Electromagnetic induction

https://imgur.com/a/bKZfMkX

I've worked out the first part 86.6 microcoulombs. And the charge increases proportionally with number of turns

I'm unsure how to work out the second part.

I tried multiplying the charge by frequency to obtain a current and using I^2R=c(rate of increase of temperature)
but that's wrong

I also tried using angular frequency 2pi f but that's no good either [how do I know when to use frequency and angular frequency? usually I just differentiate but that sometimes gives the wrong answer]

I tried working out the induced emf and using V^2/R=c x rate of increase of temp but that was also wrong

I'm out of ideas

How do I work out the 2nd part? [rate of heating] Please help me. What assumptions are there? I'm not sure about this too. no heat losses to the surroundings?

I tried working out the induced emf and using V^2/R=c x rate of increase of temp but that was also wrong.

Can you be more specific of how do you do the following calculation (as I think the method is correct and I am not sure have you done it correctly)

I will include all my workings:

1st
Charge = Change in flux / Resistance
= (2 x 10^-4 x 500 x 25 x 10^-4 x sin 60 x 2) / 50 *
= 8.66 x 10^-5 C
which is correct

2nd
Induced EMF = N x rate of change of flux linkage = NABsin60 x f x 2 - (1)
= 500 x 25 x 10^-4 x 2 x 10^-4 sin 60 x 2 x 100 = 0.0433 V

V^2/R = c x rate
Rate of heating is worked out as 2.5 x 10^-5 K s^-1 which is not correct

also tried multiplying (1) by 2pi but this is wrong too

*not sure why you have to multiply this quantity [2 x 10^-4 x 500 x 25 x 10^-4 x sin 60] by 2? Is it because we average the horizontal and vertical components?

The problem lies that you are using the max induced emf instead of r.m.s. induced emf. You need to find the r.m.s. induced emf NOT max induced emf and use the r.m.s. induced emf to compute the power of the heating effect.
Your max induced emf is also incorrect. It should be

Thanks I got it now
P.S If possible please explain why we have to multiply by 2 in the first part
Why not in the second part too? Thanks

I cannot get what you stated in the first part:
(2 x 10^-4 x 500 x 25 x 10^-4 x sin 60 x 2) / 50 ≠ 8.66 x 10^-5 C

As for first part, there are at least two methods to get the answer.
One method is what you have stated. How did you arrive the expression for the charge flows as change in flux over the resistance?
It seems that you are applying the expression without really thinking the fundemantal meaning of the quantities in the given question.
Initial flux is –NABsin(60°) (or NABsin(60°)) and the final flux is NABsin(60°) (or –NABsin(60°)).
Change in flux = NABsin(60°) – (–NABsin(60°)) = 2NABsin(60°)

…..
The first part asked for mathematical statements of:
i) Faraday's law of electromagnetic induction
ii) relation between current and pd
iii) relation between current and charge

i) E = -N d(phi)/dt
ii) V= IR
iii) Q=It
So Q= Change in flux / Resistance

…..
What is the other method ?

I doubt you can arrive such relationship without doing integration. Anyway, just hope that you really understand what you are doing.



As the induced emf can be described as $\mathcal{E}_{ind} =\omega NBA \sin(\omega t)$, so when we integrate from t= 0 to t=T/2, we would obtain the change in flux.
Compute the charge flow as change in flux over the resistance.