Fundamentals of Electromagnetics: A Two-Week, 8-Day, Intensive Course for Training Faculty in Electrical-, Electronics-, Communication-, and Computer- Related Engineering Departments by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, India Amrita Viswa Vidya Peetham, Coimbatore August 11, 12, 13, 14, 18, 19, 20, and 21, 2008

Module 2 Maxwell’s Equations In Integral Form The line integral The surface integral Faraday’s law Ampere’s circuital law Gauss’ Laws The Law of Conservation of Charge

Instructional Objectives 5. Apply Faraday's law in integral form to find the electromotive force induced around a closed loop, fixed or revolving, for a given magnetic field distribution 6. Apply Ampere's circuital law in integral form to find the magnetomotive force induced around a closed path, for a given current and/or electric field distribution 7. Apply Gauss' law for the electric field in integral form, Ampere's circuital law in integral form, the law of conservation of charge, and symmetry considerations, to find the line integral of the magnetic field intensity around a closed path, given an arrangement of point charges connected by wires carrying currents

The Line Integral (FEME, Sec. 2.1; EEE6E, Sec. 2.1)

Work done in carrying a charge from A to B in an electric field: The Line Integral Work done in carrying a charge from A to B in an electric field: a1 a2

(Voltage between A and B)

In the limit , = Line integral of E from A to B. = Line integral of E around the closed path C.

is independent of the path from A to B (conservative field) If then 2-7 If then is independent of the path from A to B (conservative field)

along the straight line paths, 2-8 Ex. For find along the straight line paths, from (0, 0, 0) to (1, 0, 0), from (1, 0, 0) to (1, 2, 0) and then from (1, 2, 0) to (1, 2, 3).

From (0, 0, 0) to (1, 0, 0), From (1, 0, 0) to (1, 2, 0),

From (1, 2, 0) to (1, 2, 3),

In fact,

1.5- The Surface Integral (FEME, Sec. 2.2; EEE6E, Sec. 2.2) 12

Flux = (B)(DS) Flux = 0 The Surface Integral 2-13 The Surface Integral Flux of a vector crossing a surface: Flux = (B)(DS) Flux = 0 a

= Surface integral of B over S. 2-14 = Surface integral of B over S.

= Surface integral of B over the closed surface S. D2.4 (a)

(b)

(c)

(d) From (c),

Faraday’s Law (FEME, Sec. 2.3; EEE6E, Sec. 2.3)

Faraday’s Law

Voltage around C, also known as electromotive force (emf) around C (but not really a force), = Magnetic flux crossing S, = Time rate of decrease of magnetic flux crossing S,

Important Considerations (1) Right-hand screw (R.H.S.) Rule. The magnetic flux crossing the surface S is to be evaluated toward that side of S a right-hand screw advances as it is turned in the sense of C.

(2) Any surface S bounded by C. The surface S can be any surface bounded by C. For example: This means that, for a given C, the values of magnetic flux crossing all possible surfaces bounded by it is the same, or the magnetic flux bounded by C is unique.

(3) Imaginary contour C versus loop of wire. There is an emf induced around C in either case by the setting up of an electric field. A loop of wire will result in a current flowing in the wire. (4) Lenz’s Law. States that the sense of the induced emf is such that any current it produces, if the closed path were a loop of wire, tends to oppose the change in the magnetic flux that produces it.

Thus the magnetic flux produced by the Thus the magnetic flux produced by the induced current and that is bounded by C must be such that it opposes the change in the magnetic flux producing the induced emf. (5) N-turn coil. For an N-turn coil, the induced emf is N times that induced in one turn, since the surface bounded by one turn is bounded N times by the N-turn coil. Thus

where  is the magnetic flux linked by one turn

y y dec. wt y inc. wB0 wt –wB0 Lenz’s law is verified.

2-28 (b)

2-29 (c)

Motional emf concept conducting rails conducting bar

This can be interpreted as due to an electric field induced in the moving bar, as viewed by an observer moving with the bar, since

where is the magnetic force on a charge Q in the bar. Hence, the emf is known as motional emf.

Ampére’s Circuital Law (FEME, Sec. 2.4; EEE6E, Sec. 2.4)

Ampére’s Circuital Law

= Magnetomotive force (only by analogy with electromotive force), 2-35 = Magnetomotive force (only by analogy with electromotive force), = Current due to flow of charges crossing S, = Displacement flux, or electric flux, crossing S,

= Time rate of increase of displacement flux crossing S, or, displacement current crossing S, Right-hand screw rule. Any surface S bounded by C, but the same surface for both terms on the right side.

Three cases fo clarify Ampére’s circuital law (a) Infinitely long, current carrying wire No displacement flux

(b) Capacitor circuit (neglect fringing of field in the capacitor)

(c) Finitely long wire

Uniqueness of

Displacement current emanating from a closed surface = – (current due to flow of charges emanating from the same closed surface)

D2.9 (a) Current flowing from Q2 to Q3. 2-42 D2.9 (a) Current flowing from Q2 to Q3.

(b) Displacement current emanating from the 2-43 (b) Displacement current emanating from the spherical surface of radius 0.1 m and centered at Q1. (c) Displacement current emanating from the spherical surface of radius 0.1 m and centered at Q3.

Interdependence of Time-Varying Electric and Magnetic Fields

Hertzian Dipole

Radiation from Hertzian Dipole

Gauss’ Laws (FEME, Secs. 2.5, 2.6; EEE6E, Sec. 2.5)

Gauss’ Law for the Electric Field Displacement flux emanating from a closed surface S = charge contained in the volume bounded by S = charge enclosed by S. r

Gauss’ Law for the Magnetic Field Magnetic flux emanating from a closed surface S = 0.

P2.21 Finding displacement emanating from a surface enclosing charge (a) Surface of cube bounded by

(b) Surface of the volume x > 0, y > 0, z > 0, and (x2 + y2 + z2) < 1.

P2.23

The Law of Conservation of Charge (FEME, Sec. 2.5; EEE6E, Sec. 2.6)

Law of Conservation of Charge Current due to flow of charges emanating from a closed surface S = Time rate of decrease of charge enclosed by S. r(t)

(1) (2) (3) (4) Summarizing, we have the following: 2-56 Summarizing, we have the following: Maxwell’s Equations (1) (2) (3) (4)

(5) Law of Conservation of Charge (4) is, however, not independent of (1), whereas (3) follows from (2) with the aid of (5). (5)

(Ampére’s Circuital Law) Example: (Ampére’s Circuital Law) (Gauss’ Law for the electric field and symmetry considerations)

(Law of Conservation of Charge)

The End