Finding the Electric field given a time varying magnetic field

In summary, a time dependent magnetic field in a circular region of space with radius R and a given equation produces an electric field for r>R which is given by option D, -(B0R2/2r)θ. The method of finding this answer can be done through the curl-of-E Maxwell equation and applying Stokes theorem.
  • #1
OONeo01
18
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Homework Statement


This is a question asked in one of my papers.

A time dependent magnetic field (t) is produced in a circular region of space, infinitely long and of radius R. The magnetic field is given as =B0t for 0≤r<R and is zero fr r>R, where B0 is a positive constant. The Electric field for r>R is:

A.)(B0R2/r)
B.)(B0R2/2r)
C.)-(B0R2/r)
D.)-(B0R2/2r)

Homework Equations


Maxwells Equations

The Attempt at a Solution


I started with ∇χE=-∂B/∂t=-B0

I can see the unit vectors are spherical in the options given. And there is a negative sign which I have gotten from the above Maxwell's Equation. So I am assuming the answer is either Option C or D.

Regardless, what should be my next step ? Can somebody just tell me which formulae(or relations) to use in what order, I would appreciate it.
 
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  • #2
You might try integrating both sides of your curl of E equation over a circular region of radius r with r > R and then apply Stokes theorem to the left side integral. Keep in mind that B only extends over the region r<R.
 
  • #3
TSny said:
You might try integrating both sides of your curl of E equation over a circular region of radius r with r > R and then apply Stokes theorem to the left side integral. Keep in mind that B only extends over the region r<R.

I was trying out this question and I used the following equation:

I found out E using this. My answer matches one of the option.

I posted this because I would like to know which method is easier. Is finding the curl easier or the one I have mentioned?
 
  • #4
Pranav-Arora said:
I was trying out this question and I used the following equation:

I found out E using this. My answer matches one of the option.

I posted this because I would like to know which method is easier. Is finding the curl easier or the one I have mentioned?

Starting from the curl-of-E Maxwell equation, integrating over a patch of area and then applying Stokes theorem leads to your starting equation. Your equation is the "integral form" of the curl equation. So your method gets to the answer with less steps. Good.
 
  • #5
TSny said:
Starting from the curl-of-E Maxwell equation, integrating over a patch of area and then applying Stokes theorem leads to your starting equation. Your equation is the "integral form" of the curl equation. So your method gets to the answer with less steps. Good.

Thanks TSny! I don't know too much about the curl and the Stokes Theorem because they never had been of much use to me. I had a look at these from engineering texts but never found them in any of the intro physics book. Is there too much of mathematics involved? As far as I remember, we have to solve a determinant for finding out the curl of E. Right?
 
  • #6
OONeo01 said:

Homework Statement


This is a question asked in one of my papers.

A time dependent magnetic field (t) is produced in a circular region of space, infinitely long and of radius R. The magnetic field is given as =B0t for 0≤r<R and is zero fr r>R, where B0 is a positive constant. The Electric field for r>R is:

A.)(B0R2/r)
B.)(B0R2/2r)
C.)-(B0R2/r)
D.)-(B0R2/2r)

Homework Equations


Maxwell's Equations

The Attempt at a Solution


I started with ∇×E=-∂B/∂t=-B0
This is true for 0 < r < R .

I can see the unit vectors are spherical in the options given. And there is a negative sign which I have gotten from the above Maxwell's Equation. So I am assuming the answer is either Option C or D.

Regardless, what should be my next step ? Can somebody just tell me which formulae(or relations) to use in what order, I would appreciate it.
Those unit vectors abr for cylindrical coordinates, not spherical.

Beyond that, see TSny's suggestions.
 
  • #7
Thanks a lot guys ! Got the Answer :-) Option D, right ?
 
  • #8
OONeo01 said:
Thanks a lot guys ! Got the Answer :-) Option D, right ?

Right!
 
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