91Ó°ÊÓ

A solenoid of length \(40.0 \mathrm{~cm}\) is made of 10000 circular coils. It carries a steady current of 12.0 A. Near its center is placed a small, flat, circular metallic coil of 200 circular loops, each with a radius of \(2.00 \mathrm{~mm}\). This small coil is oriented so that it receives half of the maximum magnetic flux. A switch is opened in the solenoid circuit and its current drops to zero in \(25.0 \mathrm{~ms}\). (a) What was the initial flux through the small coil? (b) Determine the average induced emf in the small coil during the \(25.0 \mathrm{~ms}\). (c) If you look along the long axis of the solenoid so that the initial 12.0 A current is clockwise, determine the direction of the induced current in the small inner coil during the time the current drops to zero. (d) During the \(25.0 \mathrm{~ms}\), what was the average current in the small coil, assuming it has a resistance of \(0.15 \Omega ?\)

Step by step solution

01

Calculate the Magnetic Field of the Solenoid

The magnetic field inside a solenoid is given by the formula: \( B = \mu_0 n I \), where \( \mu_0 = 4\pi \times 10^{-7} \mathrm{~T}\cdot\mathrm{m/A} \) is the permeability of free space, \( n \) is the number of loops per unit length \( (N/L) \), and \( I \) is the current. Here, \( N = 10000 \) and \( L = 0.40 \mathrm{~m} \) gives \( n = 25000 \mathrm{~m}^{-1} \). The current \( I = 12.0 \mathrm{~A} \). Thus, \( B = (4\pi \times 10^{-7}) \times 25000 \times 12.0 \).

02

Magnetic Flux Through the Small Coil

The magnetic flux \( \Phi \) through one loop of the small coil is \( \Phi = B \cdot A \cdot \cos\theta \), where \( A = \pi r^2 \) is the area of one loop with \( r = 2.00 \times 10^{-3} \mathrm{~m} \), and \( \theta = 0 \) gives \( \cos\theta = 0.5 \). Calculate \( A = \pi \times (2.00 \times 10^{-3})^2 \). Then, \( \Phi = B \cdot A \cdot 0.5 \). The total flux \( \Phi_{total} \) is \( 200 \times \Phi \) because there are 200 loops.

03

Average Induced EMF in the Coil

The average induced EMF \( \mathcal{E} \) is given by Faraday's Law: \( \mathcal{E} = - \frac{N \Delta\Phi}{\Delta t} \). Here, \( \Delta\Phi = -\Phi_{total} \) as current drops to zero and \( \Delta t = 25 \times 10^{-3} \mathrm{~s} \). Calculate \( \mathcal{E} = - \frac{200 \times \Phi}{25 \times 10^{-3}} \).

04

Direction of the Induced Current

Using Lenz's Law, the direction of the induced current is such that it opposes the change in magnetic flux. Since the original current was clockwise and is decreasing, the induced current will be such that it tries to maintain the magnetic field in the same direction, thus flowing counterclockwise when viewed along the solenoid's axis.

05

Average Current in the Small Coil

The average current \( I_{avg} \) can be found using Ohm's Law: \( \mathcal{E} = I_{avg} R \), where \( R = 0.15 \Omega \). Solve for \( I_{avg} = \frac{\mathcal{E}}{0.15} \) using the \( \mathcal{E} \) calculated in Step 3.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solenoids

A solenoid is a long coil of wire wrapped in many turns, usually in the form of a cylinder. When electric current flows through the solenoid, it creates a magnetic field inside it. The strength of this magnetic field depends on several factors:

• The number of turns of the wire (N),
• The length (L) of the solenoid, and
• The current (I) flowing through it.

The magnetic field inside a solenoid is uniform and parallel to its axis, making it an essential component in electromagnets and inductors. The formula for the magnetic field inside the solenoid is given by\[B = \mu_0 \times n \times I\]where \(\mu_0\) is the permeability of free space (a constant value \(4\pi \times 10^{-7} \mathrm{~T}\cdot\mathrm{m/A}\)), and \(n\) denotes the number of turns per unit length, calculated as \(\frac{N}{L}\). Solenoids are fundamental in producing controlled magnetic environments for research and electronic applications.

Electromagnetic Induction

Electromagnetic induction is a phenomenon where a change in magnetic field within a closed loop induces an electromotive force (EMF). This was discovered by Michael Faraday and is a principle that underpins many electrical technologies.

The process is governed by Faraday's Law of Induction, which states that the induced EMF in a loop is equal to the rate of change of magnetic flux through the loop.

• Induced EMF is proportional to the number of loops (N) and the rate at which the magnetic flux changes (\(\Delta \Phi\)).
• If the magnetic flux changes rapidly, a larger EMF is induced.

This principle is used in transformers, electric generators, and many other electromagnetic devices. It's important to note that the direction of the induced current is such that it opposes the change in magnetic flux, as per Lenz's Law. This opposition is nature's way of conserving energy by generating a current that attempts to maintain the original magnetic condition.

Magnetic Flux

Magnetic flux (\(\Phi\)) is a measure of the total magnetic field passing through a given area. It can be thought of as the number of magnetic field lines that intersect a specified area. The concept of magnetic flux is critical in understanding electromagnetic induction.

Magnetic flux is calculated as:\[\Phi = B \cdot A \cdot \cos\theta\]where \(B\) is the magnetic field, \(A\) is the area through which the field lines pass, and \(\theta\) is the angle between the field lines and the normal to the surface.

• A maximum flux occurs when the field lines are perpendicular to the surface (\(\theta = 0\)).
• If the field lines are parallel to the surface (\(\theta = 90^\circ\)), the flux is zero.

Understanding flux is crucial for calculating induced EMF, as changes in flux are what lead to the generation of electric currents in conductive loops.

Induced EMF

Induced Electromotive Force (EMF) is the voltage generated by the change in magnetic flux in a coil or circuit. It is a direct consequence of Faraday's Law of Electromagnetic Induction. The induced EMF can be calculated using the formula:\[\mathcal{E} = -N \frac{\Delta \Phi}{\Delta t}\]where \(N\) is the number of turns in the coil, \(\Delta \Phi\) is the change in magnetic flux, and \(\Delta t\) is the time over which the change occurs.

The negative sign in this equation is a representation of Lenz's Law, which states that the direction of the induced EMF and the resulting current will oppose the change in magnetic flux that produced them.

• This means the induced EMF will create a magnetic field that opposes the change in the original magnetic field.
• Understanding this opposition is key in predicting the direction and magnitude of induced currents in loops or coils.

Induced EMF has applications in designing circuits, transformers, and other electrical and electronic devices.