Problem 10 - College Physics

Question

\(\bullet\) A circular loop of wire with a radius of 12.0 \(\mathrm{cm}\) is lying flat on a tabletop. A magnetic field of 1.5 \(\mathrm{T}\) is directed vertically upward through the loop (Figure 21.49 ). (a) If the loop is removed from the field region in a time interval of 2.0 \(\mathrm{ms}\) , find the average emf that will be induced in the wire loop during the extraction process. (b) If the loop is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

Step-by-Step Solution

Answer

(a) 33.9 V; (b) Counterclockwise.

Step 1: Understand the problem

Identify the given values and the questions asked. We are given a circular loop with radius 12.0 cm and a magnetic field of 1.5 T. The magnetic field is vertical, and the loop is removed from this field in 2.0 ms. We need to find the average induced emf and determine the direction of the induced current observing from above.

Step 2: Convert units

Before calculating, convert the radius to meters since standard units for length in physics are in meters.\[\text{Radius} = 12.0\, \text{cm} = 0.12\, \text{m}\]

Step 3: Calculate the Area of the loop

Use the formula for the area of a circle, \(A = \pi r^2\), to determine the area of the loop.\[A = \pi (0.12\, \text{m})^2 = 0.0452\, \text{m}^2\]

Step 4: Calculate the initial magnetic flux

Magnetic flux \( \Phi \) is given by \( \Phi = B A \cos \theta \), where \( B \) is the magnetic field (1.5 T) and \( \theta = 0 \text{ degrees} \) since the field is perpendicular to the loop plane.\[\Phi = 1.5\, \text{T} \times 0.0452\, \text{m}^2 \times \cos 0 = 0.0678\, \text{Tm}^2\]

Step 5: Find the change in magnetic flux

The loop is removed entirely from the field, so the final magnetic flux is 0. The change in magnetic flux is the initial flux since it goes from \(0.0678 \, \text{Tm}^2 \) to 0.\[\Delta \Phi = \Phi_{initial} - \Phi_{final} = 0.0678\, \text{Tm}^2 - 0 = 0.0678\, \text{Tm}^2\]

Step 6: Calculate the induced emf using Faraday's Law

Faraday's Law states that the induced emf \( \mathcal{E} \) is the rate of change of magnetic flux. Thus, \[\mathcal{E} = \frac{-\Delta \Phi}{\Delta t}\]Given \( \Delta t = 2.0 \times 10^{-3}\, \text{s} \), calculate the emf:\[\mathcal{E} = \frac{-0.0678\, \text{Tm}^2}{2.0 \times 10^{-3}\, \text{s}} = -33.9\, \text{V}\]The magnitude of the average induced emf is \( 33.9 \, \text{V} \).

Step 7: Determine the direction of the induced current

Using Lenz's Law, the direction of the induced current will oppose the change in magnetic flux. Initially, the magnetic field is up, and its removal will cause an induced field in the same direction (upward) due to Lenz's Law. For an observer looking down, to maintain this upward-induced field, the current flows counterclockwise.

Key Concepts

Faraday's Law

Faraday's Law is a fundamental principle that explains how electrical energy is generated from magnetism. Specifically, it states that the electromotive force (emf) induced in a loop is proportional to the rate of change of magnetic flux through the loop. This principle allows us to transform changing magnetic environments into useful electrical currents. To put it simply, if you move a magnet through a coil, or vice versa, you create electricity. This is one of the core ideas behind generators and transformers seen in everyday life.

In mathematical terms, the induced emf \( \mathcal{E} \) is given as:

• \( \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \)

The negative sign is essential because it indicates the direction of the induced emf, as given by Lenz's Law, which states that the direction opposes the change in flux.
Applying Faraday's Law helps us calculate the induced emf when the magnetic environment of a conductor changes, such as when a loop is pulled out of a field.

Magnetic Flux

Magnetic flux, denoted as \( \Phi \), is a measure of the total magnetic field passing through a certain area. It can be thought of as the number of magnetic field lines that tickle a surface. The concept is crucial when discussing how magnetic fields interact with loops of wire. For our circular loop sitting flat on the table, the flux is determined not just by the strength of the magnetic field but also by the area it passes through.The formula for magnetic flux is:

• \( \Phi = B \cdot A \cdot \cos \theta \)

Where:

• \( B \) is the magnetic field strength,
• \( A \) is the area,
• \( \theta \) is the angle between the magnetic field line and the perpendicular to the surface.

In our scenario, as the loop is aligned with the magnetic field (\( \theta = 0\)), magnetic flux simplifies to \( \Phi = B \times A \). Understanding magnetic flux helps us determine how much 'magnetic presence' interacts with our loop, which is an essential step in finding the emf.

Lenz's Law

Lenz's Law provides an insightful addition to Faraday's Law, giving us the direction of the induced current. It tells us that the induced emf will always oppose the change in the magnetic environment that produced it. Essentially, it assures conservation of energy by preventing efforts or processes that could increase energy from occurring spontaneously. Imagining our exercise's scenario: when the loop is removed from the magnetic field, the system seeks to counteract this change. Since the field initially points upward, the loop generates an internal current that creates a magnetic field in the same direction—upward. This response is designed to oppose the loss of the external upward magnetic field, a classic embodiment of Lenz's Law. This principle is critical for predicting how induced currents behave and ensuring we correctly interpret the physical consequences of electromagnetic induction.

Circular Loop

A circular loop of wire serves as an ideal model for studying electromagnetic induction because its symmetry simplifies calculations and conceptual understanding. For our exercise, we're dealing with a circular loop that lies flat on a tabletop.A circle makes life easier in physics because its area calculation is straightforward:

• \( A = \pi r^2 \)

Where \( r \) is the radius of the loop. Our loop's radius was given as 12 cm, converted to meters for calculations, yielding an area of \( 0.0452 \, \text{m}^2 \). Such transformations ensure all calculations fit within the standard metric units used in physics.

Moreover, a circular geometry provides consistent interaction points for understanding how magnetic fields pass through or are removed from the loop—an essential part in calculating the induced emf and understanding current flow.

Induced Current Direction

Identifying the direction of the induced current produced in a loop is critical for practical applications of electromagnetic induction. Once we have calculated the magnitude of the induced emf, determining the direction involves understanding the response of the loop to environmental changes. For our circular loop viewed from above, we apply Lenz's Law to establish direction. When the loop is removed from an upward-pointing magnetic field, it tries to recreate this missing field upward, invoking a counteractive current flow.

To create this upward field by means of solenoidal currents, the induced current must flow in a counterclockwise direction as seen from above. Lenz's Law aids us in assuming this viewpoint, allowing us to visualize how even electronic designs impact the broader environment around them.