Induced electric field

Tags:
1. Mar 8, 2016

Titan97

1. The problem statement, all variables and given/known data
In the below figure, magnetic field is decreasing at a rate of 1.5T/s in a circular region of radius 2m. EMF developed in the rod will be?

2. Relevant equations
$$V_{\text{induced}}=-A\frac{\mathrm{d}B}{\mathrm{d}t}$$

3. The attempt at a solution
I tried using induced electric fields. They will be concentric circles and its magnitude is given by $$E=-\frac{r^2}{R^2}\frac{\mathrm{d}B}{\mathrm{d}t}$$ (where $r$ is the radius of the circular region and $R$ is the radius of the circular electric field) I took the components of this electric field along the rod and then tried to used $$V=\int E_rdr$$ But finding $E_r$ is difficult. There will be two points on the rod which cut the circular electric field. What should be my approach?

2. Mar 8, 2016

BvU

Hi Ti,

There is something wrong with $E=-\frac{r^2}{R^2}\frac{\mathrm{d}B}{\mathrm{d}t}\quad$ -- it doesn't fit dimensionally.
Where does it come from ?

I liked the expose here (although it ends abruptly) : it tells you magnitude and direction of the induced $\vec E$, so all you need to do is take the component along the rod and integrate

3. Mar 8, 2016

Titan97

Aw snap. It's $$E=\frac{r^2}{2R}\frac{dB}{dt}$$

It comes from $$\int \vec{E}\cdot\mathrm{d}\vec{l}=-A\frac{\mathrm{d}B}{\mathrm{d}t}$$

4. Mar 9, 2016

BvU

And then some trigonometry and an integral and Bob's your uncle.

5. Mar 9, 2016

Titan97

The solution says that 1/5th of flux says through OAB. They solved the question in a single step while I am still stuck. How did they get 1/5?

6. Mar 9, 2016

A ?

7. Mar 9, 2016

Titan97

The ends of rods are A and B

8. Mar 9, 2016

BvU

Very practical, with an A and a B for Area and magnetic field. Where is B ?
Well, at least that should pin down the position of B (end of rod).
Interesting that there is a one step solution. I figured we were heading for some real work. Can you post it ? (This time I don't mind a picture...)

Last edited: Mar 9, 2016
9. Mar 9, 2016

Titan97

@BvU I will post the picture soon. I don't have my Lumia with me now. My tab does not have a camera.

10. Mar 9, 2016

gracy

11. Mar 9, 2016

Titan97

@gracy I will post the pictures after 7:00 IST

12. Mar 9, 2016

Titan97

13. Mar 9, 2016

gracy

If we can somehow prove that triangle BOX = 36 degrees.
Then yes, flux passing through AOB will be 1/5th of the total flux as 72 × 5= 360
(I am not sure though)

Last edited: Mar 9, 2016
14. Mar 9, 2016

cnh1995

We can't prove that. It should be shown in the diagram. I guess there's a misprint?

15. Mar 10, 2016

BvU

We assume $\angle OBA = 54^\circ$ as well and proceed.

The bright idea (that I missed too ) is to construct a Faraday loop BOAB where you know the induced emf and where you know the contributions from BO and AO are zero (why ? ).

It would still be interesting to carry out the integral as mentioned in post #4 -- a lot of work that should give the same result.

( I use the term Faraday loop because I resent the word Faradian )

Then:
worried me. Is it clear that there is an induced E field as (for R > r) described in post #3 all over the place ?

16. Mar 10, 2016

Titan97

Yes.

17. Mar 10, 2016

The electric field can be determined for each r>R using Stokes law. (Draw a separate circle of radius r and let the circle and compute E which is tangent to the circle at each point.) Then you can find r as a function of $\theta$ where each circle of radius r crosses the line. I think the trigonometric factors in computing the integral of E*ds cancel with a simple $d\theta$ result. This longer integral approach can be shortened just by applying Stoke's law as mentioned above. It appears it doesn't even need to be a straight line path and Stokes law will give the same answer for the EMF for any path from A to B that stays outside of R.

Last edited: Mar 10, 2016
18. Mar 10, 2016

cnh1995

The electric field lines cutting OA and OB will be circular in nature, hence the field direction at each point will be perpendicular to OA and OB. So, there will be no emf in OA and OB. Hence, emf is developed only in rod AB.

Last edited: Mar 10, 2016
19. Mar 10, 2016