A conducting loop of radius 10π√cm is placed perpendicular to a uniform magnetic field of 0.5T. The magnetic field is decreased to zero in 0.5s at a steady rate. The induced emf in the circular loop at 0.25s is:
Question
A conducting loop of radius 10π√cm is placed perpendicular to a uniform magnetic field of 0.5T. The magnetic field is decreased to zero in 0.5s at a steady rate. The induced emf in the circular loop at 0.25s is:
Solution
The induced emf in a loop due to a changing magnetic field is given by Faraday's law of electromagnetic induction, which states that the induced emf is equal to the rate of change of magnetic flux.
The magnetic flux (Φ) through the loop is given by the product of the magnetic field (B), the area of the loop (A), and the cosine of the angle between the field and the normal to the loop (cosθ). In this case, the loop is perpendicular to the field, so cosθ = 1. The area of the loop is πr², where r is the radius of the loop.
Given: B = 0.5 T (initial magnetic field) B' = 0 T (final magnetic field) r = 10π√ cm = 10π cm (converted from square root cm to cm) t = 0.5 s (time taken for the field to change) t' = 0.25 s (time at which we want to find the induced emf)
First, calculate the initial and final magnetic fluxes:
Φ = B * A * cosθ Φ = B * πr² Φ (initial) = 0.5 T * π * (10π cm)² = 500π² T.cm² Φ (final) = 0 T * π * (10π cm)² = 0 T.cm²
The rate of change of flux is the change in flux divided by the change in time:
dΦ/dt = (Φ' - Φ) / (t' - t) dΦ/dt = (0 T.cm² - 500π² T.cm²) / (0.5 s - 0 s) = -1000π² T.cm²/s
The induced emf at 0.25 s is the same as the rate of change of flux, because the rate of change is constant:
emf = dΦ/dt = -1000π² T.cm²/s
The negative sign indicates that the induced emf opposes the change in the magnetic field, as per Lenz's law. So, the induced emf in the circular loop at 0.25s is 1000π² T.cm²/s.
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