The maximum emf induced in a 10 turn coil with a cross sectional area of 0.1m squared rotating at 20 rad/s in a uniform 10 Tesla magnetic field is _________.
To find the maximum EMF induced in a rotating coil, we can use Faraday's law of electromagnetic induction. According to the law, the electromotive force (EMF) induced in a coil is equal to the rate of change of magnetic flux through the coil. The formula to calculate the EMF is:
EMF = N * dΦ/dt
where:
EMF is the electromotive force
N is the number of turns in the coil
dΦ/dt is the rate of change of magnetic flux through the coil
In this case, the coil has 10 turns (N = 10), a cross-sectional area of 0.1 m², and is rotating at 20 rad/s in a uniform magnetic field of 10 Tesla.
To calculate the rate of change of magnetic flux, we need to determine the magnetic flux (Φ) and how it changes with time.
The magnetic flux (Φ) through a coil is given by:
Φ = B * A * cosθ
where:
B is the magnetic field strength
A is the cross-sectional area of the coil
θ is the angle between the magnetic field and the normal to the coil's plane
In this case, B = 10 Tesla and A = 0.1 m². Since the coil is rotating in the magnetic field, the angle (θ) between the magnetic field and the normal to the coil's plane changes with time.
Assuming the angle changes sinusoidally, we can express it as: θ = ωt, where ω is the angular speed in rad/s, and t is the time.
Now, let's calculate the rate of change of magnetic flux (dΦ/dt):
dΦ/dt = d(B * A * cosθ)/dt
= (-B * A * ω * sin(ωt)) [Using the chain rule of differentiation]
Since cosθ = cos(ωt), we can substitute it in the expression above.
Now, substituting the given values into the formula:
dΦ/dt = (-10 Tesla * 0.1 m² * 20 rad/s * sin(20t))
= -20 Tesla·m²·rad/s · sin(20t)
Finally, we can calculate the maximum EMF by multiplying the rate of change of magnetic flux by the number of turns in the coil:
EMF = N * dΦ/dt
= 10 turns * (-20 Tesla·m²·rad/s · sin(20t))
= -200 Tesla·m²·rad/s · sin(20t)
Therefore, the maximum EMF induced in the 10-turn coil rotating at 20 rad/s in a uniform 10 Tesla magnetic field is -200 Tesla·m²·rad/s. Note that the negative sign indicates the direction of the induced current.