# 29 - Chapter 29 Magnetic Fields Due to Currents In this...

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Chapter 29 Magnetic Fields Due to Currents In this chapter we will explore the relationship between an electric current and the magnetic field it generates in the space around it. We will follow a two-prong approach, depending on the symmetry of the problem. For problems with low symmetry we will use the law of Biot-Savart in combination with the principle of superposition. For problems with high symmetry we will introduce Ampere’s law . Both approaches will be used to explore the magnetic field generated by currents in a variety of geometries (straight wire, wire loop, solenoid coil, toroid coil) We will also determine the force between two parallel current carrying conductors. We will then use this force to define the SI unit for electric (29 – 1)

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This law gives the magnetic field generated by a wire segment of length that carries a current . Consider the geometry shown in the figure. Associated with the element dB ds i ds The law of Biot -Savart r we define an associated vector that has magnitude whch is equal to the length . The direction of is the same as that of the current that flows through segment . ds ds ds ds r r A 3 The magnetic field generated at point P by the element located at point A is given by the equation: . Here is the vector that connects 4 point A (location of element ) w o dB dS i ds r dB r r ds μ π = r r r r r r 7 6 2 ith point P at which we want to determine . The constant 4 10 T m/A 1.26 10 T m/A and is known as sin " ". The magnitude of is: 4 Here is the angle o o dB i ds dB dB r θ - - = = = r r permeability constant between and . ds r r r 3 4 o i ds r dB r = r r r (29 – 2)
The magnitude of the magnetic field generated by the wire at point P located at a distance from the wire is given by the equation: 2 o i B R R μ π = Magnetic field generated by a long straight wire 2 o i B R = The magnetic field lines form circles that have their centers at the wire. The magnetic field vector is tangent to the magnetic field lines. The sense for is given by the . We po B B right hand rule r r int the thumb of the right hand in the direction of the current. The direction along which the fingers of the right hand curl around the wire gives the direction of . B r (29 – 3)

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φ 2 Consider the wire element of length shown in the figure. The element generates at point P a magnetic field of
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29 - Chapter 29 Magnetic Fields Due to Currents In this...

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