F = q v B. F = q E + q v B. = q v B F B. F = q vbsinφ. Right Hand Rule. Lorentz. The Magnetic Force. More on Magnetic Force DEMO: 6B-02.

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1 Lorentz = q E + q Right Hand Rule Direction of is perpendicular to plane containing &. If q is positie, has the same sign as x. If q is negatie, has the opposite sign of x. = q = q sinφ is neer parallel to. can only change the direction of the particle not the speed. 10/3/ /3/16 2 More on Magnetic orce The magnetic force on a charged object that moes in a magnetic field does not do any work, because it s perpendicular to. The magnetic force cannot change the magnitude of the elocity of a charged object, but can change the direction of motion. = steering wheel, E = accelerator or brake pedal, so to speak The SI unit for magnetic field is tesla (T): The Magnetic orce = q x x x x x x x x x x x x x x x x x x +q +q DEMO: 6-02 The direction of the force is: +q N N N 1 T = 1 = 1 = 1 C m / s C / s m A m A common unit gauss (G): 1 G = 10-4 ~Earth s surface T field! 10/3/16 3 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x q 10/3/16 4 1

2 Magnetic orce on a Current-Carrying Wire Magnetic orce on a Current-Carrying Wire A wire 3 mm long with a current of 3 A (+x). It lies in a magnetic field of 0.02 T that is in the xy plane and makes an angle of 30 o with the x direction. In which direction & what is the magnitude of the magnetic force on the wire? 10/3/16 5 a) 1.8 x10-5 N in the y b) 1.8 x10-5 N in the +y c) 9.0 x10-5 N in the +z d) 9.0 x10-5 N in the -z e) None of the aboe 10/3/16 6 Top iew of Current-Carrying ar Sliding on two current carrying frictionless rails in a magnetic field. I V + I Motion of a Point Charge in a Magnetic ield cannot change of a charged particle. cannot change the kinetic energy of a charged particle. can only change the direction of a particle. THIS IS A ORM O ELECTRIC MOTOR, TURNING ELECTRICAL INTO MECHANICAL ENERGY I motion by reersing Direction of I, by reersing V Note that this example assumes that the magnetic field caused by the currents in the rails is negligible compared to the external magnetic field shown. The length L is the distance between the rail and is the magnetic field. The current I flows in the green bar. DEMO /3/16 7 glow of ionized gas 10/3/16 8 2

3 Motion of a Point Charge in a Magnetic ield Period of Circular Motion The period of the motion is or, the angular frequency 10/3/ /3/16 10 Mass Spectrometer Ions of different masses can hae the same charge q and the same elocity. If we shoot them to a uniform magnetic field perpendicularly, question: The circular trajectories followed by the ions once they enter the field would show: a) The same radii (the radius has to remain the same) b) Different radii (the radius of the trajectories depend on the mass) c) One cannot tell (we need more information to decide) Mass Spectrometer (Ions with same KE) The purpose of a mass spectrometer is to separate ions by mass and measure the mass of each type of ion. If positie ions start from rest and moe through a potential difference, V, the ions kinetic energy when they enter the magnetic field equals their loss in potential energy: What kind of charge do the ions in the picture hae? 10/3/ /3/

4 Mass Spectrometer (Ions with same KE) Working with both equations : irst sole for the elocity on the first one, Then substitute it on the kinetic energy equation A mass spectrometer can be improed if instead of haing ions with the same kinetic energy entering the field we hae ions with the same elocity. 10/3/16 13 Combine an Electric ield and a Magnetic ield If we shoot charged particles into a region of space that has both an electric and a magnetic field, we would end up with a net electro-magnetic force that is equal to the ector sum of the electric and magnetic forces acting on the charge: = E + = qe + q A ery interesting effect can be achieed when we apply an electric and a magnetic force to a charged particle in such a way that these forces balance. = E + = 0 10/3/16 14 Crossed E and ields Crossed E and ields Question: In which direction is the magnetic force, once the positie charge reaches the region with the field? a) Up b) Down c) into the page Question: If we would like to balance this magnetic force with an electric force, we would hae to apply an electric field in which direction? a) Up b) Down c) into the page 10/3/16 15 This deice is called a Velocity Selector. CONCLUSION: There is only one particular elocity of a + charged particle that will balance the magnetic and electric forces 10/3/

5 Torque on a Current Loop We first hae to define an unambiguous direction of the loop, perpendicular to the plane of the loop nˆ. We do this with our right hand (again) Rectangular Current Loop in a ield Curl your fingers of your right hand in the direction of the current, then your thumb should point in the direction of nˆ 10/3/16 17 ˆn wants to align with : τ = NIabsinθ = NIAsinθ wherea=ab and the formula does NOT depend on the shape of the loop, only on the area A N counts the number of turns of wire in this loop, each turn contributes. 10/3/16 18 Torque on a Magnetic Dipole lat current loop of arbitrary shape area of loop number of turns in loop τ = µ remember an electric dipole in an E field whereτ = p E. 10/3/16 19 Magnetic Dipole in a Uniform ield τ = µ τ = µ sinθ When = 0 o or 180 o, = 0. Howeer, = 180 o is unstable. When a torque is exerted through an angle, work is done. When a dipole is rotated through an angle d" dw = τ dθ = µ sinθdθ du = dw = + µ sinθdθ U = du = µ sinθ = µ cosθ + U 0 U = 0 Choose ( θ 90 ) = 0 U = µ cosθ = µ 0 10/3/

6 Potential Energy of Magnetic Dipole A magnetic dipole has its highest energy when its µ dipole moment is lined up with the magnetic field. U = µ cos180 U = + µ o A magnetic dipole has its lowest energy when its µ dipole moment is lined up with the magnetic field. U = µ U = µ cos0 = µ 10/3/