Electromagnetism Lecture#06 MUHAMMAD MATEEN YAQOOB THE UNIVERSITY OF LAHORE SARGODHA CAMPUS

Faraday’s Law The electric fields and magnetic fields considered up to now have been produced by stationary charges and moving charges respectively. Imposing an electric field on a conductor gives rise to a current which in turn generates a magnetic field. In 1831, Michael Faraday discovered that, by varying magnetic field with time, an electric field could be generated. The phenomenon is known as electromagnetic induction. Faraday’s experiment demonstrates that an electric current is induced in the loop by changing the magnetic field. The coil behaves as if it were connected to a source. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Consider a uniform magnetic field passing through a surface S The magnetic flux through the surface is given by Faraday’s law of induction may be stated as: The induced emf ε in a coil is proportional to the negative of the rate of change of magnetic flux For a coil that consists of N loops MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Lenz’s Law The direction of the induced current is determined by Lenz’s law To illustrate how Lenz’s law works, let’s consider a conducting loop placed in a magnetic field. We follow the procedure below: MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Ampere’s Law We have seen that moving charges or currents are the source of magnetism. This can be readily demonstrated by placing compass needles near a wire. As shown in Figure, all compass needles point in the same direction in the absence of current. However, when I is non zero, the needles will be deflected along the tangential direction of the circular path. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Field inside and outside a Current- carrying wire Consider a long straight wire of radius R carrying a current I of uniform current density, as shown in Figure. Find the magnetic field everywhere. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Solution MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Magnetic flux (Φ) The group of force lines going from north pole to south pole of a magnet is called magnetic flux Number of lines of force in a magnetic field determines the value of flux Unit of magnetic flux is Weber (Wb) One weber is 10 8 lines It is a huge unit; so in most of applications micro-weber (µWb) is used MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Magnetic flux density (B) It is the amount of flux per unit area perpendicular to the magnetic field Its symbol is B and its unit is Tesla (T) One tesla equals one weber per square meter (Wb/m 2 ) B = Φ / A MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Inductor An inductor is a passive element designed to store energy in its magnetic field. Inductors find numerous applications in electronic and power systems. They are used in power supplies, transformers, radios, TVs, radars and electric motors. Any conductor of electric current has inductive properties and may be regarded as an inductor. But in order to enhance the inductive effect, a practical inductor is usually formed into a cylindrical coil with many turns of conducting wire. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Inductor An inductor is made of a coil of conducting wire Inductors are formed with wire tightly wrapped around a solid central core MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Inductance Inductance (or electric inductance) is a measure of the amount of magnetic flux produced for a given electric current. The inductance has the following relationship: L= Φ/i where ◦L is the inductance in henrys, ◦i is the current in amperes, ◦Φ is the magnetic flux in webers MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

If current is allowed to pass through an inductor, it is found that the voltage across the inductor is directly proportional to the time rate of change of the current. Using the passive sign convention, where L is the constant of proportionality called the inductance of the inductor. The unit of inductance is the henry (H), named in honor of the American inventor Joseph Henry (1797–1878). MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

I-V Relation of Inductors An inductor consists of a coil of conducting wire. + - v i L Figure shows this relationship graphically for an inductor whose inductance is independent of current. Such an inductor is known as a linear inductor. For a nonlinear inductor, the plot of Eq. will not be a straight line because its inductance varies with current. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Flux in Inductors The relation between the flux in inductor and the current through the inductor is given below. i φ Linear Nonlinear MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

+ - v L where i(t 0 ) is the total current for −∞ < t < t 0 and i(−∞) = 0. The idea of making i(−∞) = 0 is practical and reasonable, because there must be a time in the past when there was no current in the inductor. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

The inductor is designed to store energy in its magnetic field The energy stored in an inductor + - v L MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Important properties of inductor When the current through an inductor is a constant, then the voltage across the inductor is zero, same as a short circuit. An inductor acts like a short circuit to dc. The current through an inductor cannot change instantaneously. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Example 1 The current through a 0.1-H inductor is i(t) = 10te -5t A. Find the voltage across the inductor and the energy stored in it. Solution: MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Assignment # 2 Date of submission: On the day of mid term paper MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Assignment # 2 Consider the circuit in Fig (a). Under dc conditions, find: (a) i, v C, and i L. (b) the energy stored in the capacitor and inductor.

Example 2 Find the current through a 5-H inductor if the voltage across it is Also find the energy stored within 0 < t < 5s. Assume i(0)=0. Solution: MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Example 2 MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Inductors in Series MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Series Inductor Applying KVL to the loop, Substituting v k = L k di/dt results in MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Inductors in Parallel MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Parallel Inductors Using KCL, But MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Example 3 Find the equivalent inductance of the circuit shown in Fig. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Example 3 Solution: MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Example 4 For the circuit in Fig, If find : MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Solution MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Applications of Capacitors and Inductors Circuit elements such as resistors and capacitors are commercially available in either discrete form or integrated-circuit (IC) form. Unlike capacitors and resistors, inductors with appreciable inductance are difficult to produce on IC substrates. Therefore, inductors (coils) usually come in discrete form and tend to be more bulky and expensive. For this reason, inductors are not as versatile as capacitors and resistors, and they are more limited in applications. However, there are several applications in which inductors have no practical substitute. They are routinely used in relays, delays, sensing devices, pick-up heads, telephone circuits, radio and TV receivers, power supplies, electric motors, microphones, and loudspeakers, to mention a few. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Introduction The term alternating indicates only that the waveform alternates between two prescribed levels in a set time sequence.

Sinusoidal AC Voltage Characteristics and Definitions Generation An ac generator (or alternator) powered by water power, gas, or nuclear fusion is the primary component in the energy-conversion process. The energy source turns a rotor (constructed of alternating magnetic poles) inside a set of windings housed in the stator (the stationary part of the dynamo) and will induce voltage across the windings of the stator. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Sinusoidal AC Voltage Characteristics and Definitions Generation Wind power and solar power energy are receiving increased interest from various districts of the world. The turning propellers of the wind-power station are connected directly to the shaft of an ac generator. Light energy in the form of photons can be absorbed by solar cells. Solar cells produce dc, which can be electronically converted to ac with an inverter. A function generator, as used in the lab, can generate and control alternating waveforms. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Sinusoidal AC Voltage Characteristics and Definitions Definitions Waveform: The path traced by a quantity, such as voltage, plotted as a function of some variable such as time, position, degree, radius, temperature and so on. Instantaneous value: The magnitude of a waveform at any instant of time; denoted by the lowercase letters (e 1, e 2 ). Peak amplitude: The maximum value of the waveform as measured from its average (or mean) value, denoted by the uppercase letters E m (source of voltage) and V m (voltage drop across a load). MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Sinusoidal AC Voltage Characteristics and Definitions Definitions Peak value: The maximum instantaneous value of a function as measured from zero-volt level. Peak-to-peak value: Denoted by E p-p or V p-p, the full voltage between positive and negative peaks of the waveform, that is, the sum of the magnitude of the positive and negative peaks. Periodic waveform: A waveform that continually repeats itself after the same time interval. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Sinusoidal AC Voltage Characteristics and Definitions Definitions Period (T): The time interval between successive repetitions of a periodic waveform (the period T 1 = T 2 = T 3 ), as long as successive similar points of the periodic waveform are used in determining T Cycle: The portion of a waveform contained in one period of time Frequency: (Hertz) the number of cycles that occur in 1 s MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Amplitude PEAK AMPLITUDE PEAK-TO-PEAK AMPLITUDE MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Instantaneous Value Instantaneous value or amplitude is the magnitude of the sinusoid at a point in time. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Average Value The average value of a sinusoid signal is the integral of the sine wave over one full cycle. This is always equal to zero. ◦If the average of an ac signal is not zero, then there is a dc component known as a DC offset. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Root Mean Square (RMS) Most equipment that measure the amplitude of a sinusoidal signal displays the results as a root mean square value. This is signified by the unit Vac or V RMS. ◦RMS voltage and current are used to calculate the average power associated with the voltage or current signal in one cycle. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Current (I) Electrical current is the rate of flow of charges where: I = current in amperes (A) Q = charge in coulombs (C) t = time in seconds (s) the rate of flow of charge. Random motion of free electrons in a material. Electrons flow from negative to positive when a voltage is applied across a conductive or semiconductive material. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Definition of Current One ampere (1 A) is the amount of current that exists when a number of electrons having a total charge of one coulomb (1 C) move through a given cross-sectional area in one second (1 s). MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Example MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Resistance (R) Resistance is the opposition to current. Definition of resistance One ohm (1 Ω) of resistance exists if there is one ampere (1 A) of current in a material when one volt (1 V) is applied across the material. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Conductance (G) The reciprocal of resistance is conductance, symbolized by G. It is a measure of the ease with which current is established. The formula is Unit is siemens. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Types of Resistor Fixed Resistor MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Carbon-composition resistor This resistor is made with a mixture of finely ground carbon, insulating filler, and a resin binder. The ratio of carbon to insulating filler sets the resistance value. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Resistor Color Code MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Resistor 4-band color code MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Example What is the resistance and tolerance of each of the four-band resistors? 5.1 k  ± 5%  k  ± 5% 47  ± 10% 1.0  ± 5% Tolerance= 0.255KΩ MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Variable Resistor Variable resistors include the potentiometer and rheostat. A potentiometer can be connected as a rheostat The center terminal is connected to the wiper MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE