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- Electromagnetic Waves Chapter 1
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- 2 Chapter Outlines Chapter 1 Electromagnetic Waves Faradays Law Transformer and Motional EMFs Displacement Current Maxwells Equations Lossless TEM Waves EM Wave Fundamental and Equations EM Wave Propagation in Different Media EM Wave Reflection and Transmission at Normal or Oblique Incidence
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- 3 Introduction In your previous experience in studying electromagnetic, you have learned about and experimented with electrostatics and magnetostatics concentrating on static, or time invariant electromagnetic fields (EM Fields). Henceforth, we shall examine situations where electric and magnetic fields are dynamic or time varying !!
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- 4 Where : In static EM Fields, electric and magnetic fields are independent each other, but in dynamic field both are interdependent. Time varying EM Fields, represented by E(x,y,z,t) and H(x,y,z,t) are of more practical value than static EM Fields. In time varying fields, it usually due to accelerated charges or time varying currents. Introduction (Contd..)
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- 5 In summary: Stationary charges electrostatic fields Steady currents magnetostatic fields Time varying currents electromagnetic fields or waves Introduction (Contd..)
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- 1.1Faradays Law According to faradays experiment, a static magnetic field produces no current flow, but a time varying field produces an induced voltage called electromotive force or emf in a closed circuit, which causes a flow of current. Faradays Law the induced emf, V emf in volts, in any closed circuit is equal to the time rate of change of the magnetic flux linkage by the circuit.
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- Faradays Law (Contd..) Where, The negative sign is a consequence of Lenz Law. If we consider a single loop, Faradays Law can be written as:
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- An increasing magnetic field out of the page induces a current in (a) or an emf in (b). (c) The distributed resistance in a continuous conductive loop can be modeled as lumped resistor R dist in series with a perfectly conductive loop. Faradays Law (Contd..)
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- Generating emf requires a time varying magnetic flux linking the circuit. This occurs if the magnetic field changes with time transformer emf or if the surface containing the flux changes with time motional emf. The emf is measured around the closed path enclosing the area through which the flux is passing, can be written as: It is clear that in time varying situation, both E and B are present and interrelated.
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- 10 Example 1 Consider the rectangular loop moving with velocity u=u y a y in the field from an infinite length line current on the z axis. Assume the loop has a distributed resistance R dist. Find an expression for the current in the loop including its direction.
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- Solution to Example 1 First calculate the flux through the loop at an instant time, Where, unit vector along the line current unit vector perpendicular from the line current to the field point Remember ? So,
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- Solution to Example 1 (Contd..) Arbitrarily choose dS in the +a x direction, So the flux can be easily calculated as:
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- Solution to Example 1 (Contd..) Then, we want to find how this flux changes with time, By chain rule, By considering u y =dy/dt,
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- Our emf is negative of this, where: Solution to Example 1 (Contd..) Since we considered dS in the +a x direction, our emf is taken counterclockwise circulation. But since the emf is negative, our induced current is apparently going in the clockwise direction with value of:
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- 1.2Transformer and motional emf The variation of flux with time as in previous equation maybe caused in three ways: By having a stationary loop in a time varying B field. (transformer emf) By having a time varying loop area in a static B field. (motional emf) By having a time varying loop area in a time varying B field.
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- Transformer and motional emf (Contd..) Stationary Loop in Time Varying B Field This is the case where a stationary conducting loop is in a time varying magnetic B field. The equation becomes: By applying Stokes Theorem in the middle term, we obtain:
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- Transformer and motional emf (Contd..) This leads us to the point or differential form of Faradays Law, Based on this equation, the time varying electric field is not conservative, or not equal to zero. The work done in taking a charge about a closed path in a time varying electric field, for example, is due to the energy from the time varying magnetic field.
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- Transformer and motional emf (Contd..) Moving Loop in static B Field When a conducting loop is moving in a static B field, an emf is induced in the loop. Recall that the force on a charge moving with uniform velocity in magnetic field, So then, we define the motional electric field E m,
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- Transformer and motional emf (Contd..) If we consider a conducting loop moving with uniform velocity u as consisting of a large number of free electrons, the emf induced in the loop is: Moving Loop in Time Varying Field The total emf would be:
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- 20 Example 2 The loop shown is inside a uniform magnetic field B = 50 a x mWb/m 2. If side DC of the loop cuts the flux lines at the frequency of 50Hz and the loop lies in the yz plane at time t = 0, find the induced emf at t = 1 ms.
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- Solution to Example 2 Since the B field is time invariant, the induced emf is motional, that is: Where,
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- As u and dL is in cylindrical coordinates, transform B field into cylindrical coordinate (Chapter 1 in Electromagnetic Theory !! ): Solution to Example 2 (Contd..) Where B 0 = 0.05, therefore:
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- And Solution to Example 2 (Contd..) To determine recall that, C is constant at because the loop is in the yz plane!
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- Solution to Example 2 (Contd..) Hence, Therefore, So that at t = 1 ms,
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- 1.3Displacement Current We recall from Amperes Circuital Law for static field, c subscript is used to identify it as a conduction current density, which related to electric field Ohms Law by: But divergence of curl of a vector is identically zero,
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- Displacement Current (Contd..) The current continuity equation, We see that the static form of Amperes Law is clearly invalid for time varying fields since it violates the law of current continuity, and it was resolved by Maxwell introduction which what we called displacement current density, Where J d is the rate of change of the electric flux density,
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- Displacement Current (Contd..) The insertion of J d was one of the major contribution of Maxwell. Without J d term, electromagnetic wave propagation (e.g. radio or TV waves) would be impossible. At low frequencies, J d is usually neglected compared with J c. But at radio frequencies, the two terms are comparable. Therefore,
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- Displacement Current (Contd..) By applying the divergence of curl, rearrange, integrate and apply Stokes Theorem, we can get the integral form of Amperes circuital Law: Do you really understand this displacement current?? Only formula and formula????
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- Displacement Current (Contd..) To have clear understanding of displacement current, consider the simple capacitor circuit of figure below. A sinusoidal voltage source is applied to the capacitor, and from circuit theory we know the voltage is related to the current by the capacitance. i(t) here is the conduction current.
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- Displacement Current (Contd..) Consider the loop surrounding the plane surface S 1. By static form of Amperes Law, the circulation of H must be equal to the current that cuts through the surface. But, the same current must pass through S 2 that passes between the plates of capacitor.
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- Displacement Current (Contd..) But, there is no conduction current passes through an ideal capacitor, (where J=0, due to =0 for an ideal dielectric ) flows through S 2. This is contradictory in view of the fact that the same closed path as S 1 is used. But to resolve this conflict, the current passing through S 2 must be entirely a displacement current, where it needs to be included in Amperes Circuital Law. So we obtain the same current for either surface though it is conduction current in S 1 and displacement current in S 2.
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- Displacement Current (Contd..) The ratio of conduction current magnitude to the displacement current magnitude is called loss tangent, where it is used to measure the quality of the dielectric good dielectric will have very low loss tangent. Other example for physical meaning: J i = current source J c = conducted current through resistor J d =displacement current through dielectric material m i = magnetic current source m d =displacement magnetic current
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- 1.4Maxwell Equations Below is the generalized forms of Maxwell Equations: Maxwell EquationsPoint or Differential Form Integral Form Gausss Law Gausss Law for Magnetic Field Faradays Law Amp