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We’ve been working with the effects of magnetic fields without considering where they come from. Today we learn about sources of magnetic fields. Today’s agenda: Magnetic Fields Due To A Moving Charged Particle. You must be able to calculate the magnetic field due to a moving charged particle. Biot-Savart Law: Magnetic Field due to a Current Element. You must be able to use the Biot-Savart Law to calculate the magnetic field of a currentcarrying conductor (for example: a long straight wire). Force Between Current-Carrying Conductors. You must be able to begin with starting equations and calculate forces between currentcarrying conductors. Magnetic Field of a Moving Charged Particle Let’s start with the magnetic field of a moving charged particle. r It is experimentally observed that a moving point charge q gives rise to a magnetic field. B r̂ + v μ0 qv rˆ B= . 2 4π r 0 is a constant, and its value is 0=4x10-7 T·m/A Remember: the direction of r is always from the source point (the thing that causes the field) to the field point (the location where the field is being measured. cross products of unit vectors We are going to be doing lots of cross products of unit vectors. Here are some handy ways to do the cross product. The “always works if you can do math” method C = A B Example: ˆi C = det A x B x ˆj Ay By ˆi ˆj ˆk ˆj = det 0 0 0 -1 kˆ Av z B z kˆ ˆ 1 = i 0 - -1 = ˆi 0 cross products of unit vectors We are going to be doing lots of cross products of unit vectors. Here are some handy ways to do the cross product. The right hand rule method link to image http://en.wikipedia.org/wiki/Cross_product cross products of unit vectors We are going to be doing lots of cross products of unit vectors. Here are some handy ways to do the cross product. The “I learned this decades ago and forgot the name for it” method i j k i j k ˆi ˆj =kˆ cross products of unit vectors i j k i j k ˆj ˆi = -kˆ i j k i j k ˆi kˆ = -jˆ i j k i j k ˆj kˆ = - -iˆ = ˆi Example: proton 1 has a speed v0 (v0<<c) and is moving along the x-axis in the +x direction. Proton 2 has the same speed and is moving parallel to the x-axis in the –x direction, at a distance r directly above the x-axis. Determine the electric and magnetic forces on proton 2 at the instant the protons pass closest to each other. y This is example 28.1 in your text. FE The electric force is v0 1 q1q2 ˆ FE = r 2 4 r E r 1 2 1 e ˆ FE = j 2 4 r Homework Hint: this and the next 3 slides! 2 z r̂ v0 x Alternative approach to calculating electric force. This is “better” because we use the concept of field to calculate both of electric and (later) magnetic forces. At the position of proton 2 there is an electric field due to proton 1. y 1 q1 ˆ 1 eˆ E1 = r= j 2 2 4 r 4 r FE E v0 This electric field exerts a force on proton 2. 1 eˆ 1 e2 ˆ FE = qE1 = e j= j 2 2 4 r 4 r 2 r 1 z r̂ v0 x To calculate the magnetic force: at the position of proton 2 there is a magnetic field due to proton 1. q1 v1 rˆ B1 = 4 r2 ev 0 ˆi ˆj B1 = 4 r2 y FE v0 ev 0 ˆ B1 = k 2 4 r 2 B1 r 1 z r̂ v0 x Proton 2 “feels” a magnetic force due to the magnetic field of proton 1. FB = q2 v 2 B1 FB = ev 0 ev 0 ˆ ˆ i k 2 4 r y FE v0 e2 v 20 ˆ FB = j 2 4 r 2 FB B1 What would proton 1 “feel?” r 1 Caution! Relativity overrules Newtonian mechanics! However, in this case, the force is “equal & opposite.” z r̂ v0 x Both forces are in the +y direction. The ratio of their magnitudes is e 2 v 20 2 4 r FB = FE 1 e 2 2 4 r y FE FB = v 20 FE v0 2 FB B1 Later we will find that r 1 1 = 2 c z r̂ v0 x FB v 20 Thus = 2 FE c If v0=106 m/s, then 10 FB -5 = 1.11 10 FE 3 108 2 6 2 y FE Don’t you feel sorry for the poor, weak magnetic force? v0 What if you are a nanohuman, lounging on proton 1. You rightfully claim you are at rest. There is no magnetic field from your proton, and no magnetic force on 2. B1 If you don’t like being confused, Another nanohuman riding on proton 2 would say “I am close your eyes at rest, so there is no magnetic force on my proton, and iscover your even though there a magnetic fieldears. from proton 1.” This calculation says there is a magnetic field and force. Who is here, right? Take Physics 2305/107 totolearn Or see here, and here for a hint about how resolvethe the answer. paradox. 2 FB r 1 z r̂ v0 x