Problem 6 - University Physics with Modern Physics

Question

A toroidal solenoid has 500 turns, cross-sectional area \(6.25 \mathrm{cm}^{2},\) and mean radius \(4.00 \mathrm{cm} .\) (a) Calcualte the coil's self-inductance. (b) If the current decreases uniformly from 5.00 \(\mathrm{A}\) to 2.00 \(\mathrm{A}\) in 3.00 \(\mathrm{ms}\) , calculate the self- induced emf in the coil. (c) The current is directed from terminal \(a\) of the coil to terminal \(b\) . Is the direction of the induced emf from \(a\) to \(b\) or from \(b\) to \(a ?\)

Step-by-Step Solution

Answer

(a) 0.981 mH; (b) -981 mV; (c) From b to a.

Step 1: Calculate the coil's self-inductance

The formula for the self-inductance of a toroidal solenoid is given by \( L = \frac{\mu_0 N^2 A}{2\pi r} \), where \( L \) is the self-inductance, \( N \) is the total number of turns, \( A \) is the cross-sectional area, \( r \) is the mean radius of the solenoid, and \( \mu_0 \) is the permeability of free space, valued at \( 4\pi \times 10^{-7} \ \mathrm{T \cdot m/A} \).Substitute the given values: \( N = 500 \), \( A = 6.25 \ \mathrm{cm}^2 = 6.25 \times 10^{-4} \ \mathrm{m}^2 \), and \( r = 4.00 \ \mathrm{cm} = 0.04 \ \mathrm{m} \).\[ L = \frac{(4\pi \times 10^{-7}) \times 500^2 \times 6.25 \times 10^{-4}}{2\pi \times 0.04} \]Simplify to find \( L = 0.981 \ \mathrm{mH} \).

Step 2: Calculate the self-induced emf

The induced emf \( \epsilon \) in the coil can be calculated using Faraday's law of induction, \( \epsilon = -L \frac{\Delta I}{\Delta t} \), where \( L \) is the self-inductance, \( \Delta I \) is the change in current, and \( \Delta t \) is the change in time.\( \Delta I = 5.00 \ \mathrm{A} - 2.00 \ \mathrm{A} = 3.00 \ \mathrm{A} \) and \( \Delta t = 3.00 \ \mathrm{ms} = 3.00 \times 10^{-3} \ \mathrm{s} \).Substitute the values: \[ \epsilon = -0.981 \times \frac{3.00}{3.00 \times 10^{-3}} \]Calculate to find \( \epsilon = -981 \ \mathrm{mV} \).

Step 3: Determine the direction of the induced emf

The direction of the induced emf is determined by Lenz's law, which states that the induced emf acts in a direction to oppose the change in current that causes it.Since the current decreased, the induced emf will try to increase the current. Therefore, if the current flows from \( a \) to \( b \), the induced emf will act from \( b \) to \( a \) to oppose the reduction in current.

Key Concepts

Toroidal Solenoid

A toroidal solenoid is a type of coil that is shaped like a donut, where the wire is wound around a circular core. The key to a toroidal solenoid is that its magnetic field is almost entirely contained within the coil. This means it can produce a strong magnetic field across its cross-section, yet it barely affects the space outside its structure.
A toroid has several key components:

• Turns: This refers to the number of loops of wire around the core. More turns result in a stronger magnetic field.
• Cross-sectional Area: This measures the size of the coil's cross-section, influencing the field strength inside the coil.
• Mean Radius: The average distance from the center of the toroid cross-section. Affects the loop length of the wire and overall properties of the solenoid.

In our exercise, we utilized the formula for self-inductance, which relies on these parameters to calculate how effectively the solenoid can store magnetic energy.

Faraday's Law of Induction

Faraday's Law of Induction is a fundamental principle describing how electric currents can be generated by changing magnetic fields. This law explains the process of electromagnetic induction, where an induced electromotive force (emf) is generated in a conductor when it experiences a change in magnetic flux.
Mathematically, the induced emf (\( \epsilon \)) is proportional to the rate of change of magnetic flux through the circuit:

• The formula is: \( \epsilon = -L \frac{\Delta I}{\Delta t} \)
• \( L \) denotes self-inductance, \( \Delta I \) represents the change in current, and \( \Delta t \) is the time over which the current changes.

The negative sign is significant as it denotes the direction of the induced emf which opposes the change in current. This feature is further elaborated with Lenz's Law, showcasing the nature of the induction process.

Lenz's Law

Lenz's Law provides the direction of the induced current generated through electromagnetic induction. It ensures that the induced current will flow in a direction that opposes the cause of its generation, i.e., the change in magnetic flux. This phenomenon can be understood through a conservation of energy perspective.
In solving the exercise:

• When the current through the toroidal solenoid decreases, Lenz's Law indicates that the induced emf will act to oppose this reduction.
• If the original current flows from terminal a to terminal b, the induced emf would therefore act from b to a, trying to maintain the current flowing in the original direction.

Lenz's Law helps us understand the behavior of circuits in response to changing currents, playing a crucial role in the operation of transformers, inductors, and other electromagnetic devices.

Change in Current

Changes in current lead to changes in magnetic fields, which is a central principle when working with electromagnetic induction. This change is often what drives the generation of an induced emf in a coil or solenoid.
Components influencing how much emf is generated include:

• Magnitude of Current Change: A larger change in current will produce a stronger emf.
• Time over which the Current Changes: Faster changes in current lead to higher electromotive forces due to a steeper rate of change.
• Self-inductance: The self-inductance of the coil also defines how sensitive the system is to these changes, with higher inductance leading to larger emfs for the same rate of current change.

In the given problem, the change in current and its effects within the specified time period were used to calculate the induced emf in the toroidal solenoid.