Please Do Course Evaluation RLC circuit RLC circuit Solution: R - - PowerPoint PPT Presentation

Please Do Course Evaluation

RLC circuit RLC circuit Solution: R - t   2L Q(t) Q e cos 0 d 2   1 R      d   LC 2L   0 under damped 2    1 R      0 critically damped   LC 2L    0 over damped Kirchhoff' s rule : Q dI d     0 IR L (I Q ) C d t dt 2 d d Q     L Q R Q 0 2 dt C dt

Damping R - t    d real: under damped 2L Q(t) Q e cos 0 d  d = 0: critically damped  d imaginary: overdamped 2   1 R      d   LC 2L

Class 43 Displacement currrent

Maxwell’s Equations Maxwell’s equations describe all the properties of electric and magnetic fields and there are four equations in it: Integral form Differential form Name of equation (optional) 1 st Equation    Electric          E d A Q E Gauss’s Law 0 enclosed 0   Magnetic       B  B d A 0 Gauss’s Law 0     Ampere’s          B d I B J Law 0 enclosed 0 (Incomplete)           B     E d - B (t) d A    E -            t  t   B Lorentz force equation is not part of Maxwell’s equations. It describes what happens when charges are put in an electric or magnetic fields:        F (q E v B ) Slide #6 of Class 36

Revisit Ampere’s Law For DC, I=0 and B=0, so there is no problem. If I is changing with time, I  0 (except at the gap) and there will be a magnetic field (changing with time also). d If the gap d is very small (d  0), there should be magnetic field everywhere surrounding the wire even though there is no physical current through the gap. The problem now is:         For surface S : B d s I I 1 0 Enclosed 0 by S Path P 1        For surface S : B d s I 0 2 0 Enclosed by S Path P 2 How to reconcile the difference?

Maxwell’s proposal We can introduce an imaginary current, called displacement current, I d within the I gap so the current now looks like continuous. With this displacement current: S 2 d     I d             For For surface surface S S : : B B d d S S I I I I 1 1 0 0 Enclosed Enclosed 0 0 by by S S P P 1 1                     P For For surface surface S S : : B B d d S S I I I I I I S 1 2 2 0 0 Enclosed Enclosed 0 0 d d 0 0 I by by S S P P 2 2 Ampere’s Law now becomes:       1  B d S (I I ) 0 Enclosed d

Displacement current But at the end what is a displacement current? I It is not a real current due to motion of charges within the gap, so we have to relate it to something that really exists in the gap: electric field. S 2 dq dV d     I d I I C (q CV) d dt dt dE   Cd (V Ed) dt P S 1   A dE A I    0 0 d (C ) d dt d d(EA)   0 dt  d   E 0 dt

Abstraction  d   E I d 0 dt I We got this idea from parallel plate capacitor. We expand this and say this is generally true for any S 2 geometry and Ampere’s Law now becomes: d I d    d       E B d S (I ) 0 Enclosed 0 dt P S 1 I   d       (I E d A ) 0 Enclosed 0 dt

Maxwell’s Equations Maxwell’s equations describe all the properties of electric and magnetic fields and there are four equations in it: Integral form Differential form Name of equation (optional) 1 st Equation    Electric          E d A Q E Gauss’s Law 0 enclosed 0   Magnetic       B  B d A 0 Gauss’s Law 0     Ampere’s          B d I B J Law 0 enclosed 0 (Incomplete)           B     E d - B (t) d A    E -            t  t   B Lorentz force equation is not part of Maxwell’s equations. It describes what happens when charges are put in an electric or magnetic fields:        F (q E v B ) Slide #6 of Class 36

Maxwell’s Equations Maxwell’s equations describe all the properties of electric and magnetic fields and there are four equations in it: Integral form Differential form Name of equation (optional) 1 st Equation    Electric          E d A Q E Gauss’s Law 0 enclosed 0   Magnetic       B  B d A 0 Gauss’s Law 0 Ampere’s                  Law       B ( J E )  B d (I E d A )  0 enclosed 0  0 0 t t (Incomplete)           B     E d - B (t) d A    E -            t  t   B Lorentz force equation is not part of Maxwell’s equations. It describes what happens when charges are put in an electric or magnetic fields:        Slide #6 of Class 36 F (q E v B )

Three different forms of Maxwell’s Equations

Linearly polarized electromagnetic Waves The wave is traveling in the E  B direction. Linearly polarized waves

Applying Maxwell’s Third Equation to Plane Electromagnetic Waves     d      E d s - B d A d t           E d s E(x dx) 0 - E(x) 0    [E(x dx) - E(x)]  E    dx  x    d      - B d A - ( B dx)  d t t  B    - dx  t   E B   -   x t

Applying Maxwell’s Fourth Equation to Plane Electromagnetic Waves     d          B d s (I E d A ) (I 0 ) 0 in 0 in dt           B d s B(x) 0 - B(x dx) 0    - [B(x dx) - B(x)]  B    - dx  x    d          E d A ( E dx)  0 0 0 0 d t t  E      dx  0 0 t   B E     -   0 0 x t