Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to CurrentsWhat Are We Going to Learn? A Road MapElectric Current: A Source of Magnetic FieldPowerPoint PresentationCasey’s Right Hand Rule!SuperpositionBiot-Savart LawThe Biot-Savart LawSlide 9Slide 10Field of a Straight WireIs the B-Field From a Power Line Dangerous?Slide 13Slide 14Lecture 22: MON 09 MARLecture 22: MON 09 MARCh.29.1–2: Magnetic Fields Due to Ch.29.1–2: Magnetic Fields Due to CurrentsCurrentsBiot-Savart Law Biot-Savart Law Physics 2102Jonathan DowlingJean-Baptiste Biot (1774-1862)Felix Savart (1791–1841)QuickTime™ and a decompressorare needed to see this picture.What Are We Going to Learn?What Are We Going to Learn?A Road MapA Road Map•Electric charge Electric force on other electric charges Electric fi eld, and electric pot ential•Moving electric charges : current •Electronic circuit components: batteries, resistors, capacitors•Electric currents Magnetic field Magnetic force on moving charges•Time-varying magnetic field Electric Field•More circuit components: inductors. •Electromagnetic waves light waves•Geometrical Optics (light rays). •Physical optics (light waves)Electric Current: Electric Current: A Source of Magnetic FieldA Source of Magnetic FieldIBWire withcurrentINTO pageB•Observation: an electric current creates a magnetic field •Simple experiment: hold a current-carrying wire near a compass needle! QuickTime™ and a decompressorare needed to see this picture.BHans Christian Oersted was a professor of science at Copenhagen University. In 1820 he arranged in his home a science demonstration to friends and students. He planned to demonstrate the heating of a wire by an electric current, and also to carry out demonstrations of magnetism, for which he provided a compass needle mounted on a wooden stand.While performing his electric demonstration, Oersted noted to his surprise that every time the electric current was switched on, the compass needle moved. He kept quiet and finished the demonstrations, but in the months that followed worked hard trying to make sense out of the new phenomenon.QuickTime™ and a decompressorare needed to see this picture.QuickTime™ and a decompressorare needed to see this picture.Casey’s Right Hand Casey’s Right Hand Rule!Rule!•Point your thumb along the direction of the current in a straight wire•The magnetic field created by the current consists of circular loops directed along your curled fingers.•The magnetic field gets weaker with distance: For long wire it’s a 1/R Law!•You can apply this to ANY straight wire (even a small differential element!)•What if you have a curved wire? Break into small elements.iBDirection of B!iSuperpositionSuperposition•Magnetic fields (like electric fields) can be “superimposed” -- just do a vector sum of B from different sources•The figure shows four wires located at the 4 corners of a square. They carry equal currents in directions indicated•What is the direction of B at the center of the square?BI-OUTI-OUTI-INI-INWhen we computed the electric field due to charges we usedCoulomb’s law. If one had a large irregular object, one broke itinto infinitesimal pieces and computed,rrdqEdˆ4120πε=rrrdqEdˆ4120πε=rWhich we write as,If you wish to compute the magnetic field due to a current in a wire, you use the law of Biot and Savart.Biot-Savart LawBiot-Savart LawThe Biot-Savart LawThe Biot-Savart Law•Quantitative rule for computing the magnetic field from any electric current•Choose a differential element of wire of length dL and carrying a current i•The field dB from this element at a point located by the vector r is given by the Biot-Savart Law304 rrLidBdrrr×=πμ304 rrLidBdrrr×=πμ0 =4x10–7 T•m/A(permeability constant)0 =4x10–7 T•m/A(permeability constant)Jean-Baptiste Biot (1774-1862)Felix Savart (1791-1841)LdrrridB€ dr E =14πε0dqr2) r € dr E =14πε0dqr2) r LdrrridB € dr B =μ04πidr L ×) r r2 € dr B =μ04πidr L ×) r r2Biot-Savart Law for B-FieldsCoulomb Law for E-FieldsBiot-Savart Requires A Right-Hand RuleBoth Are 1/r2 Laws!The has no units. € ) rBiot-Savart LawBiot-Savart Law•An infinitely long straight wire carries a current i. •Determine the magnetic field generated at a point located at a perpendicular distance R from the wire.•Choose an element ds as shown•Biot-Savart Law: dB points INTO the page•Integrate over all such elements304 rrsidBdrrr×=πμ304 rrsidBdrrr×=πμ30)sin(4 rridsdBθπμ=30)sin(4 rridsdBθπμ=∫∞∞−=30)sin(4 rrdsiBθπμ∫∞∞−=30)sin(4 rrdsiBθπμ( )∫∞∞−+=2/32204RsRdsiπμ( )∫∞+=02/32202RsRdsiπμ( )∞⎥⎥⎦⎤⎢⎢⎣⎡+=02/122202RsRsiRπμRiπμ20=rR /sin =θrR /sin =θ2/122)( Rsr +=2/122)( Rsr +=∫∞∞−=30)sin(4 rrdsiBθπμField of a Straight Field of a Straight WireWire304 rrsidBdrrr×=πμ304 rrsidBdrrr×=πμ30)sin(4 rridsdBθπμ=30)sin(4 rridsdBθπμ=Is the B-Field From a Power Line Is the B-Field From a Power Line Dangerous?Dangerous?A power line carries a current of 500 A. What is the magnetic field in a house located 100Gm away from the power line?A power line carries a current of 500 A. What is the magnetic field in a house located 100Gm away from the power line?RiBπμ20=RiBπμ20=€ =(4π ×10−7T ⋅ m / A)(500A)2π(100m)€ =(4π ×10−7T ⋅ m / A)(500A)2π(100m)= 1 T= 1 TRecall that the earth’s magnetic field is ~10–4T = 100 TRecall that the earth’s magnetic field is ~10–4T = 100 TProbably not dangerous!QuickTime™ and a decompressorare needed to see this picture.Biot-Savart LawBiot-Savart Law•A circular arc of wire of radius R carries a current i. •What is the magnetic field at the center of the loop? 304 rrdsiBdrr×=πμ304 rrdsiBdrr×=πμ203044 RiRdRidsRdBφπμπμ==203044 RiRdRidsRdBφπμπμ==0 04 4iidBR Rμ μφπ πΦ= =∫0 04 4iidBR Rμ μφπ πΦ= =∫Direction of B?? Not anot her right hand rule?!TWO right hand rules!:If your thumb points along the CURRENT, your fingers will point in the same direction as the FIELD.If you curl our fingers around direction of CURRENT, your thumb points along FIELD!
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