Final answer:
Using Faraday's law, we can calculate the induced emf in a circular wire loop due to a changing magnetic field. The emf is found by calculating the change in magnetic flux over the time interval and dividing by that interval. The result is an induced emf of approximately -0.044 V.
Step-by-step explanation:
To determine the emf induced in a single-turn circular loop of wire when the magnetic field changes, we can use Faraday's law of electromagnetic induction, which states that the induced emf (ε) in a loop is equal to the negative change in magnetic flux (ΔΦ) over time (Δt).
The magnetic flux (Φ) through a 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 field and the normal to the loop's surface. Since the loop lies perpendicular to the magnetic field, the angle θ = 0 and cos(θ) = 1, simplifying the expression for magnetic flux to Φ = BA.
Given that:
-
- The initial magnetic field (Bi) is 450 mT (0.45 T)
-
- The final magnetic field (Bf) is 700 mT (0.70 T)
-
- The radius of the loop (r) is 75 mm (0.075 m)
-
- The time interval (Δt) is 0.10 s
The area (A) of the circular loop is πr2 = π(0.075)2 m2. The change in magnetic flux (ΔΦ) is (Bf - Bi)A. Substituting the given values and solving for the induced emf (ε):
ε = - ΔΦ/Δt = - ((Bf - Bi) * πr2)/Δt = - ((0.70 T - 0.45 T) * π * 0.0752 m2)/0.10 s = - π * (0.25 T) * 0.0752 m2/0.10 s.
After calculating, the resulting induced emf is approximately -0.044 V (or 44 mV with the negative sign indicating the direction of the induced emf according to Lenz's Law).