Faraday's law and linearly time dependent B field

[razidan]

Homework Statement

A positron is moving in a circular orbit of radius r = 2cm within a uniform magnetic field B0 = 50$\mu$T. The magnetic field varies over time according to the expression:
B = 700t + Bo
and, therefore, each orbit can be considered almost circular.
(a) Calculate the electric field E induced and the velocity v of the positron after one orbit [11]

(b) Show that the magnetic flux linked with one orbit stays constant as the magnetic field varies [4]

I assumed a field going into the page, and so the positron is rotating ccw.

Homework Equations

$\int E\cdot dl=\frac{\partial \phi}{\partial t}$
$W=\int F\cdot dl$
$v=\frac{qBr}{m}$

The Attempt at a Solution

for a) my issue is that r is dependent on v, and v changes, so it's nonlinear. they do say that each orbit is almost circular, so i figure i can neglect that effect. but I got that the change in speed is 5 orders of magnitude smaller then the initial speed, so negligible. just wanted to verify.
For the inital speed i got $v_0=175,882 m/s$.
By using ampere's law I got $E \cdot 2\pi r =- \frac{700}{2 \pi r} \rightarrow \vec{E}=-7 V/m \hat{\varphi}$
That is a not so strong field, so the change in kinetic energy (after work energy theorem) comes out to be $-1.2 \cdot 10^{-15} J$.
A couple of things I'm unsure about are:

1)the sign of the change. from lenz's law, the particle should gain energy (so there is more field out of the page). so should there be a negative sign in the integral of the work energy theorem?
2) the magnitude of the change is speed is tiny, do i have a mistake?

for section b - I'm lost. I am not even sure what the exact wording of the problem means. what is "the flux linked with one orbit?"

Thanks

 

[TSny]

B = 700t + Bo

The units for the first term on the right are not specified. What are the units for the 700? Is the time t in seconds or microseconds (or something else)?

I assumed a field going into the page, and so the positron is rotating ccw.

OK

for a) my issue is that r is dependent on v, and v changes, so it's nonlinear. they do say that each orbit is almost circular, so i figure i can neglect that effect. but I got that the change in speed is 5 orders of magnitude smaller then the initial speed, so negligible. just wanted to verify.

Yes, the change in speed per orbit is very small compared to the speed itself. Also, if you can verify that the change in B per orbit is very small compared to B_o, then this would justify the assumption that the orbits are almost circular.

For the inital speed i got $v_0=175,882 m/s$.

OK

By using ampere's law I got $E \cdot 2\pi r =- \frac{700}{2 \pi r} \rightarrow \vec{E}=-7 V/m \hat{\varphi}$

Did you mean to say Faraday's law instead of Ampere's law? Your calculation does not look correct to me. How did you get $E \cdot 2\pi r = - \frac{700}{2 \pi r}$?

1)the sign of the change. from lenz's law, the particle should gain energy (so there is more field out of the page). so should there be a negative sign in the integral of the work energy theorem?

Yes, the particle will gain energy. When using Faraday's law, be sure to interpret the meaning of the negative sign correctly. If you get a negative sign in front of E, it doesn't necessarily mean that E is in a direction opposite to the velocity of the positron.

2) the magnitude of the change is speed is tiny, do i have a mistake?

The change in speed per orbit will be very small compared to the speed itself.

for section b - I'm lost. I am not even sure what the exact wording of the problem means. what is "the flux linked with one orbit?"

The flux linked with one orbit is the magnetic flux through the circular path of one orbit.

 

[razidan]

The units for the first term on the right are not specified. What are the units for the 700? Is the time t in seconds or microseconds (or something else)?

OK

Yes, the change in speed per orbit is very small compared to the speed itself. Also, if you can verify that the change in B per orbit is very small compared to B_o, then this would justify the assumption that the orbits are almost circular.

OK
Did you mean to say Faraday's law instead of Ampere's law? Your calculation does not look correct to me. How did you get $E \cdot 2\pi r = - \frac{700}{2 \pi r}$?

Yes, the particle will gain energy. When using Faraday's law, be sure to interpret the meaning of the negative sign correctly. If you get a negative sign in front of E, it doesn't necessarily mean that E is in a direction opposite to the velocity of the positron.

The change in speed per orbit will be very small compared to the speed itself.The flux linked with one orbit is the magnetic flux through the circular path of one orbit.

Hi.
1. The question is posted as is given to me. Units are not specified.
2. I did mean Faraday's law, thanks.
The integration of dl will give 2πr and the integration of ds will give πr². What am I getting wrong?
3. If I have a minus sign for the field, does it not mean the direction is -φ? This is clockwise, and will deccelerate the position, producing the opposite result given by lenz's law...
4. The flux through a circular orbit does change. That's why we used Faraday's law...

 

[TSny]

1. The question is posted as is given to me. Units are not specified.

OK. If t is in seconds, then I guess the 700 is in μT.

2. I did mean Faraday's law, thanks.
The integration of dl will give 2πr and the integration of ds will give πr². What am I getting wrong?

Please show detailed steps so we can see specifically where you are making a mistake.

3. If I have a minus sign for the field, does it not mean the direction is -φ?

No. You are probably going to be better off using lenz's law to get the direction of the induced electric field. There are other ways of interpreting the minus sign in Faraday's law that you can use to get the direction of E. For example see:
https://en.wikipedia.org/wiki/Faraday's_law_of_induction#Maxwell–Faraday_equation
But you can typically get by without these formalities and just use Lenz's law.

4. The flux through a circular orbit does change. That's why we used Faraday's law...

The change in flux during one orbit is very small because the positron completes the orbit very quickly. So, when calculating the "flux linked with one orbit", you can treat the orbit as circular and the B field as constant. You have to wait for many orbits before the magnetic field and the orbital radius change significantly. They want you to show that no matter how long you wait, the flux linked by one orbit is still the same as for the first orbit (to a good approximation).