A single loop of wire with an area of 0.0900 m^2 is in a uniform magnetic field that has an initial value of 3.80 T, is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 T/s. (a) What emf is induced in this loop?

Verified step by step guidance

1. Identify the given values: magnetic field (B) = 3.80 T, area of the loop (A) = 0.0900 m^2, rate of change of magnetic field (dB/dt) = -0.190 T/s.
2. Recall Faraday's Law of Electromagnetic Induction, which states that the induced electromotive force (emf) in a loop is equal to the negative rate of change of magnetic flux through the loop. The formula is given by \( \mathcal{E} = -\frac{d\Phi_B}{dt} \).
3. Calculate the magnetic flux (\( \Phi_B \)) through the loop. Since the magnetic field is perpendicular to the plane of the loop, the magnetic flux is given by \( \Phi_B = B \times A \).
4. Substitute the rate of change of magnetic field into Faraday's Law to find the induced emf. Use \( \mathcal{E} = -A \times \frac{dB}{dt} \) since the area of the loop is constant.
5. Ensure the sign of the induced emf is correct according to Lenz's Law, which states that the induced emf will act in a direction to oppose the change in magnetic flux that produced it.

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that the electromotive force (emf) induced in a closed loop is directly proportional to the rate of change of magnetic flux through the loop. Mathematically, it is expressed as emf = -dΦ/dt, where Φ is the magnetic flux. This principle is fundamental in understanding how changing magnetic fields can generate electric currents.

Magnetic Flux

Magnetic flux (Φ) is defined as the product of the magnetic field (B) and the area (A) through which the field lines pass, taking into account the angle (θ) between the field and the normal to the surface. It is given by the formula Φ = B * A * cos(θ). In this scenario, since the magnetic field is perpendicular to the loop, the angle is 0 degrees, simplifying the calculation.

Lenz's Law

Lenz's Law states that the direction of the induced emf and the resulting current will be such that it opposes the change in magnetic flux that produced it. This law is a consequence of the conservation of energy and helps determine the polarity of the induced emf. In this case, as the magnetic field decreases, the induced current will flow in a direction that attempts to maintain the original magnetic field within the loop.