# SJSU EE 140 - ch28_B_field (73 pages)

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## ch28_B_field

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- Pages:
- 73
- School:
- San Jose State University
- Course:
- Ee 140 - Principles of Electromagnetic Fields

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Chapter 28 Sources of the Magnetic Field 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 r The magnetic field is dB at some point P The length element is r ds The wire is carrying a steady current of I Please replace with fig 30 1 Biot Savart Law Observations r r ds The vector dB is perpendicular to both and r ds to the unit vector r directed from toward P r The magnitude of dB is inversely proportional r to r2 where r is the distance from ds to P Biot Savart Law Observations cont r The magnitude of dB is proportional to the current andr to the magnitude ds of the length element ds r The magnitude of dB is proportional to sin r where is the angle between the vectors ds and r Biot Savart Law Equation The observations are summarized in the mathematical equation called the Biot Savart law r I dsr r dB o 4 r 2 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 Permeability of Free Space The constant o is called the permeability of free space o 4 x 10 7 T m A Total Magnetic Field r dB is the field created by the current in the length segment ds To find the total field sum up the r contributions from all the current elements I ds r I dsr r B o 2 4 r The integral is over the entire current distribution Biot Savart Law Final Notes The law is also valid for a current consisting of charges flowing through space r ds 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 r r B Compared to E 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 r r B Compared to E 2 Direction The electric field created by a point charge is radial in direction The magnetic field created by a current element is r perpendicular to both the length element ds and the unit vector r r r B Compared to E 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 r B for a Long Straight Conductor The thin straight wire is carrying a constant current r ds r dx sin k Integrating over all the current elements gives o I B cos d 4 a o I sin 1 sin 2 4 a 2 1 r B for a Long Straight Conductor Special Case If the conductor is an infinitely long straight wire 1 2 and 2 2 The field becomes o I B 2 a r B for a Long Straight Conductor Direction The magnetic field lines are circles concentric with the wire The field lines lie in planes perpendicular to to wire The magnitude of the field is constant on any circle of radius a The right hand rule for determining the direction of the field is shown r B for a Curved Wire Segment Find the field at point O due to the wire segment I and R are constants o I B 4 R will be in radians r B for a Circular Loop of Wire Consider the previous result with a full circle 2 o I o I o I B 2 4 a 4 a 2a This is the field at the center of the loop r B for a Circular Current Loop The loop has a radius of R and carries a steady current of I Find the field at point P Bx o I a 2 2 a x 2 2 3 2 Comparison of Loops Consider the field at the center of the current loop At this special point x 0 Then Bx o I a 2 2 a x 2 2 3 2 o I 2a This is exactly the same result as from the curved wire Magnetic Field Lines for a Loop Figure a shows the magnetic field lines surrounding a current loop Figure b shows the field lines in the iron filings Figure c compares the field lines to that of a bar magnet Magnetic Force Between Two Parallel Conductors Two parallel wires each carry a steady current r The field B2 due to the current in wire 2 exerts a force on wire 1 of F1 I1 B2 PLAY ACTIVE FIGURE Magnetic Force Between Two Parallel Conductors cont r Substituting the equation for B2 gives o I1 I 2 F1 l 2 a Parallel conductors carrying currents in the same direction attract each other Parallel conductors carrying current in opposite directions repel each other Magnetic Force Between Two Parallel Conductors final The result is often expressed as the magnetic force between the two wires FB This can also be given as the force per unit length FB o I1 I 2 2 a l Definition of the Ampere The force between two parallel wires can be used to define the ampere When the magnitude of the force per unit length between two long parallel wires that carry identical currents and are separated by 1 m is 2 x 10 7 N m the current in each wire is defined to be 1 A Definition of the Coulomb The SI unit of charge the coulomb is defined in terms of the ampere When a conductor carries a steady current of 1 A the quantity of charge that flows through a cross section of the conductor in 1 s is 1 C Andre Marie Amp re 1775 1836 French physicist Created with the discovery of electromagnetism The relationship between electric current and magnetic fields Also worked in mathematics Magnetic Field of a Wire A compass can be used to detect the magnetic field When there is no current in the wire there is no field due to the current The compass needles all point toward the Earth s north pole Due to the Earth s magnetic field Magnetic Field of a Wire 2 Here the wire carries a strong current The compass needles deflect in a direction tangent to the circle This shows the direction of the magnetic field produced by the wire Use the active figure to vary the current PLAY ACTIVE FIGURE Magnetic Field of a Wire 3 The circular magnetic field around the wire is shown by the iron filings Ampere s Law r r The product of B ds can r be evaluated for small length elements ds on the circular path defined by the …

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