Magnetic FieldsMagnetismMagnetic FieldCathode Ray TubeDefinition of the Magnetic FieldDirection of the Magnetic FieldRight-Hand RuleMagnetic Fields and WorkLorentz Force Equatione/m Determination of the Electronthe Carrier of ElectricityReview: Charge of the ElectronMagnetic Force and Circular MotionOrbital FrequencyCyclotrons and SynchrotronsHall EffectMagnetic Force on a Current-Carrying WireTorque on a Current LoopPHY2061 Enriched Physics 2 Lecture Notes Magnetic FieldsMagnetic FieldsDisclaimer: These lecture notes are not meant to replace the course textbook. The content may be incomplete. Some topics may be unclear. These notes are only meant to be a study aid and a supplement to your own notes. Please report any inaccuracies to the professor.MagnetismIs ubiquitous in every-day life!- Refrigerator magnets (who could live without them?)- Coils that deflect the electron beam in a CRT television or monitor- Cassette tape storage (audio or digital)- Computer disk drive storage- Electromagnet for Magnetic Resonant Imaging (MRI)Magnetic FieldMagnets contain two poles: “north” and “south”. The force between like-poles repels (north-north, south-south), while opposite poles attract (north-south). This is reminiscent of the electric force between two charged objects (which can have positive or negative charge). Recall that the electric field was invoked to explain the “action at a distance” effect of theelectric force, and was defined by:elq=FEwhere qel is electric charge of a positive test charge and F is the force acting on it.We might be tempted to define the same for the magnetic field, and write:magq=FBwhere qmag is the “magnetic charge” of a positive test charge and F is the force acting on it. However, such a single magnetic charge, a “magnetic monopole,” has never been observed experimentally! You cannot break a bar magnet in half to get just a north pole ora south pole. As far as we know, no such single magnetic charges exist in the universe, D. Acosta Page 1 1/14/2019PHY2061 Enriched Physics 2 Lecture Notes Magnetic Fieldsalthough we continue to look. Thus, we must look for other interactions with magnetic force to define the magnetic field.It turns out that an electrically charged object can also be accelerated by a magnetic force, and through that interaction we can define the magnetic field. In fact, the electric and magnetic force share a much deeper relation. They are really manifestations of the same force, and can be shown to be related by transformations in Einstein’s theory of Special Relativity. But here let us discuss the historical perspective. Cathode Ray TubeConsider a “Crooke’s Tube”, which is otherwise known as a Cathode Ray Tube (CRT) – aprimitive version of what is in a television. Such a CRT has an electron gun that accelerates electrons between two electrodes with a large electric potential difference between them (and a hole in the far plate). The resulting beam of electrons can be rendered visible with a phosphorous screen, and then we can observe the effects on the deflection of the beam in magnetic fields.From such experiments we can determine several characteristics of electrically charged particles interacting with magnets:- The force depends on the direction of the magnetic field (i.e. whether it emanates from a north pole or a south pole).- The force is perpendicular to both the velocity and magnetic field directions- The force is zero if the particle velocity is zero (and depends on the sign of v)- The force depends on the sign of the electric chargeDefinition of the Magnetic FieldThus, we will converge on the following relation for the magnitude of the magnetic force on a charged object:sinBq f=F v Bor, turned around, allows us to define the magnitude of the magnetic field as:sinBq f=FBvD. Acosta Page 2 1/14/2019e-BPHY2061 Enriched Physics 2 Lecture Notes Magnetic Fieldswhere - is the angle between the velocity, v, and the magnetic field, B.The units of the magnetic field are N N N T (Tesla)m CA-mC ms s= = �Another unit based on the cgs metric system is the Gauss, where 41 G = 10 T-. The Earth’s magnetic field has a magnitude of approximately 0.5 G.Direction of the Magnetic FieldWhat direction do we assign to the magnetic field to insert into the force equation? We imagine field lines the are directed outward from the north pole of a magnet, and inward to the south pole.Now in full vector form, we write the expression for the magnetic force acting on a electrically charged particle as:Bq= �F v BWe can see that it will satisfy all the empirical observations noted earlier.The vector cross-product is defined by:( )( )( )ˆ ˆ ˆˆ ˆ ˆdetsinx y z y z y z x z x z x y x yx y za a a a b b a a b b a a b b ab b bab f� =- � = = - - - + -� =x y za b b a x y za bD. Acosta Page 3 1/14/2019From HRW 7/ePHY2061 Enriched Physics 2 Lecture Notes Magnetic FieldsRight-Hand RuleIf we consider the following simple example, that 0ˆv=v x and 0ˆB=B y, then we see that the magnetic force direction is given by a right-hand rule: 0 0ˆF qv B= zMagnetic Fields and WorkLet’s calculate the work done by the magnetic force acts on a moving charged particle that moves from point 1 to point 2:21W d� ��F sWe can re-write d dt=s v to get:( )2 21 1W dt q dt� � = � �� �F v v B vNow with what we learned about the magnetic force, it is always perpendicular to the velocity vector, so in fact the vector quantity ( )q �v B is zero. So W=0 and no work is done by the magnetic field! You can remember this simply as “Magnetic fields do no work.” So apparently the magnetic force is on the physics welfare system!This implies that there is no change in energy of a charged particle being accelerated by amagnetic force, only a change in direction. We will come back to this when we discuss magnetic fields and circular motion.Lorentz Force EquationWe can combine what we learned about the electric force to that we just learned about themagnetic force into one equation, the Lorentz force equation:( )q= + �F E v BWe can apply this to some historical work done by J.J. Thompson, who determined that the carrier of electricity was an electrically charged particle dubbed the “electron.”D. Acosta Page 4 1/14/2019PHY2061 Enriched Physics 2 Lecture Notes Magnetic Fieldse/m Determination of the Electronthe Carrier of ElectricityIt was determined by J.J. Thompson in 1897 that cathode rays are charged particles emitted
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