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. (b) If the loop has a resistance of 0.600 Ω, find the current induced in the loop.

Verified step by step guidance

1. Determine the rate of change of the magnetic field ($\frac{dB}{dt}$). In this problem, it is given as -0.190 T/s, indicating that the magnetic field is decreasing.
2. Calculate the electromotive force (emf) induced in the loop using Faraday's Law of Electromagnetic Induction, which is given by $\text{emf}=-N\frac{d{\mathrm{\Phi }}_B}{dt}$. Since there is only one loop (N=1), and the magnetic flux (${\mathrm{\Phi }}_B$) is given by $B\times A$, where A is the area of the loop.
3. Substitute the values into the Faraday's Law equation to find the emf. Use $\text{emf}=-1\times (A\times \frac{dB}{dt})$, where A is the area of the loop (0.0900 m^2) and $\frac{dB}{dt}$ is the rate of change of the magnetic field (-0.190 T/s).
4. Calculate the current induced in the loop using Ohm's Law, which states that $I=\frac{\text{emf}}{R}$, where R is the resistance of the loop.
5. Substitute the calculated emf and the given resistance (0.600 Ω) into the Ohm's Law equation to find the induced current.

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a loop induces an electromotive force (EMF) in the loop. The induced EMF is proportional to the rate of change of the magnetic flux, which can be calculated as the product of the area of the loop and the change in magnetic field over time. This principle is fundamental in understanding how electric currents can be generated in conductive materials.

Ohm's Law

Ohm's Law relates the voltage (V), current (I), and resistance (R) in an electrical circuit, expressed as V = I × R. In the context of induced EMF, the induced voltage across the loop can be used to determine the current flowing through it when the resistance is known. This relationship is crucial for calculating the current induced in the loop due to the changing magnetic field.

Magnetic Flux

Magnetic flux is defined as the product of the magnetic field strength (B) and the area (A) through which the field lines pass, taking into account the angle between the field and the normal to the surface. It is measured in Weber (Wb) and is essential for understanding how changes in the magnetic field affect the induced EMF in a loop. In this scenario, the decreasing magnetic field leads to a change in flux, which induces a current in the wire loop.