Wednesday, Mar. 29, 2006 PHYS 1444-501, Spring 2006Dr. Jaehoon Yu1PHYS 1444 – Section 501Lecture #17Wednesday, Mar. 29, 2006Dr. Jaehoon Yu• Solenoid and Toroidal Magnetic Field• Biot-Savart Law• Magnetic Materials• B in Magnetic Materials•HysteresisToday’s homework is #9, due 7pm, Thursday, Apr. 13!!Wednesday, Mar. 29, 2006 PHYS 1444-501, Spring 2006Dr. Jaehoon Yu2Announcements• Reading assignments– CH28 – 7, 28 – 8, 28 – 9 and 28 – 10 • Two Colloquia you must attend– Dr. H. Weerts, director of High Energy Physics Division at Argonne National Laboratory• Friday, Apr. 21• International Linear Collider: The Physics and Its Challenges– Dr. I. Hinchcliff, Lorentz Berkley Laboratory• Wednesday, Apr. 26• Title: Early Physics with ATLAS at the LHC • Term exam #2– Date and time: 5:30 – 6:50pm, Wednesday, Apr. 5– Location: SH103– Coverage: Ch. 25 – 4 to Ch. 28Wednesday, Mar. 29, 2006 PHYS 1444-501, Spring 2006Dr. Jaehoon Yu3Solenoid and Its Magnetic Field• What is a solenoid?– A long coil of wire consisting of many loops– If the space between loops are wide• The field near the wires are 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 uniform– For a 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 the ends– The same number of field lines spread out to an open spaceSolenoid AxisWednesday, Mar. 29, 2006 PHYS 1444-501, Spring 2006Dr. Jaehoon Yu4Solenoid Magnetic Field• Now let’s use Ampere’s law to determine the magnetic field inside a very long, densely packed solenoid• 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 onaÆb is 0.– Now the field B is perpendicular to the bc and da segments. So these integrals become 0, also. Bdl⋅=∫GGvbaBdl⋅+∫GGcbBdl⋅+∫GGdcBdl⋅∫GGadBdl⋅∫GGWednesday, Mar. 29, 2006 PHYS 1444-501, Spring 2006Dr. Jaehoon Yu5Solenoid 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⋅=∫GGvBl =0BnIµ=dcBdl⋅=∫GGBl0NIµWednesday, Mar. 29, 2006 PHYS 1444-501, Spring 2006Dr. Jaehoon Yu6Example 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⋅=∫GGvSolving 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 and B = µ0nI.(b) Outside the solenoid, the field is 0 since the net enclosed current is 0.()2Brπ=0 enclIµ=0NIµ02NIrµπWednesday, Mar. 29, 2006 PHYS 1444-501, Spring 2006Dr. Jaehoon Yu7Biot-Savart Law• Ampere’s law is useful in determining magnetic field utilizing symmetry• But sometimes it is useful to have another method of using infinitesimal current segments for B field– 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–• r is the displacement vector from the element dl to the point P• Biot-Savart law is the magnetic equivalent to Coulomb’s law02ˆ4Idl rdBrµπ×=GGBiot-Savart LawB field in Biot-Savart law is only that by the current, nothing else.Wednesday, Mar. 29, 2006 PHYS 1444-501, Spring 2006Dr. Jaehoon Yu8Example 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==∫vWhere dy=dl and r2=R2+y2and since we obtaincotyRθ=−B=The same as the simple, long straight wire!! It works!!02ˆ4dl rIrµπ+∞−∞×=∫G02sin4yIdyrµθπ+∞=−∞∫2cscRdθθ+=2sinRdθθ=()2RdRrθ=2rdRθ02sin4yIdyrµθπ+∞=−∞=∫001sin4IdRπθµθθπ==∫001cos4IRπµθπ−=012IRµπWednesday, Mar. 29, 2006 PHYS 1444-501, Spring 2006Dr. Jaehoon Yu9Magnetic 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 aligned reduce– This gives magnetization to the material• How do we demagnetize a bar magnet?– Hit the magnet hard or heat it over the Curie temperatureWednesday, Mar. 29, 2006 PHYS 1444-501, Spring 2006Dr. Jaehoon Yu10B in Magnetic Materials• What is the magnetic field inside a solenoid?•– Magnetic field in a long solenoid is directly proportional to the current.– This is valid only if air is inside the coil• What do you think will happen to B if we have something
View Full Document
Unlocking...