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chapter30 - Chapter 30 Sources of the Magnetic Field...

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Chapter 30 Sources of the Magnetic Field

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Biot-Savart Law – Introduction z Biot and Savart conducted experiments on the force exerted by an electric current on a nearby magnet z They arrived at a mathematical expression that gives the magnetic field at some point in space due to a current
Biot-Savart Law – Set-Up z The magnetic field is at some point P z The length element is z The wire is carrying a steady current of I Please replace with fig. 30.1 d B r d s r

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Biot-Savart Law – Observations z The vector is perpendicular to both and to the unit vector directed from toward P z The magnitude of is inversely proportional to r 2 , where r is the distance from to P d B r r ˆ d B r d s r d s r d s r
Biot-Savart Law – Observations, cont z The magnitude of is proportional to the current and to the magnitude ds of the length element z The magnitude of is proportional to sin θ, where θ is the angle between the vectors and d s r r ˆ d s r d B r d B r

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z The observations are summarized in the mathematical equation called the Biot-Savart law : z The magnetic field described by the law is the field due to the current-carrying conductor z Don’t confuse this field with a field external to the conductor Biot-Savart Law – Equation 2 4 o μ d d π r × = sr B r r ˆ I
Permeability of Free Space z The constant μ o is called the permeability of free space z o = 4 π x 10 -7 T . m / A

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Total Magnetic Field z is the field created by the current in the length segment ds z To find the total field, sum up the contributions from all the current elements I z The integral is over the entire current distribution d B r 2 4 o μ d π r × = sr B r r ˆ I d s r
Biot-Savart Law – Final Notes z The law is also valid for a current consisting of charges flowing through space z represents the length of a small segment of space in which the charges flow z For example, this could apply to the electron beam in a TV set d s r

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Compared to B z Distance z The magnitude of the magnetic field varies as the inverse square of the distance from the source z The electric field due to a point charge also varies as the inverse square of the distance from the charge r E r
Compared to , 2 z Direction z The electric field created by a point charge is radial in direction z The magnetic field created by a current element is perpendicular to both the length element and the unit vector r ˆ d s r B r E r

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Compared to , 3 B z Source z An electric field is established by an isolated electric charge z The current element that produces a magnetic field must be part of an extended current distribution z Therefore you must integrate over the entire current distribution r E r
for a Long, Straight Conductor z The thin, straight wire is carrying a constant current z z Integrating over all the current elements gives () 2 1 12 4 4 θ o θ o μ B θ d θ π a μ θθ π a =− I cos I sin sin ( ) sin dd x θ ×= sr k r ˆ ˆ B r

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for a Long, Straight Conductor, Special Case z If the conductor is an infinitely long, straight wire, θ 1 = π /2 and 2 = - π /2 z The field becomes 2 I o μ B π a = B r
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chapter30 - Chapter 30 Sources of the Magnetic Field...

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