Souces of Magnetic Fields (chap 8) In chapte 7, we consideed the magnetic field effects on a moving chage, a line cuent and a cuent loop. Now in Chap 8, we conside the magnetic fields that ae ceated by moving chages line cuent elements (iot-savat Law) line of cuent cuent loops
Magnetic Fields fom a moving chage Magnetic fields ae ceated by a moving chage. If the chage is stationay, thee is NO magnetic field. If the chage is moving, the magnetic field law is μ qv 0 ˆ P q ˆ v μ 0 10-7 T m / A Use ight hand to get diection of field
Magnetic Fields fom a moving chage at a position pependicula to velocity μ qv 0 ˆ P sinθ 1 μ q v 0 1 q ˆ v v is out of sceen. field lines have cicula path.
Eample A q6μc point chage at the oigin is moving with a v810 6 m/s in the +y diection. What is the field it poduces at the following points? A) 0.5m, y0, z0 v q z ˆ y p μ0 qv ˆ (6 10 μ0 1.9 10 5 6 T kˆ C)(8 10 (0.5m) μ 0 10-7 T m / A 6 m s ˆ) j iˆ ) 0m, y0.5, z0m 0 v and ˆ since paallel.
Magnetic Fields fom a cuent element Suppose we have a cuent element of length dl, coss sectional aea A and paticle density n, #chages/volume. The total chage, dq is q dl A dq n q A dl. And ecall that cuent is I n q v A. The field becomes, d μ 0 dq v ˆ μ ˆ ˆ 0 v μ0 Idl nqa Vecto, dl, is the length of the cuent element in the diection of the cuent flow. This is called the iot-savat Law Since n above is vey lage, a cuent (element) will give a much bigge magnetic field than that of a single chage.
iot-savat Law θ ˆ dl d X d μ I dl ˆ 0 μ 0 10 7 N A I The magnetic field ciculates aound the wie Use ight-hand ule: thumb along I, finges cul in diection of.
Magnetic Fields fom a abitay wie dl θ ˆ X d If we have an abitay wie with cuent I flowing, and we wish to get the at a specific point fom the entie wie, we integate the fomula d μ Idl 0 ˆ I and obtain a line integal of a coss poduct μ0 dl ˆ I
Peflight 14: A cuent caying wie (with no emakable symmety) is oiented in the -y plane. Points A,, & C lie in the same plane as the wie. The z- ais points out of the sceen. 5) In what diection is the magnetic field contibution fom the segment dl at point A. Check all non-zeo components. + - +y -y +z -z 6) In what diection is the magnetic field contibution fom the segment dl at point. Check all non-zeo components. + - +y -y +z -z 7) In what diection is the magnetic field contibution fom the segment dl at point C. Check all non-zeo components. + - +y -y +z -z
d points in the diection of dl μ0i dl d 3 A: dl is to the ight, and is up d is out of the page : dl is to the ight, and is up and ight d is out of the page C: dl is to the ight, and is down and ight d is into the page Conclusion: at evey point above the wie, d is. elow the wie, d is
Peflight 14: 9) Would any of you answes fo questions 5-7 change if we integated dl ove the whole wie? NO! Why o why not? μ0i dl d 3 d points in the diection of dl At point A: dl is to the ight, and is up d is out of the page At point : dl is to the ight, and is up and ight d is out of the page At point C: dl is to the ight, and is down and ight d is into the page Fo evey point in the -y plane and evey piece of wie dl: evey dl and evey ae always in the -y plane Since d must be pependicula to and dl, d is always in the ±z diection! Conclusion: At evey point above the wie, the d due to evey piece dl is. elow the wie, the d due to evey piece dl is
Magnetic Fields fom a long wie An infinitely long wie along the y-ais with the cuent moving +y. What is the magnetic field at position on the -ais? The field is in the z diection. μ0 dl ˆ μ0 sinφ I I dl( zˆ ) 4 π 4 π zi ˆ μ0 sinφ Since, sin(φ) dy; sin(π φ), we have sinφ dy sin ( π φ) dy + 3 dy + 1 ( ) y + 3/ dy 1 y y + + μ I 0 π zˆ ( )
Magnetic Fields fom a long wie What ae the magnetic fields at points a,b, and c? μ 0I π c c a a b b