A magnetic field is applied to the plain of a flat coil of copper wire. The time variation of the magnetic flux density is given by Bosinwt.At which of the following values of t is the magnitude of the emf induced in the coil a maximum.(a)T/8(b)3T/8(c)3T/4(d)T/2
To determine at which value of t the magnitude of the induced emf is a maximum, we need to find the value of t when the rate of change of magnetic flux is maximum.
The induced emf in a coil is given by Faraday's law of electromagnetic induction, which states that the emf is equal to the negative rate of change of magnetic flux.
Given that the time variation of the magnetic flux density is given by B = B₀sin(ωt), where B₀ is the maximum magnetic flux density and ω is the angular frequency, we can calculate the rate of change of magnetic flux as follows:
dΦ/dt = d/dt (B₀sin(ωt))
= B₀ωcos(ωt)
The magnitude of the induced emf (ε) is equal to the absolute value of the rate of change of magnetic flux:
|ε| = |dΦ/dt| = |B₀ωcos(ωt)|
To find the maximum value of |ε|, we need to find the value of ωt that maximizes |cos(ωt)|. Since the cosine function has a maximum value of 1 when the argument is 0, we can set the argument of the cosine function to zero:
ωt = 0
Now, we know that ω = 2πf, where f is the frequency, and the period T is equal to 1/f. Therefore, we can rewrite the equation as:
2πft = 0
Since the phase factor doesn't change the magnitude of the emf, we can disregard it. Hence, we have:
t = 0
Therefore, the magnitude of the induced emf is maximum at t = 0.
None of the given options (a) T/8, (b) 3T/8, (c) 3T/4, or (d) T/2 correspond to t = 0. Therefore, none of these values represent the maximum magnitude of the induced emf.