Question

\(\mathrm{A}\) coil of inductance \(\mathrm{L}\) has an inductive reactance of \(\mathrm{X}_{\mathrm{L}}\) in an AC circuit in which the effective current is \(\mathrm{I}\). The coil is made from a super-conducting material and has no resistance. The rate at which power is dissipated in the coil is. (a) 0 (b) \(\mathrm{IX}_{\mathrm{L}}\) (c) \(I^{2} \mathrm{X}_{\mathrm{L}}\) (d) \(\mathrm{IX}^{2} \mathrm{~L}\)

Short answer

The rate at which power is dissipated in the superconducting coil is 0.

Step-by-step solution

Understand the power formula in an AC circuit

In an AC circuit, the total power (P) can be represented as the product of the effective voltage (V) and the effective current (I) and the power factor (pf). The power factor represents the phase angle between the voltage and current in the circuit and can be calculated as: \[pf = \cos(\phi) \] where \(\phi\) is the phase angle between voltage and current. In this case, since the coil has no resistance, the phase angle will be 90 degrees, as the voltage and current are perpendicular to each other in an inductive circuit.

Calculate the power factor

Since the phase angle between voltage and current in an inductive circuit is 90 degrees: \[pf = cos(90 ^\circ) = 0\]

Calculate the power dissipation in the coil

Using the power formula with the calculated power factor, we can find the power dissipation in the coil: \[P = VI \cdot pf\] \[P = VI \cdot 0\] \[P = 0\]

Choose the correct answer

Based on our calculations, the rate at which power is dissipated in the coil is 0. Therefore, the correct answer is: (a) 0

Key concepts

Superconducting Coil

A superconducting coil is a special type of coil made from a superconducting material, which is characterized by having zero electrical resistance. This unique property sets it apart from regular coils. When a material becomes superconducting, typically when cooled below a certain critical temperature, it allows current to flow through it without any loss of energy due to resistance.

Key features of superconducting coils include:

• No energy dissipation: Because they have no resistance, superconducting coils do not convert electrical energy into heat. Instead, all of the electrical energy remains as current flow.
• Inductive reactance: Although the coil experiences no resistive losses, it still has inductive properties, meaning it can oppose changes in the current due to its inductance.
• Magnetism: Superconducting coils can generate strong magnetic fields, which can be useful in applications such as MRI machines and other electromagnetic devices.

In an AC circuit, even without resistance, an inductive reactance exists, represented by the property known as inductive reactance \(X_L\). This quality allows the coil to momentarily store energy in its magnetic field. Thus, even with high current, the superconducting coil does not dissipate power.

Power Dissipation

In electrical circuits, power dissipation refers to the process by which an electrical component, or a circuit, converts electric power into another form of energy, such as heat. For example, in resistive circuits, power is dissipated as heat due to the resistance offered by the circuit components. This dissipative power can be calculated using the formula \(P = IV\), where \(P\) is the power, \(I\) is the current, and \(V\) is the voltage.

However, in the case of a superconducting coil, the scenario changes dramatically. Due to the absence of electrical resistance, the power dissipation in such a coil equals zero. In an AC circuit involving a superconducting coil, although the coil does have inductive reactance, it does not convert electrical energy into other forms, leading to zero power dissipation. This results in no energy loss, allowing for the very efficient transmission of electricity, which is one of the critical advantages of superconductivity.

By having zero power dissipation, superconductors enable applications that require high-efficiency energy transfers, reducing energy waste and heat production drastically. This attribute is particularly beneficial for applications in power grids and high-performance computing.

AC Circuit Analysis

Analyzing AC (Alternating Current) circuits involves understanding how different circuit components interact with the alternating current. In an AC circuit, the voltage and the current vary with time, typically sinusoidally. It's crucial to consider the reactance and resistance within the circuit to understand its behavior fully.

For a coil in an AC circuit, inductive reactance represents the property by which the coil resists changes in current. It can be expressed as \(X_L = 2\pi fL\), where \(X_L\) is the inductive reactance, \(f\) is the frequency of the AC source, and \(L\) is the inductance of the coil. This reactance plays a significant role because it adds a phase difference between the current and voltage within the circuit.

Given the coil is superconducting and has no resistance, the phase difference between the voltage and current is 90 degrees, resulting in a power factor of zero, as \(pf = \cos(90^\circ) = 0\). This effectively means that, although current and voltage are present, the coil does not dissipate any real power. Thus making the power dissipation formula \(P = VI \cdot pf\) equal to zero, explaining why no energy is lost in the circuit.

In AC circuit analysis, understanding such non-dissipative components like superconducting coils is crucial for designing systems that aim for high efficiency and low energy loss.