Magnetic Field Question

1. The problem statement, all variables and given/known data

A planar 50-turn coil of area ## 0.20m^2 ## is rotated in a uniform ## 0.15 ## Tesla magnetic field by a motor at a constant angular velocity ## \omega = 315 s^-1 ## The axis of rotation is in the plane of the coil and is perpendicular to the magnetic field. At time ## t = 0 ##, the magneticfield is perpendicular to the plane of the coil.

B.
(i) Plot the total magnetic flux through the coil as a function of time. [3 marks]
(ii) Find the maximum emf generated from the coil. [3 marks]
(iii) What is value of the root-mean-square emf? [3 marks]

C.
If the rotating coil in part (b) forms part of an electrical circuit with total resistance ##30 \Omega ##(including the coil),

(i) What is the average power generated? [3 marks]
(ii) Plot the instantaneous power as a function of time. [3 marks]

2. Relevant equations

## \epsilon = -N \frac{d\Phi_B}{dt} ##

## P = I^2R ##

## \Phi_B = BAcos(\omega t) ##

3. The attempt at a solution

(i) I need help with this part, I know Flux goes on the y-axis and Time on the x-axis. I’m not sure how to go about plotting the points. I think ## \omega ## depends on the amplitude of the flux.

(ii) Maximum emf ## (\epsilon) ## when ## \theta = 90° ## i.e when ## \omega t = 90° ##

## \Phi = NBA(cos\theta) ##

## \epsilon = -N \frac{d\Phi}{dt} = -NBA \frac{d}{dt}(cos \theta) = -NBA \frac{d}{dt}(cos (\omega t)) ##

## \epsilon = -NBA(-\omega sin(\omega t)) ## , where ## \omega t = 90° ##

## \epsilon = NBA\omega = (50)(0.15)(0.2)(314) = 471 V ##

(iii) ## \epsilon_{rms} = \sqrt{\frac{NBA\omega}{2}} ##

## = \sqrt{\frac{(50)(0.15)(0.2)(314)}{2}} = 15.35 V ##

C.

(i) ## P = I^2R ## Ohm’s Law states ## E=IR \Rightarrow I^2 = \frac{E^2}{R^2} ##

Now, ## P = \frac{E^2}{R} = \frac{(471)^2}{30} = 7394.7 W ##

(ii) As you can see I really struggle with graphs, how do I go about plotting the instantaneous powers as a function of time?

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