1.Down the page 2.Up the page 3.To the left 4.To the right 5.Out of the page 6.Into the page The magnetic field points down the page, the velocity of the positively charged particle points to the right. If we want to determine the direction of the deflection of the particle, which direction(s) can we automatically rule out? 5/15/151Oregon State University PH 213, Class #21

The magnetic field points up the page, and the velocity of the negatively charged particle points up the page, which way is the particle deflected? 1.Left 2.Right 3.Out of the page 4.Into the page 5.Not at all 5/15/152Oregon State University PH 213, Class #21

Considering a positive charge in a uniform magnetic field: A.What will be the direction of the force if the velocity is in the x direction and the B-field is in the –x direction? B.How much work is done on a particle moving in a magnetic field? Does it depend on which way it moves? C.Is it possible for a charged particle to move in a straight line when in a B-field, if so, how? 5/15/153Oregon State University PH 213, Class #21

5/15/15Oregon State University PH 213, Class #214 The Motion and Energy of a Charge in a B-Field The magnetic force, F mag, is always perpendicular to the velocity of the moving charge. Therefore, F mag can act by itself as a centripetal force—causing circular motion: F mag = q(v·sin  )B = F C = mv 2 /r where m is the mass of the particle with charge q.

Applications: circular motion F B = ma = mv 2 /rf = v/2πr 5/15/15 5Oregon State University PH 213, Class #21

5/15/15Oregon State University PH 213, Class #216 The Force on a Current in a Magnetic Field What if there are many charges moving through the same magnetic field? Each experiences a force due to the field (assuming they are not moving entirely parallel to the field).

5/15/15Oregon State University PH 213, Class #217 How much is this total force on a current-carrying wire? Look at a steady current in a wire of length L. If some amount of charge,  q, travels that length in a time  t, we know the force on that bit of charge: F =  q(v·sin  )B But v = L/  t, so F =  q[(L/  t)sin  ]B And I =  q/  t, so F = ILBsin 

5/15/15Oregon State University PH 213, Class #218 Application: An Electric Motor One useful application of the force on a current in a wire is the common electric motor. The wire is formed into the shape of a loop, and this loop is placed in a magnetic field:

Figure /15/159Oregon State University PH 213, Class #21

5/15/15Oregon State University PH 213, Class #2110 Magnetic Fields Produced by Currents A charge must be moving (with its motion at least somewhat perpendicular to the field) in order to feel a force from an external magnetic field. But every moving charge also causes a magnetic field—due to its own motion. While that charge cannot feel its own field (i.e. it can-not exert a force on itself), its field can affect other moving charges. How do we describe this mathematically? It’s rare that we would need to measure the magnetic field caused by the motion of an individual point charge (an electron or proton), but we can easily measure the magnetic effect of a whole lot of moving charges—a current.…

The Source of the Magnetic Field: Moving Charges The magnetic field of a charged particle q moving with velocity v is given by the Biot-Savart law: where r is the distance from the charge and θ is the angle between v and r. The Biot-Savart law can be written in terms of the cross product as 5/15/1511Oregon State University PH 213, Class #21

Applications 5/15/15 12Oregon State University PH 213, Class #21

5/15/15Oregon State University PH 213, Class #2113 Magnetic Field of a Current in a Straight Wire At any point in space, the field magnitude B caused by a current I flowing in a long straight wire, is proportional to I and inversely proportional to the radial distance r from the point to the wire: B =  0 I/(2  r) The proportionality here,  0 /2 , uses the definition of the coulomb (since I =  q/  t), and it also reflects the geometry and properties of space itself (much as k does for electric fields: E = kq/r 2 ).  0 = 4  x T·m/A What is the direction of the field due to current in a straight wire? Use Right-Hand Rule # 2: Point your right-hand thumb along the (positive) current direction. Your fingers will then curl around the wire in the direction of the B-field that current is causing. As with all magnetic fields, these lines are closed loops.

Right-hand rule #2: For fields 5/15/1514Oregon State University PH 213, Class #21

The positive charge is moving straight out of the page. What is the direction of the magnetic field at the position of the dot? 1.Left 2.Right 3.Up 4.Down 5/15/1515Oregon State University PH 213, Class #21

The magnetic field at the position P points… A. Into the page. B. Out of the page. C. Down. D. Up. 5/15/1516Oregon State University PH 213, Class #21

5/15/1517Oregon State University PH 213, Class #21

5/15/15Oregon State University PH 213, Class #2118 Here again is the field strength due to current in a straight wire: B =  0 I/(2  r) On a 3-D set of coordinate axes: If a wire along the y-axis carries a current in the positive y-direction, which of these identical charges q, each moving with the same speed v, will experience the greatest magnetic force magnitude? 1. q at (2,0,0), moving parallel to the x-axis. 2. q at (3,0,0), moving parallel to the x-axis. 3. q at (3,0,0), moving parallel to the y-axis. 4. q at (2,0,0), moving parallel to the z-axis. 5. q at (1,0,0), moving parallel to the z-axis.