Finding the Electric field given a time varying magnetic field

[OONeo01]

Homework Statement

This is a question asked in one of my papers.

A time dependent magnetic field $\vec{B}$(t) is produced in a circular region of space, infinitely long and of radius R. The magnetic field is given as $\vec{B}$=B₀t$\hat{z}$ 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)$\hat{r}$
B.)(B₀R²/2r)$\hat{θ}$
C.)-(B₀R²/r)$\hat{r}$
D.)-(B₀R²/2r)$\hat{θ}$

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.

 

[TSny]

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.

 

[Saitama]

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:
$$\int \vec{E} \cdot {\vec{dl}}=-\frac{d \phi}{dt}$$
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?

 

[TSny]

I was trying out this question and I used the following equation:
$$\int \vec{E} \cdot {\vec{dl}}=-\frac{d \phi}{dt}$$
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.

 

[Saitama]

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?

 

[SammyS]

Homework Statement

This is a question asked in one of my papers.

A time dependent magnetic field $\vec{B}$(t) is produced in a circular region of space, infinitely long and of radius R. The magnetic field is given as $\vec{B}$=B₀t$\hat{z}$ 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)$\hat{r}$
B.)(B₀R²/2r)$\hat{θ}$
C.)-(B₀R²/r)$\hat{r}$
D.)-(B₀R²/2r)$\hat{θ}$

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.

 

[OONeo01]

Thanks a lot guys ! Got the Answer :-) Option D, right ?

 

[TSny]

Thanks a lot guys ! Got the Answer :-) Option D, right ?

Right!