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

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RLC circuit

C Q Q dt d R Q dt d L ) Q dt d (I t d dI L IR C Q : rule s Kirchhoff'

2 2

       

RLC circuit Solution:

                        damped

• ver

damped critically damped under 2L R LC 1 2L R LC 1 cos e Q Q(t)

2 2 d d t 2L R

• 



Damping

2 d d t 2L R

• 2L

R LC 1 cos e Q Q(t)           

d real: under damped d = 0: critically damped d imaginary: overdamped

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 (optional)

Name of equation

1st Equation Electric Gauss’s Law Magnetic Gauss’s Law Ampere’s Law (Incomplete)

enclosed

Q A d E  



   A d B  



    E     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:

) B v E (q F        I d B

enclosed

  



   J B      

              

B

A d (t) B t

• d

E



    

 



t B

• E

      

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). 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: How to reconcile the difference? d

I s d B : S surface For I I s d B : S surface For

2 1

S by Enclosed P Path 2 S by Enclosed P Path 1

     

 

      

Maxwell’s proposal

We can introduce an imaginary current, called displacement current, Id within the gap so the current now looks like continuous. With this displacement current:

I I I S d B : S surface For I I S d B : S surface For

d S by Enclosed P 2 S by Enclosed P 1

2 1

           

 

   

d I I Id S2 S1 P

Ampere’s Law now becomes:

I I I S d B : S surface For I I S d B : S surface For

d S by Enclosed P 2 S by Enclosed P 1

2 1

           

 

    ) I (I S d B

d Enclosed

1 

 



  

Displacement current

But at the end what is a displacement current? 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.

dt d dt d(EA) ) d A (C dt dE d d A Ed) (V dt dE Cd CV) (q dt dV C dt dq I I

E d

               

d I I Id S2 S1 P

Abstraction

dt d I

E d

  

d I I Id S2 S1 P We got this idea from parallel plate capacitor. We expand this and say this is generally true for any geometry and Ampere’s Law now becomes:

) A d E dt d (I ) dt d (I S d B

Enclosed E Enclosed

          

 

   

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 (optional)

Name of equation

1st Equation Electric Gauss’s Law Magnetic Gauss’s Law Ampere’s Law (Incomplete)

enclosed

Q A d E  



   A d B  



    E     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:

) B v E (q F        I d B

enclosed

  



   J B      

              

B

A d (t) B t

• d

E



    

 



t B

• E

      

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 (optional)

Name of equation

1st Equation Electric Gauss’s Law Magnetic Gauss’s Law Ampere’s Law (Incomplete)

enclosed

Q A d E  



   A d B  



    E     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:

) B v E (q F       

) A d E t (I d B

enclosed

          

 

 

) E t J ( B           

              

B

A d (t) B t

• d

E



    

 



t B

• E

      

Slide #6 of Class 36

Three different forms of Maxwell’s Equations

Linearly polarized electromagnetic Waves

Linearly polarized waves

The wave is traveling in the E B direction.

Applying Maxwell’s Third Equation to Plane Electromagnetic Waves

t B

• x

E dx t B

• dx)

B ( t

• A

d B t d d

• dx

x E E(x)]

• dx)

[E(x E(x)

• dx)

E(x s d E A d B t d d

• s

d E                             

   

             

Applying Maxwell’s Fourth Equation to Plane Electromagnetic Waves

t E x B

• dx

t E dx) E ( t A d E t d d dx x B

• B(x)]
• dx)

[B(x

• dx)

B(x

• B(x)

s d B ) (I ) A d E dt d (I s d B

in in

                              

   

                       