wire. Answer: B = 0I/(2a), out of the page. PHY 1361 7 Magnetic field surrounding a long, straight current-carrying wire Direction of B: Magnitude of B: B = 0I/(2a) Dr. Jie Zou 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. B is constant on any circle of radius a. PHY 1361 8 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. Dr. Jie Zou Answer: B = 0I/(4R), 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 = 0I/(2R), PHY 1361 9 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. Dr. Jie Zou PHY 1361 10