A metal ring 4.50 cm in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic field. These magnets produce an initial uniform field of 1.12 T between them but are gradually pulled apart, causing this field to remain uniform but decrease steadily at 0.250 T/s. (a) What is the magnitude of the electric field induced in the ring?

Verified step by step guidance

1. Identify the given values: diameter of the ring (d = 4.50 cm), initial magnetic field (B = 1.12 T), and rate of change of the magnetic field (dB/dt = -0.250 T/s).
2. Convert the diameter of the ring to meters (d = 0.045 m) to use in calculations involving SI units.
3. Calculate the area (A) of the ring using the formula for the area of a circle, A = \(\pi r^2\), where r is the radius of the ring. Since the radius is half of the diameter, r = 0.0225 m.
4. Use Faraday's Law of Electromagnetic Induction, which states that the magnitude of the induced electromotive force (emf) in a circuit is equal to the rate of change of magnetic flux through the circuit, \(|emf| = |\frac{d\Phi_B}{dt}|\).
5. Calculate the induced electric field (E) around the ring using the relationship between emf and electric field in a circular path, which is given by \(emf = E \cdot 2\pi r\). Solve for E to find the magnitude of the induced electric field.

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a closed loop induces an electromotive force (EMF) in that loop. The induced EMF is directly proportional to the rate of change of the magnetic flux. In this scenario, as the magnetic field decreases, the changing flux through the metal ring generates an electric field within it.

Magnetic Flux

Magnetic flux is defined as the product of the magnetic field strength and the area through which the field lines pass, taking into account the angle between the field lines and the normal to the surface. It is measured in Weber (Wb). In this case, the area of the ring and the strength of the magnetic field are crucial for calculating the change in magnetic flux as the field decreases.

Induced Electric Field

The induced electric field arises from the EMF generated by the changing magnetic flux. According to the relationship between EMF and electric field, the induced electric field can be calculated using the formula E = EMF / L, where L is the length of the loop. This electric field is responsible for driving currents if the circuit is closed, and its magnitude is directly related to the rate of change of the magnetic field.