Chapter 26 - The Magnetic Field

Magnetism Chapter 26 - The Magnetic Field ● Magnetic Field - caused by moving charges and exerts a force on other moving charges. A vector field, given by direction of the north pole of a compass at a given point in space. ● 3D Vector notation - dots indicate vectors coming out of the page and crosses indicate vectors going into the page (perpendicular to the surface). ● Magnetic Field ( ) around a wire circles the current. As viewed from above, when the current is going into the page, the magnetic field circles clockwise around the wire. When the current is going out of the page, the magnetic field circles counterclockwise around the wire. ● Right hand rule for currents - thumb goes along the conventional current; fingers wrap in the direction of the magnetic field. Dots and X’s make such diagrams clearer by indicating 3D vectors with more specificity. ● Field Strengths - and but . The current length ( the source of the magnetic field. In the case of a wire: where ) is , (called the permeability of free space -- note that is the permittivity of free space). ● SI Unit for magnetic field strength - the Tesla (T) --> ● Bar magnets - certain solids can be magnetic in the absence of an overall current: ferromagnetic materials (iron, nickel, cobalt, neodymium) ● Electrons appear to be little magnets (property of magnetic spin). "spin up" or "spin down" ● Domain theory: 1. In some atoms, the electrons have an overall magnetic field, so the atom itself "looks" like a magnet. 2. Domain - a group of atoms magnetically aligned 3. Typically, a lump of iron has many small domains randomly aligned weakly magnetized overall. 4. If you put an object in a strong magnetic field, the domains get bigger and tend to line up, increasing its magnetic field strength. ● What do magnetic fields look like? ● Breaking a magnet creates two smaller ones (no magnetic monopoles). ● Magnetic field lines are always closed loops. They do not begin or end anywhere (unlike electric field lines). There are no magnetic "monopoles" like there are for an electric charge. ● Magnetic force on a current. Since magnetic field from , we get the direction of the (cross product between the length vector and the magnetic field vector). ● To determine the direction of the force, use the right hand slap rule. Fingers align up with the magnetic field and the thumb aligns along the currentlength. Perpendicular from your palm is the direction of the force. ● Since , we can substitute and get . This is the force on a moving charge in a magnetic field. ● In all cases, if the current or charge is negative, reverse the vector effect (direction of the magnetic field or force). ● Cyclotron effect - magnetic force is a centripetal force because it is always perpendicular to the velocity. Magnetic forces don't do work on moving charges as they only affect the direction of a particle's motion, not speed. If a particle moves at an angle to the magnetic field, it will spiral along the magnetic field line. Overall direction of motion is the component of the original velocity that was parallel to the magnetic field. ● Recall the centripetal force (sum of radial forces only). For a charged particle in a magnetic field: ● , thus . Velocity Selector - crossed electric and magnetic fields. In the image below, the magnetic field deflects a positive charge down. The electric field deflects a positive charge up (with parallel plates). For the charge to remain undeflected, through undeflected is , and thus . , and thus the velocity that will pass ● You can either calculate or select out particles of a certain velocity by adjusting the electric and magnetic fields. ● The electric field, velocity, and magnetic fields must all be mutually perpendicular. ● Mass Spectrometer - used to calculate mass of atoms or molecules. ● Particles are often singly or doubly ionized (charge of +e or +2e). Acceleration across some voltage will produce some final velocity. Since then . If you can't calculate this, use , on the velocity selector. Fire into the magnetic field, so cyclotron effect yields mass and thus solving for and combining with once can find the the mass from and then solving for . , , Current Loops and torques - motor A current loop behaves like a bar magnet. It thus tends to line up with an external field. We can wrap our right hand fingers around the loop to get the magnetic field generated by the loop (i.e. - up). OR, you can look at it as having an upward force on the left and a downward force on the right. In a stable state: Close and far wires go opposite now so everything is balanced (note how they are not parallel to the magnetic field now). There is maximum torque when the area vector is perpendicular to the magnetic field. Here, theta is the angle between the area vector and the magnetic field. but there are two forces, and length here is and generally, for a coil of ● number of loops. Hall Effect - if a current bearing conductor is placed in a magnetic field, moving charges will be deflected to one side setting up a voltage, the conductor. Consider a conducting strip width , across the width of , and thickness . Electrons (assuming that they are moving) set an electric field across the width. Quickly, the electric and magnetic forces balance out. ● We know from velocity selectors that in perpendicular magnetic and electric fields, our drift velocity will be defined by Voltage can be defined as follows: . We also know the Hall . Combining gives . This is a simple way to find drift velocity if you know the hall voltage, the magnetic field, and the width. ● If your material is a particular sort of semiconductor, the hall voltage reverses across the width. These are called p-type materials. Implication: some materials have "positive charge carriers", not electrons, transferring current. ● The Hall Effect is also used to measure weak magnetic fields. Recall for chargecarrier density in a material: equation, . Combining with the previous , which solved for magnetic field is .