We discussed in lectures that traveling Electromagnetic waves in vacuum of the form

Question

E⃗ =E0xˆcos(kz−ωt),B⃗ =B0yˆcos(kz−ωt)
satisfy all 4 Maxwell's equations. In lectures, I showed that an application of the generalized Ampere's Law (closed loop surrounding area A2, see below), leads to: B0=ε0μ0cE0, and I mentioned that independently it follows from an application of Faraday's Law that B0=E0/c. Combining these two results then leads to the fantastic result that the “speed of light" in vacuum c=1/ε0μ0‾‾‾‾‾√. I want you to show that Faraday's Law indeed leads to the result B0=E0/c. You can show this by choosing a similar special area as we did in lectures:

We define the normal of the surface A1 in the figure above to point in the +yˆ direction. In the following , assume that ℓ=1 m, E0=1 V/m, f=610×1012 Hz, where f is the frequency of oscillation.

(a) We define f1(t)=∮E⃗ ⋅dℓ⃗ , where ∮E⃗ ⋅dℓ⃗ is the closed loop integral taken along the contour of the area A1 in the figure above. Evaluate the function f1(t) in Volts for the following value of t
t=2e-16 sec

(b) Consider the function f2(t)=ϕB(t). Following the method used in Lecture 27, calculate the function f2(t), and evaluate it (in Volts seconds) for the following time t
t=2e-16 sec

(c) Consider again the function f2(t) and evaluate −df2(t)dt in Volts for the following time t
t=2e-16 sec

Answer 1

a)0.197142857148
c)0.197142857148

b) any idea for this?

Answer 2

Formula, please Flu

and have you any part of Problems 1 and 2?

Answer 3

What are the answers when t=2.2e-16??
And do u have answers for the 4th question?

Anonymous, I have the answers for the 1st and 2nd problems, can u tell me the specific questions for whom u want the answers

Answer 4

Sorry Anonymous, just put the numbers in I don't have the formula.

Problem 2:
a)?
b)1.5
c)First answer: (n*E_0)/c
d)all 0

Do you have a) and second and third answer for c) phy?

Answer 5

Phy,
1 c,d
2 a,b

Answer 6

Anonymous, do you have 1 f?

Answer 7

No, Sorry.

Answer 8

No prob Anonymous.

1)b) is 1.5 by the way, if you have slightly different values, make some guesses, it is normally around that value.
with other values it is the same, it must be around that.

Answer 9

But 4th: (2)
dB eB both (E_0/c)*cos(pi/4)

Answer 10

Thanks Anonymous I have all other just some part from 1, 2 and 2 missing.
If Phy or someone figures out please let us have it or the formula.
thanks!

Answer 11

sorry meant some missing from Problem 1, 2 and 3 missing.

Answer 12

Phy 4 Problem:
0
cos(pi/4)
cos(pi/4)
0
-(E_0/c)*cos(pi/4)
(E_0/c)*cos(pi/4)
0
cos(pi/4)
-cos(pi/4)
0
(E_0/c)*cos(pi/4)
(E_0/c)*cos(pi/4)

PHy can you give the remaining answers for Problem 1 and 2?

Answer 13

Anyone please?

Answer 14

Thanks a lot Flu for the 4th prob

2nd prob-
a) Wavelength= 2*pi/k
Frequency= omega/(2*pi)

Answer 15

1st prob-
c) 1.451e8 and
1.6711e8

d) 1.936e8 and
2.229e8

If you want answers of any other probs or formulas, please let me know

Answer 16

And can anyone tell me the answer of
1st (g)??

Answer 17

And Flu, from where did you find out the answers for 3rd prob?? My grader isn't accepting the the answer which u told, can u please tell me the formula, or from where u found out the answer?

Answer 18

I problem ans not coming right.

Answer 19

Q1 part f and part g formulae plz.....!

Answer 20

Q3 part b answer plz...!

Answer 21

neyone who has solved Q7 n Q8 ??

Answer 22

Q1, f)? any ideas

Answer 23

Q1, f)? any ideas
ThE PROBLEM IS WITH b0

Answer 24

Phy, sorry no formula, was an accident to get the value. But our values are not far off so try to guess a little it must be around my value.
Your value did not work for me too.
Could you tell me what the value for k and omega is please Phy?

Answer 25

question 3 i have no idea how to begin.

Answer 26

Can someone help with SNELL'S LAW?

Answer 27

OK, got a) can you help with c) for second problem Phy?
maybe the value or formula.

Answer 28

B0= E0/ speed of case 1 or case 2 u found in abuv parts !

Answer 29

Please tell Q1 e and b.

Answer 30

neyone who has solved Q7 n Q8 ? Q3 part b , Q1 part g..........????

Answer 31

Rohan, is E_0=8.85418782*10^-12?

I don't get the result with the speed of case 1 and 2.

Answer 32

no ur eqn is of the form E= Eo(kx- wt), sooo u can guess nw ! okkay do u hav answers for Q3 part b, Q 1 part g n Q 8 ???

Answer 33

neyone who has got the answer for Q8 part a and Q3 part b ?????

Answer 34

Q8)
a)0.89
b)1.34

Rohan, experiment with some guesses if it does not fit, as you can see the number is quite narrow.

I could not figure B_0 out but with your formula managed +-omega in both cases, can you help me with the formula please Rohan?

Answer 35

Also somebody could help with Problem 2:
What is the value of k in radians per meter?

Answer 36

B_0 in both cases seem difficult cannot get it, please help!

Answer 37

Q3help!!

Answer 38

@ FLU :- Q8 1st part, I tried everythin but didn't get green tick, mine alpha angle is 70 degree and n=1.5

Also what is Q3 part b answer ???

Answer 39

OK try for b) 1.5 as n was in my part the answer.

try for a) 0.60 and go up some decimals, 0.62.0.64 etc. it must be around that, because values are not very far away from that.

Answer 40

Q3 part b I do not have yet, so if anyone has please help.

Do you have Q1) f) B_0 for both cases? value and/or formula Anonymous?

Answer 41

Pls tell ans of Q1 b,c,d,e,f,g

Answer 42

B_0 for Q1?

Answer 43

Did anyone got B_0 correct for Problem 1) f)?

Answer 44

PLease B_0 for Problem 1 both cases!

Answer 45

For B_0 the formula seems to be :
E_0/c
However, I cannot get the result. can anybody work with this or help?

Answer 46

Flu pls. tell ans of Q1 part b wavenumber, part c and e

Answer 47

Phy pls. tell ans of Q1 part b wavenumber, part c and e and 3b

Answer 48

Problem 3)b) please value or formula?

Answer 49

Help with Problem 3b please!

Answer 50

wavenumber = (2*pi)/(Wavelength)

Answer 51

it is analog W

Answer 52

leGum can you help with Problem 3 b) please?

Answer 53

Anyone managed Problem 3) b) formula or value please?

Answer 54

Pls. tell ans of freq in Q1 and part e and g.

Answer 55

B_0 for Problem 3 please!?

Answer 56

Hard, I didn't make Q3.

Give pliz somebody formulas on all questions from Q3 and explain as to find B_0 in Q1

Answer 57

Anyone for Q3 for B_0 both cases please!

Answer 58

Please Problem 3 someone!?

Answer 59

Problem 3!

Answer 60

Part a and Part c:
d/dt (l (sin((k lambda)/4-t omega)+sin(t omega)))/k where t=2.4*10^-16; omega=3.83*10^15; k=12775810.12; lambda=4.91*10^-7;l=1

Put this value in wolframalpha and divide the result by speed of light

and for Part b: Integrate[Cos[k z - omega t], {z, 0, lambda/4}, {x, 0, l}] where t=2.2*10^-16; omega=3.83*10^15; k=12775810.12; lambda=4.91*10^-7;l=1

Put this value in wolframalpha and divide the result by speed of light

Answer 61

MAny thanks X10! I don't sem to get the result. could you give ,e your value, so I can find an approximation for mine?

Answer 62

Meant value for b) X10 please?

Answer 63

Got it X10 thanks vey much!

can you maybe help with Problem 1) f) case 2? I cannot get the B_0 for case 2.

A formula and the value, I would be greatful.

Answer 64

Problem 1: part f: Bo= E_0/speed u got in case 1 or case2

Answer 65

formula for Q#8 part 1 or answer for α=84∘?

Answer 66

FLU pl tell ans for value t=2.4e-16

Answer 67

Pl tell ans for prob 1 part c case 2
1.6711e8 is showing wrong

Answer 68

Thanks X10, have same problem like P.
I don't get it. is this the formula, my speed case value is:200500000

(8.85418782*10^-12)/(200500000)

Is there a mistake with my E_0 value?

Answer 69

X10 please tell value for 1)f)
how did you calculate step by step? having problems with solution

Answer 70

X10, the formula did not give a right answer, could you give us the value for yours? and is E_0= 8.85418782*10^-12?

Answer 71

Q3 a,c (put your time)
-((3.83*10^15)/(2.99e8))*integral from 0 to ((4.91*10^-7)/4) of sin(12775810.12*z-3.83*10^15*2.6*10^-16) dz

Answer 72

for b

Answer 73

gggg, for question 1)f) B_0 what is it the formula?
Could you give us the value, as the formula from X10 did not work for me.

thanks!

Answer 74

Anyone for 1)f) B_0 please?

formula didn't work, please help!

Answer 75

B_0 for 1)f) please?

Answer 76

is E_0=8.85418782*10^-12 value this?
It does not work with case 1/2 speed, please help!

Answer 77

FLu sorry for late reply. My Internet not working for past 20 hrs. Anyhow I'm sorry for the time up. But the mistake u were doing is that E_o is ur maximum amplitude value in the given equation not permitivity constant.like in my case E_o= 55 in Ey=55sin(4.71x-1.09e10t)

Answer 78

No problem X10, thanks for it anyway!