Question

An electromagnet consists of 200 loops and has a length of \(10.0 \mathrm{~cm}\) and a cross-sectional area of \(5.00 \mathrm{~cm}^{2}\). Find the resonant frequency of this electromagnet when it is attached to the Earth (treat the Earth as a spherical capacitor)

Short answer

Answer: The resonant frequency of the electromagnet when attached to the Earth is approximately 3.122 MHz.

Step-by-step solution

Calculate the inductance of the electromagnet.

To calculate the inductance (L) of the electromagnet, we will use the following formula: L = µ₀ * N² * A / length where µ₀ = 4π × 10⁻⁷ Tm/A (permeability of free space) N = 200 (number of loops) A = 5.00 cm² = 5.00 × 10⁻⁴ m² (cross-sectional area in m²) length = 10.0 cm = 0.100 m (length in meters) Now, let's plug in the given values: L = (4π × 10⁻⁷ Tm/A) * (200)² * (5.00 × 10⁻⁴ m²) / (0.100 m) L ≈ 2.513 × 10⁻³ H (henrys)

Calculate the capacitance of the Earth.

To calculate the capacitance (C) of the Earth as a spherical capacitor, we will use the following formula: C = 4πε₀R where ε₀ = 8.854 × 10⁻¹² F/m (permittivity of free space) R = 6.371 × 10⁶ m (radius of the Earth) Now, let's plug in the values: C = (4π × 8.854 × 10⁻¹² F/m) * (6.371 × 10⁶ m) C ≈ 7.112 × 10⁻¹⁰ F (farads)

Calculate the resonant frequency.

To find the resonant frequency (f) of the LC circuit, we will use the following formula: f = 1 / (2π√(LC)) where L = 2.513 × 10⁻³ H (henrys) C = 7.112 × 10⁻¹⁰ F (farads) Now, let's plug in the values: f = 1 / (2π√((2.513 × 10⁻³ H) * (7.112 × 10⁻¹⁰ F))) f ≈ 3.122 × 10⁶ Hz (hertz) So, the resonant frequency of the electromagnet when attached to the Earth is approximately 3.122 MHz.

Key concepts

Inductance Calculation

Inductance is a fundamental concept in electromagnetism which signifies the ability of a coil to store energy in the magnetic field. For a straight coil, like our electromagnet, the inductance depends on several factors:

• Number of turns (N): More turns result in a higher inductance.
• Cross-sectional area (A): A larger area increases inductance.
• Length of the coil: A longer coil typically decreases inductance.
• Permeability of the core (µ₀): Air or vacuum permeability is often considered, using the constant \(4π × 10^{-7} \, Tm/A\).

To calculate the inductance ( \(L\)) of our given electromagnet, we use the formula:%\[L = \frac{μ₀N²A}{\text{length}}%\]Given:- \(μ₀ = 4π × 10^{-7} \, Tm/A\)- \(N = 200 \, \text{turns}\)- \(A = 5.00 \times 10^{-4} \, \text{m}^2\)- \(\text{length} = 0.100 \, \text{m}\)After computation, we get an inductance \(L ≈ 2.513 × 10^{-3} \, H\) (henrys). These calculations show how physical attributes of a coil influence its electromagnetic properties.

Capacitance of Earth

The Earth is often treated as a spherical capacitor when studying large-scale electromagnetic effects. The capacitance of a spherical object is calculated differently than flat plates. This gets into how capacitance is defined:

• Permittivity (\(ε₀\)): This is the measure of resistance in forming an electric field. Earth's capacitance is based on the permittivity of free space, which is approximately \(8.854 × 10^{-12} \, F/m\).
• Radius of the Sphere (R): For Earth, this is roughly \(6.371 × 10^{6} \, m\).

The formula for capacitance \(C\) of Earth is:\[C = 4πε₀R\]Substituting Earth's radius and permittivity, the calculated capacitance of Earth is \(7.112 × 10^{-10} \, F\). Though small, this defines how electromagnetic waves might be influenced by the Earth's property.

LC Circuit

An LC circuit is a simple type of electrical circuit that combines inductance (L) and capacitance (C). These components work together to produce oscillations, or alternating current at a specific frequency, called the resonant frequency. Understanding this helps in various applications from radios to medical instruments. Let's explore it:

• Inductance (L): Stores energy in a magnetic field. Here, it's around 2.513 millihenrys (\(H\)).
• Capacitance (C): Stores energy in an electric field. Our Earth as a capacitor is approximately 7.112 nanofarads (\(F\)).

The resonant frequency \(f\) in an LC circuit is obtained from:\[f = \frac{1}{2π\sqrt{LC}}\]After substituting the previously calculated values for \(L\) and \(C\), the resulting resonant frequency is about \(3.122 \, \text{MHz}\). This is a crucial frequency where the circuit naturally oscillates, showing a combination of Earth's properties and the electromagnet working in harmony.