These notes are a brief synopsis of the electricity and magnetism course based on the Physics for Scientists and Engineers by Serway and Beichner book. Magnetism: A magnetic field is in many ways similar to an electric field. It has a strength and direction. It is measured in Teslas T. Its force dies off with the square of the distance r. A magnetic field can come from a permanent magnet or it can be generated by a moving charge. The magnetic field is measured by the force it generates on a moving charge. The force depends on the velocity of the charge and its magnitude and its angle of motion relative to the field. FB q vBsin N N 1T C m s A m The magnetism created by a moving charge i.e. current flowing in a wire is called electromagnetism. Electromagnetism is the basis of many modern technologies such as microphones, loudspeakers, motors, generators and transformers. One of the key points to remember is that magnetism is caused by the movement of electric charges which in turn is caused by electric fields. Conversely moving magnetic fields will cause electric fields to be generated under the right circumstances. The strength and direction of a magnetic field caused by a current in a wire can be calculated using the Biot-Savart law. o Ids r ˆ 0 4 10 7 T.m / A dB 4 r 2 This law by itself tells us the strength and direction of the field at some point near a current carrying conductor due to the current flowing in a very small segment ds. It does not tell us about the effect of the current in the whole wire. To find this effect we need to integrate the contributions to the field from all of the segments of the wire. In the general case this is fairly difficult but there are a few specific cases where the integration can be done analytically. These cases are the straight wire and the circular segment. For the straight wire the equation becomes: 0 I B cos1 cos 2 4a where a is the perpendicular distance from the wire to the point of interest and 1 2 are the angles between the ends of the wire and the point of interest. Remember to always measure these angles clockwise from 0. The direction of the field is given by the right hand rule. Grasp the wire in your right hand with your thumb pointing in the direction of the current flow and your fingers will point in the direction of the field. Field directions are indicated with X to indicate it is going away from you (imagine the tail flight of an arrow) and a dot to indicate the field is coming toward you (imagine the point of the arrow) Field lines point from North poles to South poles. In the case of a very long straight wire the equation above reduces to 0 I B 2a because the angles 1 2 tend towards 0 and 180 respectively. In the case of the circular loop the field at the centre of the loop is given by : 0 I B 4R Where is the portion of the full circle in radians. For a full circle loop therefore the equation becomes : 0 I B 2R The Biot-Savart law applies to the general case but can be very difficult to apply at times. Ampere’s Law can be used to simplify many situations. Ampere’s name was given to the definition of the Amp which is the unit of current. Symbol I. When two parallel conductors carry a current they both create a magnetic field. The current in one will experience a force due to the field of the other and vice versa. The size of this force is given by: II F o 1 2 l This force is used as in the definition of the Amp 2a When the force per unit length between two long conductors separated by 1m is 2 x10-7 N/m the current in the conductors is 1 amp. Ampere’s law then goes on to show that the line integral of the magnetic field along any closed path is dependant only on the total current passing through the area enclosed by that path and is given by oI. This law can be used to deduce the magnetic field inside and outside a conductor. It can also be used to deduce the field inside a toroid and a solenoid. Inside the ring there is no field because no current is enclosed. Outside the whole device there is equal current flowing into and out of the plane of the page so the total is zero also. Along any line within the body of the toroid the otal current enclosed is given by the current I x the number of turns N. Inside an ideal solenoid the magnetic field is uniform in strength and direction. The strength is given by: B 0 nI Where n is the number of turns per unit length of the solenoid. An ideal solenoid is one which is much longer than it is wide and where each turn is exactly touching the previous one. Magnetic Flux: This is the term used to describe the total amount of magnetic field passing through a surface. It is measured in Webers Symbol W and is Tesla per meter squared. Flux depends on the strength of the field and the area of interest and on the cos of the angle between the area and the direction of the field thus: B BAcos Magnetism in Materials: When a magnetic field interacts with materials other than vacuum the strength of the field will be altered by a certain amount. This amount depends on the magnetic susceptibility of the material. The material can enhance the field if it is Paramagnetic or decrease the field if it is diamagnetic. This effect works by changing the 0 term in the equation. m = 0(1+susceptibility). For ferrous materials the effect is thousands of times stronger and it is then called ferromagnetism. Ferromagnetism is not a linear effect however because the contribution to the field strength will saturate when the strength gets too high. Faraday’s Law: Faraday discovered that a changing magnetic flux will induce a voltage in a nearby conductor. The size of this voltage or emf depends on the rate of change of flux and also on the number of turns of wire involved in the case of a coil. dB E N dt This law becomes the basis for many other effects. Lenz’s Law, self induction, mutual induction and transformer action all spring directly from this. Lenz’s law says that a changing magnetic flux will induce a current in a loop in such a sense as to oppose the changing flux. d B E ds dt This also gives rise to another idea which says that a changing magnetic flux will produce a circular electric field around itself even when there is no material present. This will become an important part of electromagnetism later on. The changing flux around a coil in which the current is changing will create a voltage in that coil which opposes the change in current. This voltage is called a back emf and is the basic cause of self inductance L. L Vs L 1H 1 dI dt A Resistors: Resistors in series add. Rtotal = R1 +R2 +R3… Resistors in parallel add as their inverses. Rtotal = 1/(1/ R1 + R2 + R3 + …) For pairs of resistors the Product/Sum rule can be used as a short cut for calculation. Resistors are measured in Ohms symbol Resistor values are usually tens, hundreds or thousands of Ohms. The abbreviations used are k for thousands(103) and M for millions (106). Sometimes simply k or M are used without the symbol. Sometimes the k or M are used in place of a decimal point so 1k5 stands for 1500 Ohms. Power in a resistor is measured in Watts or Joules per second. This can be calculated in several ways. P = VxI P = I2R P = V2/R where I stands for current and V stands for voltage. In a circuit containing a mixture of parallel and series resistors you can calculate the total resistance by reducing each parallel combination to a single resistor first and then combining the result with the series components. Capacitors: Capacitors in series add as their inverses. See parallel resistors above. Capacitors in parallel add. Capacitance is measured in Farads F Capacitance values are usually very small, much less than a Farad. Typical abbreviations used are milli mF 10-3, micro F 10-6, nano nF 10-9, pico pF 10-12. Capacitors act as energy stores and the energy is measured in Joules and is given by: 1 U CV 2 2 The voltage on a capacitor does not rise to its final value instantly. Current must flow into it through a resistor before the voltage on the plates rises. The charge or voltage on a capacitor at any time after the switch has been closed can be calculated by: t q( t ) Q(1 e RC ) where Q is the maximum charge = C x V and R is in Ohms and C is in Farads. Maximum current flows in at switch on and is simply V/R. The value RC is called the time constant of the circuit and is measured in seconds. The capacitor is said to be fully charged after 5 time constants i.e. 5RC. Capacitors will allow an alternating current to pass through. Because the voltage is constantly changing the capacitor is continually charging and discharging so a current is passing. The size of the current is determined by a property of the capacitor called the reactance XC . This is measured in Ohms like a resistor and to all intents a capacitor acts like a resistor for AC. Capacitive reactance XC = 1/2FC = 1/C. It falls with rising frequency. The current that flows in a capacitor in an AC circuit is 90o out of phase with the voltage. The current leads the voltage. Inductors: Inductors in series add. Inductors in parallel add as their inverses. See resistors above. The unit of inductance is the Henry symbol L. Milli, micro etc. apply as for capacitors. Inductors can be used as energy stores. The energy is stored in the magnetic field around a current carrying inductor. The energy is measured in Joules and is given by: 1 U LI 2 2 The current through an inductor does not rise immediately to its maximum value. Work must be done to change the current against the back emf. The current flowing in an inductor at any time t after switch on can be calculated by: t L I t I (m ax)e R where I(max) is the maximum current that will flow in the circuit and is given by V/R. L is in Henries and R is in Ohms and t is in seconds. The current rises in exactly the same way as the charge in the capacitor. (See above). The term L/R is called the time constant for the circuit and the current will reach its maximum value in 5L/R. Inductors will oppose the flow of AC current and behave like resistors in this respect. They have a property called inductive reactance XL which is measured in Ohms and is given by XL = 2FL = L. XL rises with rising frequency. The current in an inductor in an AC circuit will be out of phase with the applied voltage. The current in an inductor lags the applied voltage by 90o RCL Circuits: When R,C and L are combined we generally use phasor diagrams to calculate the overall opposition to the flow of AC current. The term used for the combination of Reactance and resistance is called impedance. Impedance will have a value measured in Ohms and a phase term measured in degrees or radians. Impedance is given by: Z R 2 ( XL XC) 2 X XC And the phase term is given by : tan 1 ( L ) R The result of these ideas leads to resonance of an LC circuit. There are two ways to imagine resonance in this case. 1) Energy stored in a capacitor is analogous to the potential energy in a pendulum displaced from its equilibrium position and energy stored in an inductor is analogous to the kinetic energy of a pendulum as it moves through its arc. Energy swaps back and forth between the two components and gradually dissipates due to resistance (friction). In either case if we add small bursts of energy at the correct rate then the strength of the oscillation will build up. The 1 natural resonance frequency for an LC circuit is given by 0 . LC The other way to think of resonance is that when XC and XL are equal they cancel out and the only component of the impedance is the resistance R. At the frequency where they cancel current in the circuit will rise to a maximum. Transformers: If you pass an AC current through a coil then you will produce a changing magnetic field in its vicinity. If this field intersects a second coil it will induce an AC voltage in that coil. The size of the induced voltage depends on the rate of change of flux and on the number of turns in the second coil. This arrangement is called a transformer and is used to convert voltages from one level to another. I1V1 I 2 V2 N2 V2 V1 N1 The interesting point here is that the power (VI) on both sides is the same. So if the voltage goes down the current goes up and vice versa. When power is transmitted electrically some is wasted as heat in the resistance of the connecting wires. Maximising the voltage minimises the current and so minimises the power loss. This is why very high voltage is used to transmit power around the country on the national grid. Transformers are used in your local area to reduce this voltage to a safe level of 230VAC for use in your house. These Notes should be used as a guide to study of the Physics text by Serway. Sample Questions are available on the Experimental Physics web site.
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