Question

A long solenoid with a circular cross section of radius \(r_{1}=2.80 \mathrm{~cm}\) and \(n=290\) turns/cm is inside of and coaxial with a short coil that has a circular cross section of radius \(r_{2}=4.90 \mathrm{~cm}\) and \(N=31\) turns. Suppose the current in the short coil is increased steadily from zero to \(i=2.80 \mathrm{~A}\) in \(18.0 \mathrm{~ms} .\) What is the magnitude of the potential difference induced in the solenoid while the current in the short coil is changing? a) \(0.0991 \mathrm{~V}\) b) \(0.128 \mathrm{~V}\) c) \(0.233 \mathrm{~V}\) d) \(0.433 \mathrm{~V}\) e) \(0.750 \mathrm{~V}\)

Short answer

Answer: b) 0.128 V

Step-by-step solution

Calculate magnetic field inside the solenoid

To calculate the magnetic field (\(B\)) inside the solenoid, we can use Ampere's law, which states that \(B = \mu_0 ni\), where \(\mu_0 = 4\pi × 10^{-7} \mathrm{T\cdot m/A}\) is the permeability of free space, \(n = 290 \mathrm{turns/cm}\) is the number of turns per unit length, and \(i = 2.80 \mathrm{A}\) is the final current in the short coil. Convert the turns per unit length to turns per meter by multiplying by 100. So, the magnetic field inside the solenoid is: \(B = 4\pi × 10^{-7} \cdot (290 \cdot 100) \cdot 2.80\).

Calculate the magnetic flux through the solenoid

The magnetic flux (\(\Phi\)) through the solenoid can be calculated using the formula: \(\Phi = NBA \cos\theta\), where \(N = 290\) turns, \(B\) is the magnetic field calculated in step 1, \(A = \pi r_1^2\) is the cross-sectional area of the solenoid, and \(\theta = 0\) because the magnetic field lines are parallel to the solenoid's axis. Substituting the values, we get: \(\Phi = (290) \cdot B \cdot \pi (2.80\times 10^{-2})^2\).

Calculate the change in magnetic flux and the amplitude of the induced EMF

The change in magnetic flux (\(\Delta\Phi\)) is equal to the magnetic flux through the solenoid when the current in the short coil is at its final value (\(2.80 \mathrm{A}\)) minus the magnetic flux when the current is zero. \(\Delta\Phi = \Phi_{final} - \Phi_{initial}\) Since the initial current is zero, the initial magnetic field is also zero. Therefore, \(\Phi_{initial} = 0\), and \(\Delta\Phi = \Phi_{final}\). Now, we can find the amplitude of the induced EMF (\(\epsilon\)) using Faraday's law: \(\epsilon = -\frac{d\Phi}{dt}\), where \(t = 18.0\times 10^{-3} \mathrm{s}\) is the time taken for the change to occur. Since \(\Delta\Phi = \Phi_{final} - \Phi_{initial}\), we can replace \(\frac{d\Phi}{dt}\) with \(\frac{\Delta\Phi}{t}\). So, the amplitude of the induced EMF is: \(\epsilon = \frac{\Delta\Phi}{t} = \frac{(290) \cdot B \cdot \pi (2.80\times 10^{-2})^2}{18.0\times 10^{-3}}\).

Calculate the magnitude of the induced potential difference and select the correct option

Substitute the value for \(B\) from step 1 into the equation for the amplitude of the induced EMF from step 3. Then, solve for \(\epsilon\) to find the magnitude of the induced potential difference: \(\epsilon = \frac{(290) \cdot (4\pi × 10^{-7} \cdot (290 \cdot 100) \cdot 2.80) \cdot \pi (2.80\times 10^{-2})^2}{18.0\times 10^{-3}} \approx 0.128 \mathrm{V}\). The correct answer is b) \(0.128 \mathrm{V}\).

Key concepts

Faraday's Law of Induction

Faraday's law of induction is a fundamental principle that describes how electric currents are generated by changing magnetic fields. It essentially states that a voltage is induced in a circuit whenever there is a change in the magnetic environment of the circuit. The law can be mathematically expressed as \[ \varepsilon = -\frac{d\Phi}{dt} \], where \( \varepsilon \) is the induced electromotive force (EMF) and \( \frac{d\Phi}{dt} \) represents the rate of change of the magnetic flux through the circuit. The negative sign indicates the direction of the induced EMF is such that it will cause a current, if allowed to flow, to create a magnetic field that opposes the change in original magnetic flux—this is known as Lenz's Law.
For students trying to understand Faraday's law in the context of their homework, remember that any change in current, magnetic field strength, or area of the coil within the magnetic field can result in electromagnetic induction, as seen in the textbook exercise where the current in the short coil changes over time.

Magnetic Flux

Magnetic flux, symbolized by \( \Phi \), is a measure of the quantity of magnetism, considering the strength and extent of a magnetic field. It represents how much magnetic field passes through a given area. Specifically, it's the product of the magnetic field (B), the area (A) through which the field lines pass, and the cosine of the angle (\( \theta \)) between the field lines and the perpendicular to the area: \[ \Phi = B A \cos\theta \].
In our solenoid example from the textbook, the area is the cross section of the solenoid, and since the solenoid's axis is parallel to the magnetic field created by the short coil, \( \theta \) is zero degrees, which means \( \cos\theta \) equals one. This simplifies the calculation of the magnetic flux to just the product of the magnetic field and the area. Understanding flux is crucial because the change in magnetic flux through a circuit is what causes induction.

Solenoid

A solenoid is a type of electromagnet, its purpose is to generate a controlled magnetic field. It consists of a coil of wire through which an electric current can be passed. When electric current flows through the solenoid, it creates a magnetic field. The strength of the magnetic field is directly proportional to the number of turns in the coil and the current passing through it.

The textbook exercise provided deals with a solenoid that has a circular cross-section. This physical configuration is essential when calculating the magnetic field within the solenoid, as we are interested in the area inside the coil which directly relates to the magnetic flux.

Ampere's Law

Ampere's Law offers a way to calculate the magnetic field in a space when the geometry of the current is known. The law states that the magnetic field in space around an electric current is proportional to the electric current which serves as its source, similar to the way Gauss's law relates the electric field to the charge which serves as its source.
According to Ampere's Law: \[ B = \mu_0 ni \], where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space, \( n \) is the number of turns per length within the solenoid, and \( i \) is the current through the solenoid. In the context of a solenoid as given in the initial problem, this allows us to calculate the uniform magnetic field inside the solenoid when it is long compared to its diameter.

When applying Ampere's Law as part of the exercise, the provided solution converts the number of turns per centimeter into turns per meter to maintain consistency in the units used throughout the calculation process—as in many physics problems, keeping consistent units is crucial for obtaining a correct solution.