A conducting circular loop is placed in a uniform magnetic field, B=0.025 TB=0.025 T with its plane perpendicular to the direction of the magnetic field. The radius of the loop is made to shrink at a constant rate of 1 mm s−11 mm s-1. Find the emf induced in the loop when it's radius is 2 cm2 cm.
Question
A conducting circular loop is placed in a uniform magnetic field, B=0.025 TB=0.025 T with its plane perpendicular to the direction of the magnetic field. The radius of the loop is made to shrink at a constant rate of 1 mm s−11 mm s-1. Find the emf induced in the loop when it's radius is 2 cm2 cm.
Solution
The emf induced 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 (θ) between the magnetic field and the normal to the loop. In this case, since the magnetic field is perpendicular to the loop, θ = 0 and cos(θ) = 1.
So, Φ = B * A.
The area of a circular loop is given by πr², where r is the radius of the loop.
So, Φ = B * πr².
The rate of change of magnetic flux is then given by dΦ/dt.
dΦ/dt = B * d(πr²)/dt.
Since the radius of the loop is shrinking at a constant rate, dr/dt = -1 mm/s = -0.001 m/s (we convert from mm to m for consistency of units).
So, d(πr²)/dt = 2πr * dr/dt = 2πr * -0.001.
Substituting this back into the equation for dΦ/dt gives:
dΦ/dt = B * 2πr * -0.001.
The induced emf (ε) is then given by the magnitude of dΦ/dt, so:
ε = |dΦ/dt| = B * 2πr * 0.001.
Substituting the given values B = 0.025 T and r = 2 cm = 0.02 m gives:
ε = 0.025 * 2π * 0.02 * 0.001 = 3.14 x 10^-6 V or 3.14 µV.
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