Problem 32 - University Physics with Modern Physics

Question

A metal ring 4.50 \(\mathrm{cm}\) in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic ficld. 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 \(\mathrm{T} / \mathrm{s}\) (a) What is the magnitude of the electric field induced in the ring? (b) In which direction (clockwise or counterclockwise) does the current flow as viewed by someone on the south pole of the magnet?

Step-by-Step Solution

Answer

(a) 2.815 mV/m; (b) Counterclockwise current.

Step 1: Determine the Change in Magnetic Flux

The magnetic flux \( \Phi \) through the ring is given by the formula:\[ \Phi = B \cdot A \]where \( B \) is the magnetic field and \( A \) is the area of the ring. The area \( A \) of the ring is given by:\[ A = \pi \left( \frac{d}{2} \right)^2 \]where \( d = 4.50 \text{ cm} = 0.0450 \text{ m} \). Therefore,\[ A = \pi \left( \frac{0.0450}{2} \right)^2 = 1.59 \times 10^{-3} \text{ m}^2 \].The rate of change of the magnetic field is given as \( \frac{dB}{dt} = -0.250 \text{ T/s} \). So, the rate of change of magnetic flux is:\[ \frac{d\Phi}{dt} = A \cdot \frac{dB}{dt} = 1.59 \times 10^{-3} \text{ m}^2 \cdot (-0.250 \text{ T/s}) = -3.975 \times 10^{-4} \text{ Tm}^2/s \].

Step 2: Calculate the Induced Electric Field

From Faraday's law of electromagnetic induction, the magnitude of induced electromotive force (emf) \( \mathcal{E} \) is given by:\[ \mathcal{E} = - \frac{d\Phi}{dt} \]Thus,\[ \mathcal{E} = 3.975 \times 10^{-4} \text{ Tm}^2/s \].The induced electric field \( E \) in the ring is related to the emf by:\[ \mathcal{E} = E \cdot 2\pi r \]where \( r \) is the radius of the ring, \( r = \frac{d}{2} = 0.0225 \text{ m} \).Thus,\[ E = \frac{\mathcal{E}}{2\pi r} = \frac{3.975 \times 10^{-4}}{2\pi \times 0.0225} = 2.815 \times 10^{-3} \text{ V/m} \].

Step 3: Determine the Direction of the Induced Current

Using Lenz's Law, the direction of the induced current will oppose the change in magnetic flux. As the field decreases, the induced current will create a magnetic field to oppose this reduction. Viewed from the south pole, the initial field points north. A counterclockwise current is needed to create a magnetic field pointing north to oppose the field's decrease. Therefore, the current flows counterclockwise as viewed from the south pole.

Key Concepts

Faraday's Law

Faraday's Law of electromagnetic induction is a cornerstone of electromagnetism. It explains how a changing magnetic field within a closed loop induces an electromotive force (emf).
This principle is mathematically expressed in the equation:\[ \mathcal{E} = - \frac{d \Phi}{dt} \]where:

• \( \mathcal{E} \) is the induced emf.
• \( \Phi \) represents the magnetic flux.

The negative sign indicates the direction of the induced emf opposes the change in magnetic flux, a reflection of Lenz's Law.
Faraday's Law plays a crucial role in electrical generation, where mechanical energy is converted into electrical energy. In our exercise, Faraday’s Law helps compute the induced electric field as the magnetic field changes over time.

Lenz's Law

Lenz's Law gives us an intuitive understanding of the direction of induced currents. It states that the induced current will always circulate in a way that opposes the change in magnetic flux.
In simple terms:

• If the magnetic flux through a loop increases, the induced current will flow in a direction to create a magnetic field opposing this increase.
• If the magnetic flux decreases, as in our exercise, the induced current creates a field to supplement the flux.

In this particular scenario, since the magnetic field is decreasing, Lenz’s Law predicts that the induced current will be in a counterclockwise direction viewed from the south pole. This direction opposes the decrease, trying to maintain as much of the original magnetic field as possible.

Magnetic Flux

Magnetic flux quantifies the amount of magnetic field passing through a certain area. It is given by the equation:\[ \Phi = B \cdot A \]where:

• \( B \) is the magnetic field strength.
• \( A \) is the area vector perpendicular to the field.

In the problem, the magnetic flux changes because the magnetic field strength decreases over time. Understanding this change is key to determining the induced emf.
The close relationship between magnetic flux and emf highlights the dynamic nature of electromagnetic systems, where changes in one element like magnetic field strength have direct effects on others like induced current.

Induced Electric Field

An induced electric field arises when a magnetic field changes over time. This is a crucial concept in electromagnetism, illustrating how electric fields are not just produced by static charges but also by dynamic magnetic conditions.
The formula linking induced electric field \( E \) to the electromotive force \( \mathcal{E} \) is:\[ E = \frac{\mathcal{E}}{2\pi r} \]where:

• \( \mathcal{E} \) is the induced emf calculated through Faraday's Law.
• \( r \) is the radius of the circular path through which the field is measured.

The induced electric field acts along the path of the conductor, causing charges to move and creating a current. In the exercise, the decreasing magnetic field induces an electric field that is responsible for driving this current within the metal ring.