Physics for Scientists and Engineers 8ed - ch30 - PowerPoint Slides

# Physics for Scientists and Engineers 8ed - ch30 - PowerPoint Slides

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Chapter 30 Sources of the Magnetic Field

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Biot-Savart Law – Introduction Biot and Savart conducted experiments on the force exerted by an electric current on a nearby magnet 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 The magnetic field is at some point P The length element is 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 The vector is perpendicular to both and to the unit vector directed from toward P The magnitude of is inversely proportional to r 2 , where r is the distance from to P d B r r ö d B r d s r d s r d s r
Biot-Savart Law – Observations, cont The magnitude of is proportional to the current and to the magnitude ds of the length element The magnitude of is proportional to sin θ , where θ is the angle between the vectors and d s r r ö d s r d B r d B r

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

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

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

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Compared to , 3 Source An electric field is established by an isolated electric charge The current element that produces a magnetic field must be part of an extended current distribution Therefore you must integrate over the entire current distribution B r E r
for a Long, Straight Conductor The thin, straight wire is carrying a constant current Integrating over all the current elements gives ( 29 2 1 1 2 4 4 S o P o P B P dP ¶a P P P ¶a = - = - ò I cos I sin sin ( 29 sin d dx P ´ = s r k r ö ö B r

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for a Long, Straight
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