1 Anything new Magnetic field lines and magnetic fluxThe magnetic field is a vector field We can introduce magnetic field lines in the same we introduced electric field lines Let’s recall: Properties of field lines: Imaginary curve such that tangent at any point is along the B-field in this point density of field lines in a given region allows to picture the magnitude of the B-field At any particular point in space the B-field has a well defined direction Only one field line can pass through each point Field lines never cross Anything new Click Wastson

2 Examples of magnetic field linesMagnetic field lines are not force lines ! So what do they visualize then: Magnetic field lines have the direction that a compass needle would point at each location Examples of magnetic field lines Right hand rule gives direction of field Current through wire Permanent magnet

3 Clicker question What happens if you cut a permanent magnet in half?You get a separate north pole and south pole similar to electric plus and minus charges 2) There is no magnetic charge. Any permanent magnet has two poles, if you cut a magnetic dipole in half you end up with two dipoles.

4 Magnetic flux The magnetic field is a vector fieldwe can define a magnetic flux Remember: Flux : scalar quantity, , which results from a surface integration over a vector field. magnetic flux through surface vector field The SI unit of magnetic flux surface Wb read Weber in honor of Wilhelm Weber

5 Is there something like Gauss’ law for magnetic fluxYes, and it surprisingly simple with deep fundamental meaning Remember Gauss law for electric flux and let’s apply it by enclosing electric dipoles: Since the total enclosed charge is zero we have has never been observed (and that is why physicists keep looking for it -> “I want my Nobel prize” ) Since there is no such thing as magnetic monopoles The magnetic flux through any closed surface is zero (magnetic monopoles have never been observed, magnetic field lines always close ) If you don’t feel sufficiently confused yet read also “Have physicists seen magnetic monopoles?”

6 Magnetic force on a current carrying conductorLet’s consider a conducting wire carrying a current I=jA in a B-Field  to the current density vector j Lorentz force on an individual charge q with drift velocity vd (remember Drude model) reads For N charges we have therefore a total force In the wire of volume V=l A we have N=n lA charges q using the transport expression for the current density If B makes an angle  with the wire

7 If the conductor is not straightWhere l is a vector pointing in the direction of the current and has the magnitude l If the conductor is not straight consider infinitesimal short segments contributing with Example: Magnetic force on a curved conductor Let’s find the total magnetic force on the conductor Start with the straight segment:

8 Curved part: The force a straight wire of length l+2R would experience. That makes sense when considering the symmetry. Luke, the Force runs strong in your family. Pass on what you have learned. If you think you deserve a break see also

9 Force and torque on a current loopwith r2 with Image from our textbook Young and Freedman Magnitude of torque on current loop

10 Let’s generalize into a full vector notation=: the absolute value of the magnetic moment of the loop We are used to assign a vector to an area It appears natural to define a vector A is normal to the loop area. Its direction is determined by the right hand rule Since  is the angle between and B Vector torque of on current loop This equation is in wonderful analogy to the torque on an electric dipole in an E-field

11 The direct current motorClick here for a java applet https://nationalmaglab.org/education/magnet-academy/watch-play/interactive/dc-motor From the right hand rule we get direction of the magnetic moment B-field points from N to S Determines the direction of 