Question

In a classroom demonstration, an iron core is inserted into a large solenoid connected to an AC power source. The effect of the core is to magnify the magnetic field in the solenoid by the relative magnetic permeability, \(\kappa_{\mathrm{m}},\) of the core (where \(\kappa_{\mathrm{m}}\) is a dimensionless constant, substantially greater than unity for a ferromagnetic material, introduced in Chapter 28 ) or, equivalently, to replace the magnetic permeability of free space, \(\mu_{0}\), with the magnetic permeability of the core, \(\mu=\kappa_{\mathrm{m}} \mu_{0}\). a) The measured root-mean-square current drops from approximately \(10 \mathrm{~A}\) to less than \(1 \mathrm{~A}\) and remains at the lower value. Explain why. b) What would happen if the power source were \(\mathrm{DC}\) ?

Short answer

Answer: The rms current decreases due to the increased inductance caused by the insertion of the iron core. This increase in inductance leads to a higher impedance, ultimately reducing the rms current as per Ohm's law.

Step-by-step solution

Discuss the impact of inserting the core

When an iron core is inserted into the solenoid, the magnetic permeability of the core increases. This causes a substantial increase in the magnetic field inside the solenoid as given by the equation \(\mu=\kappa_{\mathrm{m}} \mu_{0}\). The inductance of the solenoid will also increase due to the increase in the magnetic permeability.

Explain the effect of increased inductance on impedance

The impedance (\(Z\)) of a solenoid is given by the equation: \(Z = \sqrt{R^2 + (ωL)^2}\), where \(R\) is the resistance, \(ω\) is the angular frequency, and \(L\) is the inductance. Since the inductance has increased due to the insertion of the iron core, the impedance will also increase.

Discuss the relationship between impedance and current

The relationship between the root-mean-square (rms) current (\(I_{rms}\)), the rms voltage (\(V_{rms}\)), and the impedance is given by Ohm's law: \(I_{rms} = \frac{V_{rms}}{Z}\). As the magnetic permeability of the core increases with the insertion of the iron core, the inductance of the solenoid increases. Due to this increase in inductance, the impedance also increases, and as a result, the rms current decreases significantly, dropping to a value less than \(1 A\). b) What would happen if the power source were DC? In the case of a DC power source, the magnetic field inside the solenoid would still increase due to the insertion of the iron core. However, with a DC source, the impedance equation simplifies to: \(Z = R\). This is because the inductive reactance (\(ωL\)) does not play a role in determining the impedance with a DC source. Therefore, there would be no substantial change in the current when the iron core is inserted, as the impedance remains unchanged.

Key concepts

Solenoid

A solenoid is a coil wound into a tightly packed helix. When an electric current passes through it, it creates a uniform magnetic field inside the coil. The strength of this magnetic field is directly proportional to the number of turns in the solenoid and the electric current flowing through it.

When a material like iron—which has a high relative magnetic permeability—is introduced into the solenoid, the magnetic field strength inside the solenoid is greatly increased. This is because the iron core amplifies the magnetic field generated by the coil, a principle exploited in electromagnets, inductors, and transformers.

AC and DC Power Sources

Alternating Current (AC) and Direct Current (DC) are two types of electrical power sources. AC power is characterized by a voltage and current that change direction periodically. This is the type of power typically supplied to homes and businesses. In AC circuits, factors like inductance and capacitance come into play, which affect the circuit's impedance.

DC power, on the other hand, provides a constant voltage and current that flows in one direction. It is used in batteries, electronics in vehicles, and more. In DC circuits, impedance is essentially the same as resistance since inductance is not a factor in a static field scenario.

Impedance in Solenoid

Impedance, denoted as global variable `Z`, is the effective resistance of an AC circuit to the flow of alternating current. It is a combination of resistance (global variable `R`) and reactance, which includes both inductive reactance (global variable `omega L` where `omega` is the angular frequency and `L` is the inductance) and capacitive reactance.

In the context of a solenoid, when an AC current is applied, the solenoid's inductance plays a significant role in its impedance. Introducing an iron core with a high magnetic permeability to the solenoid increases its inductance, which subsequently increases its impedance. This results in a reduction of the AC current through the solenoid—a key concept in controlling current in AC-powered devices.

Inductance

Inductance is a property of an electrical conductor—such as a solenoid—by which a change in current flowing through it induces an electromotive force (emf) in both the conductor itself (self-inductance) and in any nearby conductors (mutual inductance). This phenomenon is known as electromagnetic induction. The unit of inductance is the Henry (global variable `H`).

The presence of a material with high magnetic permeability, such as an iron core, in the center of the solenoid significantly enhances its inductance. This is due to the core's ability to concentrate the magnetic field, leading to a greater interaction with the coil's windings and thus a greater induced emf for any given change in current.