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UT Arlington PHYS 1444 - PHYS 1444 Lecture Notes

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PHYS 1444 – Section 004 Lecture #16Ampére’s LawSolenoid and Its Magnetic FieldSolenoid Magnetic FieldSolenoid Magnetic FieldExample 28 – 8 Biot-Savart LawExample 28 – 9 Magnetic Materials - FerromagnetismB in Magnetic MaterialsB in Magnetic MaterialsHysteresisHysteresisInduced EMFElectromagnetic InductionElectromagnetic InductionWednesday, April 4, 2007 1PHYS 1444-004, Spring 2007Dr. Andrew BrandtPHYS 1444 – Section 004Lecture #16Wednesday, April 4 2007Dr. Andrew Brandt• Solenoid and Toroidal Magnetic Field• Biot-Savart Law• Magnetic Materials• B in Magnetic Materials•Hysteresis• Induced emfHW7 due Mon 4/9 at 11 pmTest moved back one week to Weds 4/18Wednesday, April 4, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt2Ampére’s Law– How do you obtain B in the figure at any point?• Vector sum of the field by the two currents– The result of the closed path integral in Ampere’s law for green dashed path is still μ0I1. Why?– While B for each point along the path varies, the integral over the closed path still comes out the same whether there is the second wire or not.• Since Ampere’s law is valid in general, B in Ampere’s law is not necessarily just due to the current Iencl.• B is the field at each point in space along the chosen path due to all sources– Including the current I enclosed by the path but also due to any other sources0 enclBdl Iμ⋅=∫rrWednesday, April 4, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt3Solenoid and Its Magnetic Field• What is a solenoid?– A long coil of wire consisting of many loops– If the space between loops is wide• The field near the wires is nearly circular• Between any two wires, the fields due to each loop cancel• Toward the center of the solenoid, the fields add up to give a field that can be fairly large and uniformSolenoid Axis–For long, densely packed loops•The field is nearly uniform and parallel to the solenoid axes within the entire cross section•The field outside the solenoid is very small compared to the field inside, except at the endsWednesday, April 4, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt4Solenoid Magnetic Field• Now let’s use Ampere’s law to determine the magnetic field inside a very long, densely packed solenoidBdl⋅=∫rrbaBdl⋅+∫rrcbBdl⋅+∫rrdcBdl⋅∫rradBdl⋅∫rr•Let’s choose the path abcd, far away from the ends–We can consider four segments of the loop for integral––The field outside the solenoid is negligible. So the integral on aÆb is 0.–Now the field B is perpendicular to the bc and da segments. So these integrals become 0, also.Wednesday, April 4, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt5Solenoid Magnetic Field– So the sum becomes:– If the current I flows in the wire of the solenoid, the total current enclosed by the closed path is NI•Where N is the number of loops (or turns of the coil) enclosed – Thus Ampere’s law gives us – If we let n=N/l be the number of loops per unit length, the magnitude of the magnetic field within the solenoid becomes–• B depends on the number of loops per unit length, n, and the current I– Does not depend on the position within the solenoid but uniform inside it, like a bar magnetBdl⋅=∫rrBl =0BnIμ=dcBdl⋅=∫rrBl0NIμWednesday, April 4, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt6Example 28 – 8 Toroid. Use Ampere’s law to determine the magnetic field (a) inside and (b) outside a toroid, (which is like a solenoid bent into the shape of a circle). (a) How do you think the magnetic field lines inside the toroid look? Since it is a bent solenoid, it should be a circle concentric with the toroid.If we choose path of integration one of these field lines of radius r inside the toroid, path 1, to use the symmetry of the situation, making B the same at all points on the path, we obtain from Ampere’s lawBdl⋅=∫rrSolving for BB=So the magnetic field inside a toroid is not uniform. It is larger on the inner edge. However, the field will be uniform if the radius is large and the toroid is thin (B = μ0nI ).(b) Outside the solenoid, the field is 0 since the net enclosed current is 0.()2Brπ=0 enclIμ=0NIμ02NIrμπWednesday, April 4, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt7Biot-Savart Law• Ampere’s law is useful in determining magnetic field utilizing symmetry• But sometimes it is useful to have another method to determine the B field such as using infinitesimal current segments – Jean Baptiste Biot and Feilx Savart developed a law that a current I flowing in any path can be considered as many infinitesimal current elements– The infinitesimal magnetic field dB caused by the infinitesimal length dl that carries current I is–02ˆ4Idl rdBrμπ×=rrBiot-Savart LawThe B field in the Biot-Savart law is only that due to the current •r is the displacement vector from the element dl to the point P•Biot-Savart law is the magnetic equivalent to Coulomb’s lawWednesday, April 4, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt8Example 28 – 9 B due to current I in a straight wire. For the field near a long straight wire carrying a current I, show that the Biot-Savarat law gives the same result as the simple long straight wire, B=μ0I/2πR. What is the direction of the field B at point P? Going into the page.All dB at point P has the same direction based on right-hand rule.dy =Integral becomes The magnitude of B using Biot-Savart law isBdB==∫Where dy=dl and r2=R2+y2and since we obtaincotyRθ=−B=The same as the simple, long straight wire!! It works!!02ˆ4dl rIrμπ+∞−∞×=∫r02sin4yIdyrμθπ+∞=−∞∫2cscRdθθ+=2sinRdθθ=()2RdRrθ=2rdRθ02sin4yIdyrμθπ+∞=−∞=∫001sin4IdRπθμθθπ==∫001cos4IRπμθπ−=012IRμπWednesday, April 4, 2007 PHYS 1444-004, Spring 2007Dr. Andrew Brandt9Magnetic Materials - Ferromagnetism• Iron is a material that can turn into a strong magnet– This kind of material is called ferromagneticferromagnetic material• In microscopic sense, ferromagnetic materials consists of many tiny regions called domainsdomains– Domains are like little magnets usually smaller than 1mm in length or width• What do you think the alignment of domains are like when they are not magnetized?– Randomly arranged• What if they are magnetized?– The size of the domains aligned with the external magnetic field direction grows while those of the domains not


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