Electromagnetism Lecture#8-11 Instructor: Engr. Muhammad Mateen Yaqoob

Circuit Excitation Source-free circuits (free of independent sources) DC Source excitation (independent sources) MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

FIRST-ORDER CIRCUIT Three passive elements (resistors, capacitors, and inductors) individually, Circuits having various combinations of two or three of the passive elements. RC and RL circuits. Analysis of RC and RL circuits by applying Kirchhoff’s laws. The differential equations resulting from analyzing RC and RL circuits are of the first order. Hence, the circuits are collectively known as first-order circuits. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

THE SOURCE-FREE RC CIRCUIT MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation. The voltage response of the RC circuit. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

The time constant of a circuit is the time required for the response to decay by a factor of 1/e or 36.8 percent of its initial value. The voltage response of the RC circuit. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

The Key to Working with a Source - free RC Circuit is Findin g : 1. The initial voltage v(0) = V 0 across the capacitor. 2. The time constant τ. v C (t) = v(t) = v(0)e −t/τ other variables Capacitor current i C Resistor voltage v R Resistor current i R can be determined. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Applications of RC and RL circuits RC and RL Circuits have many applications in the field of Electrical, Electronics, Communication, Computer Engineering, Signal Processing and so on These circuits have many practical applications, some of their major applications are listed below: 1.Amplifiers 2.Oscillators 3.Filters 4.Switching Regulator 5.Tuned Amplifiers 6.Radio Transmitter and Receiver 7.TV Receiver MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Magnetic Fields In 1269 a Frenchman named Pierre de Maricourt found that directions of a needle near a spherical natural magnet formed lines that encircled sphere and passed through two points diametrically opposite each other, which he called poles of magnet. Subsequent experiments showed that every magnet, regardless of its shape, has two poles, called north (N) and south (S) poles, that exert forces on other magnetic poles similar to way that electric charges exert forces on one another. Poles received their names because of way a magnet, such as that in a compass, behaves in presence of Earth’s magnetic field. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Magnetic Fields Although force between two magnetic poles is otherwise similar to force between two electric charges, electric charges can be isolated (witness electron and proton) whereas a single magnetic pole has never been isolated. That is, magnetic poles are always found in pairs. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Magnetic Fields and Forces In our study of electricity, we described interactions between charged objects in terms of electric fields. In addition to containing an electric field, region of space surrounding any moving electric charge also contains a magnetic field. A magnetic field also surrounds a magnetic substance making up a permanent magnet. Symbol B has been used to represent a magnetic field. Direction of magnetic field B at any location is direction in which a compass needle points at that location. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Magnetic Fields and Forces We can represent magnetic field by means of drawings with magnetic field lines. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Defining a magnetic field We can define a magnetic field B at some point in space in terms of the magnetic force F B that field exerts on a charged particle moving with a velocity v, which we call test object. Experiments on various charged particles moving in a magnetic field give following results: 1.Magnitude F B of magnetic force exerted on particle is proportional to charge q and to speed v of particle. 2.Magnitude and direction of F B depend on velocity of particle and on magnitude and direction of magnetic field B. 3.When a charged particle moves parallel to magnetic field vector, magnetic force acting on particle is zero. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Defining a magnetic field 4.When particle’s velocity vector makes any angle θ (non-zero) with magnetic field, magnetic force acts in a direction perpendicular to both v and B; that is, F B is perpendicular to plane formed by v and B. 5.Magnetic force exerted on a positive charge is in direction opposite direction of magnetic force exerted on a negative charge moving in same direction. 6.Magnitude of magnetic force exerted on moving particle is proportional to sinθ, where θ is angle particle’s velocity vector makes with direction of B. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Defining a magnetic field We can summarize these observations by writing magnetic force in form: SI unit of magnetic field is newton per coulomb-meter per second, which is called Tesla (T) A non-SI magnetic-field unit in common use, called gauss (G), is related to tesla through conversion 1T= 10 4 G 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

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

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

Example#1: 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