# Finding the Electric field given a time varying magnetic field

• OONeo01
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.
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.

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.

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

Pranav-Arora said:
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.

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?

OONeo01 said:

## 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.

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

OONeo01 said:
Thanks a lot guys ! Got the Answer :-) Option D, right ?

Right!

## 1. How do I find the electric field given a time varying magnetic field?

To find the electric field, you can use Faraday's law of induction which states that the electric field is directly proportional to the rate of change of the magnetic field over time. This can be represented as E = -dΦ/dt, where E is the electric field, Φ is the magnetic flux, and t is time.

## 2. What is the relationship between the electric field and the time varying magnetic field?

The electric field and the time varying magnetic field are closely related. A changing magnetic field induces an electric field, and vice versa. This phenomenon is known as electromagnetic induction and is one of the fundamental principles of electromagnetism.

## 3. Can the electric field be zero even when there is a time varying magnetic field?

Yes, it is possible for the electric field to be zero even when there is a time varying magnetic field. This can occur when the magnetic field is changing at a constant rate, as the induced electric field would also be constant and cancel out to zero. However, if the magnetic field is changing at a non-constant rate, the electric field will not be zero.

## 4. How does the direction of the electric field relate to the direction of the time varying magnetic field?

The direction of the electric field is perpendicular to the direction of the time varying magnetic field. This can be seen in Faraday's law, where the negative sign indicates that the electric field is in the opposite direction of the change in magnetic field.

## 5. What are some real-world applications of finding the electric field given a time varying magnetic field?

The principles of finding the electric field from a time varying magnetic field are used in various technologies, such as generators, transformers, and electric motors. It is also used in medical imaging techniques like magnetic resonance imaging (MRI) and in telecommunications equipment like antennas for wireless communication.

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