1 MAGNETISM A magnetic field is a vector field that permeates space and which can exert a magnetic force on moving electric charges and on magnetic dipoles. We define the magnitude of the magnetic field by measuring the magnetic force on a moving cha rge q: B = 1 T(Tesla) = Direction at any location is the direction in which the north pole of the compass needl e points at that location. Outside magnet: N → S Inside magnet: S → N (always closed loops) 1. An electric charge experiences a magnetic force when moving in a magnetic field. Magnetic force acting on a charge q Magnetic force on a wire carrying current I in a magnetic Field B: F = qvB sinin a magnetic field B: F = I LB sin q = charge [C] I = current [A] v = velocity [m/s] L = length [m] B = magnetic field [Tesla T] B = magnetic field [T] = angle between v and B = angle between I and B R-H-R 1: The direction of the magnetic force on a charge/current is given by the right-hand rule 1: Outstretch fingers in the direction of v (or current I). Curl fingers as if rotating vector v (I ) into vector B. Magnetic force on a positive charge (or I) is in the direction of the thumb. Magnetic force on a negative charge points in opposite direction. Charge q in elec. field E and mag. field B The electric force: Felec = Eq The magnetic force: Fmag = qvB sin ● is always parallel to the direction of the ● is always perpendicular to the direction of the electric field. magnetic field ● acts on a charged particle only when the particle is in motion ● acts on a charged particle independent of the and only if v and B do not point in the same or opposite particle’s velocity (even at rest). direction (sin 00 = sin 1800 = 0). ● does the work when moving charge. ● Force is perpendicular to the direction of the motion, The work, W = Fel d cos θ1, is converted into kinetic so the work done by magnetic force is zero. energy which is, in the case of conductors, 0 W = Fmag d cos θ1 = 0 (cos 90 = 0). transferred to thermal energy through collisions W = ΔKE = 0 with the lattice ions causing increased amplitude of vibrations seen as temperature rise. Hence change in kinetic energy of the charge is 0, and that means that mag. force cannot change the speed of the charge. Magnetic force can only change direction of the velocity – therefore it acts as centripetal force. θ1 is angle between F and direction of motion 2 Examples of the Lorentz Force Two important applications of the Lorentz force are 1) the trajectory of a charged particle in a uniform magnetic field and 2) the force on a current-carrying conductor. 1) The trajectory of a charge q in a uniform magnetic field B ● Force is perpendicular to B,v ● Magnetic force does no work! (W = F d cos θ1 = 0 ) ● Speed is constant (W = Δ KE = 0 ) ● Circular motion Centripetal force definition: Motion in a circle represents accelerated motion, and requires a force directed toward the center of the circle. This force is called the centripetal force which means "center seeking" force. It is forcing mass m to move in the circle of radius R with the speed v . Centripetal force force has the magnitude Charged particle in a magnetic field when v B: In the case the charge q is subject to the uniform field B, centripetal force Fc is magnetic force forcing the charge to move in a circle: Positive charge q in magnetic field B B = magnetic field [T] ● massive or fast charges – large circles is represented by the crosses – into the page ● large charges and/or large B – small circles R =is the radius of the path F is magnetic force on the charge directed toward the centre of the circular path m = mass [kg] v = velocity [m/s] q = charge [C] 3 APPLICATION: A mass spectrometer – the path shown is for positive ions Step 0 – if we want to find a mass of molecules or atoms we have to ionize them first. Step 1 – Acceleration through potential difference ΔV ΔU = q ΔV = ½ mv2 v = tep 2 - the velocity selector – Crossed fields The ions emerge from the acceleration stage with a range of speeds (different masses – different accelerations). Velocity selector/crossed fields is designed to allow ions of only a particular velocity to pass through undeflected. Two forces are acting on the charge q Felec = qE and Fmag = qvB if qvB = qE the charge moves through crossed fields undeflected so only ions with speed v = E/B are undeflected and will move in the straight line Step 3 - mass separation All these ions, with the same charge and velocity, enter the mass separation stage - a region with a uniform magnetic field. Magnetic field will force the ion entering it with velocity perpendicular to it, to travel in a circular path. The only thing different for these particles is the mass, so the heavier ions travel in a circular path of larger radius than the lighter ones. Charged particle in a magnetic field when v B: mv 2 qvB R massive or fast charges – large circles large charges and/or large B – small circles mv R qB 2. A moving charge produces a magnetic field. R-H-R 2: The direction of the magnetic field produced by electric current is given by the right-hand rule 2: If a wire is grasped in the right hand with the thumb in the direction of current flow, the fingers will curl in the direction of the magnetic field. 4 Magnetic field B around a wire with current I Magnetic Field B Inside of a Solenoid B = B = 0 n I -7 = the permeability of free space 4×10 T·m/A The magnetic field is concentrated I = current [A] into a nearly uniform field in the r = distance from the center of the conductor centre of a long solenoid. The field n = N/L number of turns of wire per unit length outside is weak and diverging F F I I Magnetic Force per unit length between Two Parallel Wires: 1 2 0 1 2 L L 2 d Current I1 creates magnetic field B1 Place a wire with current I2 at distance d from the wire at distance d from I1 B1 = The force on I2 is: F2 = L I2 B1 = L I2 F2 = L (a) two parallel currents (b) two antiparallel curren ts – the force between them is attractive – the force between them is repulsive One Ampere is defined as that current flowing in each of two infinitely-long parallel wires of negligible cross-sectional area separated by a distance of one metre in a vacuum that results in a force of exactly 2 x 10-7 N per metre of length of each wire. Electricity can be produced from many different types of energy, and all of these methods make use of a generator to convert mechanical energy to electrical energy. Electrical generator Electrical energy is usually produced by rotating coil in a uniform magnetic field. Free electrons in a wire will experience magnetic force if the wire moves in a magnetic field; the electrons inside it experience a force causing them to move to one end of the wire. This causes a potential difference along the wire which can be used to create current in a circuit. We say that current has been induced in the wire. 5 A generator uses the same principle, but instead of the wire moving in a straight line, a coil rotates in the magnetic field. As the coil rotates, resulting in a current that keeps changing direction, called an alternating current or AC for short. To prevent the wires from getting twisted when the coil rotates, there must be sliding contacts between the coil and circuit. However, friction between these sliding components will eventually wear away the contacts. An alternative arrangement is for the magnets to rotate inside the coils, this also results in an alternating current but has no sliding parts Electric Motor You find it everywhere: turntables, washing machined, any fan, air conditioners, wherever something is moving there is an electric motor. There are many coils of wires, but b/c of simplicity we drew only one. Each end of the coil is attached to a metallic half-ring. Rubbing against each of the half-rings is a graphite contact called a brush. Half–rings rotate with the coil; the graphite brushes remain stationary. Left picture: the current from the battery enters the coil through the left brush, goes around the coil, and leaves through the right half-ring and brush. Forces on two vertical sides result in rotation (turning) of the coil. The coil rotates until it riches the position shown in the middle picture. In this position the half rings momentarily lose electrical contact with the brushes: no current – no force, no ability of the force to cause rotation (so called torque). But, b/c of its rotational inertia (moving object does not stop momentarily) the coil continues to rotate, and half-rings reestablish contact with the brushes. So there is again current in the coil, but now in opposite direction (the side that had current upwards, now has current downwards b/c it is connected to different terminal). The split ring ensures that the current is always in the proper direction such that the torque produces a continuous rotation of the coil. 6 PRACTICE MAGNETISM 1 1.