I have been referring research paper "Design of a Coreless Induction Furnace for Melting Iron" by M. M. Ahmed, M. Masoud and A. M. El-Sharkawy
(Refer https://s3.us-east-1.amazonaws.com/foundrygate-uploads/artigo_1678130244006_Design%20of%20a%20Coreless%20Induction%20Furnace%20for%20Melting%20Iron.pdf) (refer page 3)
Referring the paper I am stuck at this particular formula for effective value of emf created in element path of induction melting charge (refer marked in image)
I tried to break it down as below:
$$e=-\frac{dφ}{dt}$$
Where φ=BA, Assuming a sinusoidal variation of the magnetic field, $$B = B_m sin(2πft)$$
Considering flux linkage of area related to displacement $x^2$, $$φ=B_m x^2 sin(2πft)$$
Differentiating by dt, $$e=B_m x^2•(2πf)cos(2πft)$$
Considering e(t)=E_m cos(2πft) and E=E_m/√2 The peak EMF E_m $$E_m= 2πfx^2 B_m$$
Hence by that RMS of E, $$E= \frac{E_m}{√2} = \frac{2πfx^2 B_m}{√2}$$
However I cant gather the derivation of the another π in π^2 referred in the equation (shown in the pic) $$E= \frac{E_m}{√2} = \frac{2π^2fx^2 B_m}{√2}$$
for effective value of emf in subjected element path in the research paper as shown in the pic.
Would that have anything to do with sinusoidal motion displacement of the melting charge?
Any help is appreciated.
