Presentation on theme: "Sources of the Magnetic Field"— Presentation transcript:
1 Sources of the Magnetic Field Chapter 30Sources of the Magnetic Field
2 Force on or I in the filed A whole picture helpsCharge q as sourceCurrent I as sourceGauss’s LawAmpere’s LawFaraday’s LawElectric field EMagnetic field BAmpere-Maxwell LawForce on q in the fieldForce on or I in the filedSummarized inLorentz forceMaxwell equations
3 Math -- review Vector cross product: Magnitude of the vector : θ θ Determine the direction. IfThe right-hand rule:Four fingers follow the first vector.Bend towards the second vector.Thumb points to the resultant vector.
4 Sources of Electric field, magnetic field From Coulomb's Law, a point charge dq generates electric field distance away from the source:From Biot-Savart's Law, a point current generates magnetic field distance away from the source:Difference:A current segment , not a point charge dq, hence a vector.Cross product of two vectors, is determined by the right-hand rule.
5 Total Magnetic Fieldis the field created by the current in the length segment , a vector that takes the direction of the current.To find the total field, sum up the contributions from all the current elementsThe integral is over the entire current distributionmo = 4p x 10-7 T. m / A , is the constant mo is called the permeability of free space
6 Example: from a Long, Straight Conductor with current I The thin, straight wire is carrying a constant current IConstructing the coordinate system and place the wire along the x-axis, and point P in the X-Y plane.Integrating over all the current elements gives
8 Now make the wire infinitely long The statement “the conductor is an infinitely long, straight wire” is translated into:q1 = p/2 and q2 = -p/2Then the magnitude of the field becomesThe direction of the field is determined by the right-hand rule. Also see the next slide.
9 For a long, straight conductor the magnetic field goes in circles The magnetic field lines are circles concentric with the wireThe field lines lie in planes perpendicular to to wireThe magnitude of the field is constant on any circle of radius aA different and more convenient right-hand rule for determining the direction of the field is shown
10 Example: at the center of a circular loop of wire with current I YXFrom Biot-Savart Law, the field at O from isThis is the field at the center of the loopOff center points are not so easy to calculate.
11 How about a stack of loops? Along the axis, yes, the formula isN is the number of turns.When the loop is sufficiently long, what can we say about the field inside the body of this electric magnet?We need Gauss’s LawOops, typo.We need Ampere’s Law. Sorry, Mr. Ampere.
12 Ampere’s Law Connects B with I Ampere’s law states that the line integral of around any closed path equals moI where I is the total steady current passing through any surface bounded by the closed path:This is a line integral over a vector field
13 from a long, straight conductor re-calculated using Ampere’s Law Choose the Gauss’s Surface, oops, not again!Choose the Ampere’s loop, as a circle with radius a.Ampere’s Law saysis parallel with , so
14 When the wire has a size: with radius R Outside of the wire, r > RInside the wire, we need I’, the current inside the ampere’s circle
15 Plot the results The field is proportional to r inside the wire The field varies as 1/r outside the wireBoth equations are equal at r = R
16 Magnetic Field of a Toroid The toroid has N turns of wireFind the field at a point at distance r from the center of the toroid (loop 1)There is no field outside the coil (see loop 2)
17 Magnetic Field of a Solenoid A solenoid is a long wire wound in the form of a helixA reasonably uniform magnetic field can be produced in the space surrounded by the turns of the wireThe field lines in the interior arenearly parallel to each otheruniformly distributedThe interior of the solenoid field people use.
18 Magnetic Field of a Tightly Wound Solenoid The field distribution is similar to that of a bar magnetAs the length of the solenoid increasesthe interior field becomes more uniformthe exterior field becomes weaker
19 Ideal (infinitely long) Solenoid An ideal solenoid is approached when:the turns are closely spacedthe length is much greater than the radius of the turnsApply Ampere’s Law to loop 2:n = N / ℓ is the number of turns per unit length
20 Magnetic Force Between Two Parallel Conductors Two parallel wires each carry steady currentsThe field due to the current in wire 2 exerts a force on wire 1 of F1 = I1ℓ B2Substituting the equation for givesCheck with right-hand rule:same direction currents attract each otheropposite directions currents repel each otherThe force per unit length on the wire isPLAYACTIVE FIGUREAnd this formula defines the current unit Ampere.
21 Definition of the Ampere and the Coulomb The force between two parallel wires is used to define the ampereWhen 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 AThe SI unit of charge, the coulomb, is defined in terms of the ampereWhen a conductor carries a steady current of 1 A, the quantity of charge that flows through a cross section of the conductor in 1 second is 1 C
22 Magnetic Flux, defined the same way as any other vector flux, like the electric flux, the water (velocity) fluxThe magnetic flux over a surface area associated with a magnetic field is defined asThe unit of magnetic flux is T.m2 = Wb (weber)
23 Magnetic Flux Through a Plane A special case is when a plane of area A, its direction makes an angle q withThe magnetic flux isFB = BA cos qWhen the field is perpendicular to the plane, F = 0 (figure a)When the field is parallel to the plane, F = BA (figure b)PLAYACTIVE FIGURE
24 Gauss’ Law in Magnetism Yes, that is Gauss’s Law, but in Magnetism.Magnetic fields do not begin or end at any pointThe number of lines entering a surface equals the number of lines leaving the surfaceGauss’ law in magnetism says the magnetic flux through any closed surface is always zero:
25 Example problemsA long, straight wire carries current I. A right-angle bend is made in the middle of the wire. The bend forms an arc of a circle of radius r. Determine the magnetic filed at the center of the arc.Formula to use: Biot-Savart’s Law, or more specifically the results from the discussed two examples:For the straight sectionFor the arcThe final answer:magnitudedirection pointing into the page.