Question

A short coil with radius \(R=10.0 \mathrm{~cm}\) contains \(N=30.0\) turns and surrounds a long solenoid with radius \(r=8.00 \mathrm{~cm}\) containing \(n=60\) turns per centimeter. The current in the short coil is increased at a constant rate from zero to \(i=2.00 \mathrm{~A}\) in a time of \(t=12.0 \mathrm{~s}\). Calculate the induced potential difference in the long solenoid while the current is increasing in the short coil.

Short answer

Question: Calculate the induced potential difference in a long solenoid due to the increasing current in a surrounding short coil from 0 to i in a time t. Answer: The induced potential difference in the long solenoid can be calculated using Faraday's Law of Electromagnetic Induction: \(\text{Potential Difference} = -\frac{d\Phi_{\text{total}}}{dt}\) Where \(\Phi_{\text{total}}\) is the total magnetic flux through the solenoid due to the current in the short coil and can be found using the steps provided in the solution.

Step-by-step solution

Calculate the magnetic field inside the solenoid due to the current in the short coil

First, we will calculate the magnetic field inside the solenoid due to the current in the short coil using the Biot-Savart Law. The magnetic field inside a solenoid is given by: \(B = \frac{\mu_0 ni}{2}\) Where \(\mu_0\) is the permeability of free space \((4\pi \times 10^{-7} T\text{m}/\text{A})\), \(n\) is the number of turns per centimeter in the solenoid, and \(i\) is the current in the short coil.

Calculate the magnetic flux through a single turn of the solenoid

Next, we will calculate the magnetic flux through a single turn of the solenoid. The magnetic flux is given by: \(\Phi = BA\) Where \(A\) is the area of the solenoid's cross-section, which can be calculated using the formula: \(A = \pi r^2\)

Calculate the total magnetic flux through the solenoid

Now, we will calculate the total magnetic flux through the solenoid by multiplying the magnetic flux through a single turn by the total number of turns N in the short coil: \(\Phi_{\text{total}} = N\Phi\)

Calculate the rate of change of magnetic flux

To calculate the induced potential difference, we need to find the rate of change of the magnetic flux. As the current in the short coil is increasing at a constant rate with time, we can calculate the rate of change of magnetic flux as follows: \(\frac{d\Phi_{\text{total}}}{dt} = N\frac{d\Phi}{dt}\)

Calculate the induced potential difference in the long solenoid

Now, we can use Faraday's Law of Electromagnetic Induction to calculate the induced potential difference in the long solenoid: \(\text{Potential Difference} = -\frac{d\Phi_{\text{total}}}{dt}\) Substitute the values from steps 1 to 4 into the above equation to find the induced potential difference in the long solenoid.

Key concepts

Biot-Savart Law

The Biot-Savart Law is crucial for understanding how currents produce magnetic fields. In essence, this law is used to calculate the magnetic field produced at a particular point in space by a segment of current-carrying wire. The point magnetic field contribution can be thought of as:

• Each infinitesimal segment of current contributes to the magnetic field.
• The total magnetic field is the vector sum of all these contributions.
• For a solenoid, a special application of this law simplifies the calculation since the field inside the solenoid is uniform.

When dealing with solenoids, the Biot-Savart Law is often simplified to: \[B = \frac{\mu_0 ni}{2}\] where \(B\) is the magnetic field, \(\mu_0\) is the permeability of free space, \(n\) is the turn density (turns per unit length), and \(i\) is the current. This formula indicates that the magnetic field depends directly on both the number of turns per centimeter and the current passing through the coil.

Magnetic Flux

Magnetic Flux, often denoted by \(\Phi\), is a measure of the quantity of magnetism, being the total magnetic field passing through a given surface. It is calculated by the product of the magnetic field \(B\) and the area \(A\) through which it passes, taking into account the angle between them:

• Magnetic Flux through a solenoid is expressed as \(\Phi = BA\).
• \(A\) is typically the cross-sectional area of the solenoid, and can be calculated as \(A = \pi r^2\) for a circle.
• The magnetic flux highlights how much magnetic field penetrates through a surface.

In a solenoid, the flux through a single loop is compounded by the number of loops, resulting in significant induced effects when changes occur, such as changing the current.

Faraday's Law

Faraday's Law of Electromagnetic Induction is a cornerstone for understanding how changing magnetic fields can induce currents in a conducting loop. The law states:

• A changing magnetic flux through a loop induces an electromotive force (EMF) in the loop.
• The EMF is directly proportional to the negative rate of change of magnetic flux.

Mathematically, it is given by: \[\text{EMF} = -\frac{d\Phi}{dt}\] where \(\Phi\) represents the magnetic flux. This negative sign, known as Lenz's Law, indicates that the induced EMF will generate a current whose magnetic field opposes the original change in flux. When applying Faraday's Law to a series of coils or a solenoid, the formula becomes \(\text{EMF} = -N \frac{d\Phi}{dt}\), where \(N\) represents the number of turns of the coil.

Induced Potential Difference

The induced potential difference is a result of Faraday's Law, occurring when there is a change in the magnetic flux through a loop. This concept connects closely with the notion of electromotive force (EMF). To find the actual induced potential difference in a physical situation, one should:

• Calculate or observe the change in magnetic field within the region of interest.
• Measure the effective cross-sectional area intersected by the changing field.
• Compute the rate of change of magnetic flux.
• Apply Faraday's Law, considering the number of turns \(N\), to obtain the total induced EMF.

In the exercise reviewed, understanding the rate at which the current is changing helps predict the potential difference. It is this potential difference that can prompt the flow of current, showing the dynamic interplay of fields and forces in purely conductor wire loops.