Physics 11b Lecture #10 Magnetic Fields S&J Chapter 29
What We Did Last Time Electromotive forces (emfs) atteries are made of an emf and an internal resistance Resistor arithmetic R = R + R + R + + R series 1 2 3 n Opposite to capacitors! Kirchhoff s rules I in = Loop rule requires attention to the polarity RC circuits out Time-dependence as I Junction Rule exp loop 1 1 1 1 1 = + + + + Rparallel R1 R2 R3 R n V= 0 t RC Loop Rule We didn t have enough time for this one
Another RC Circuit Initially, the switch is closed and there is steady current Current through R is I = E/R C is holding Q = CE Open the switch at t = 0 C starts to discharge through R Q decreases V decreases I decreases dq() t dt I( t = 0) =E = I() t R V It () = = R Q() t RC + + E I +Q R C Q E R I C +Q Q V Current carries away the charge
Another RC Circuit Combine two equations into one differential equation dq() t 1 = Qt () dt RC I Qt () = Q0 exp t RC We know the initial condition Q(0) = Q = CE Qt () = CE exp 0 t RC oth Q and I are exponentially decreasing + E R C +Q Q dq() t E t It () = = exp dt R RC V
Time Constant Simple RC circuits relax toward the stable equilibrium exponentially Time-dependence of the current and charge has a exp( t/rc) form Product RC has the dimension of time (really?) It s called the time-constant of the circuit Real-world circuits contain Rs and Cs everywhere How fast the circuits can perform their functions often determined by the RC time constant RC circuits are also used in timers, frequency filters, power regulators, etc. We will do more of it with the AC circuits
Today s Goals Magnets North poles and south poles Magnetic field Similar to electric field E Field lines Lorentz force Force on a moving charge in a magnetic field Cyclotron motion What s a cyclotron, anyway? Van Allen belts Force on a current in a magnetic field
Magnets We all know: magnets have N and S poles Opposite poles attract, same poles repel, each other N points north, S points south That means North Pole is S, and South pole is N Similar to electric charges N +, S Difference: magnetic N-S poles cannot be separated A single magnetic charge (= monopole) does not exist Cutting a magnet in the middle results in two magnets, each with N-S poles N S N S N S
Magnetic Field Just like electric field, we can define magnetic field from the force on the N pole of a test magnet Magnetic field lines run from N poles to S poles This is cumbersome because we don t have a monopole Alternative definition of magnetic field uses force on a charged particle in motion That s the Lorentz Force N S N S
Lorentz Force A particle with charge q moving with a velocity v in a magnetic field receives a force F = qv cross-product F v Cross product of two vectors is a vector A = C The length of A is A = C sinθ, where θ is the opening angle between and C. A is perpendicular to both and C, and satisfies the right-hand rule
Lorentz Force Lorentz force is Proportional to the charge q, velocity v, and magnetic field Perpendicular to both v and Use right-hand: thumb along v, index finger along and the middle finger points the direction of F Proportional to sinθ Maximum if v and are perpendicular Zero if v and are parallel Unit of magnetic field is Tesla (T) N From F = qvsinθ Tesla = = Cm/s F = qv N Am F θ v
Cyclotron Motion A charged particle is flying in a uniform field First, assume v and are perpendicular is coming out of the screen Constant sideways acceleration turns the velocity around Particle flies in a circle Centripetal force m F = qv = ma 2 v r = qv r mv = q This circular motion is called the cyclotron motion F q q q v
Cyclotron Frequency How long does it take the particle to make a full circle? T 2π r 2π m = = v q The angular frequency is v q ω = = r m Called the cyclotron frequency It depends on q/m and, but not on the velocity Particle physics experiments use uniform field to make the charged particles curve Radius momentum mv Direction sign of the charge r = F mv q q v
Cyclotron Two D shaped metal containers in a uniform field Apply high voltage between the D s Charged particles between the D s gain velocity, circle around, and come back to the gap after T 2 = π m q same for the same species of particles Switch the polarity of the power supply every T/2 seconds Particles accelerated each time they traverse the gap High-speed particles exit outside the D s + E
Helical Motion What if v is not perpendicular to? Define z axis parallel to F = qv = ( qv, qv,0) y x If you ignore v z, the particle does the same cyclotron motion in the x-y plane No F z v z is constant Particle moves along a helix Angular frequency is unchanged x z θ v = (v x, v y, v z ) y F = (F x, F y, 0) ω = q m
Magnetic ottle Charged particles travel in helix wrapped around magnetic field lines It is also possible to confine the path by designing the field that s weak in the middle and strong at the ends Magnetic bottle is used in nuclear fusion research for confining plasma There is also a big one out there in the sky
Van Allen elt Earth s magnetic field bends and traps charged particles from the space (= cosmic rays) We d die from radiation without it Trapped particles form the Van Allen belt Discovered by Van Allen using a Geiger counter aboard Explorer 1 satellite Edge of the belt closer to Earth near the poles Auroras
Magnetic Force on Current Current I flows in a wire in a uniform field Current carriers are moving in a field Lorentz force carrier F = qv The wire contains nal carriers wire F = nalqv We know from Lecture #8 I = nqva + F v L I F = vˆ = L wire IL I Force on current is proportional to I, L and Direction is found with the right-hand rule
Curved Wires Previous result F = IL applies only to straight wires Curved wires? Cut it into small pieces (as usual) df = Ids Integrate along the wire F = I ds b a a df ds is inside the integral It may not be constant along the wire! b
Uniform Field If field is uniform, the integral simplifies b a ( b ) a F I d I d I = s = s = L L is the vector from a to b Net force on a current due to a uniform field is identical for any path with the same end points Corollary: net force on a closed loop current from a uniform field is zero a I I L b
Summary Magnetic fields are similar to electric fields Only difference: no single magnetic pole Lorentz force Moving charge in a magnetic field is deflected Uniform field Cyclotron motion Cyclotron frequency depends only on q/m and Non-uniform field can trap charged particles Van Allen belt F = qv Lorentz force on current Straight wire: F, generally: = IL If is uniform F = IL again F = I ds b a v ω = = r q m