Problem 6 - Physics

Question

A flat coil of wire is used with an LC-tuned circuit as a receiving antenna. The coil has a radius of \(0.25 \mathrm{~m}\) and consists of 450 turns. The transmitted radio wave has a frequency of \(1.2 \mathrm{MHz}\). The magnetic field of the wave is parallel to the normal to the coil and has a maximum value of \(2.0 \times 10^{-13} \mathrm{~T}\). Using Faraday's law of electromagnetic induction and the fact that the magnetic field changes from zero to its maximum value in one-quarter of a wave period, find the magnitude of the average emf induced in the antenna during this time.

Step-by-Step Solution

Answer

The average induced emf is approximately \( 8.48 \times 10^{-5} \mathrm{~V} \).

Step 1: Calculate the Wave Period

First, find the period of the radio wave, given the frequency. Use the formula for period: \[ T = \frac{1}{f} \]Here, the frequency \( f = 1.2 \times 10^6 \text{ Hz} \). Substitute into the formula to find \( T \).\[ T = \frac{1}{1.2 \times 10^6} \approx 8.33 \times 10^{-7} \text{ s} \]

Step 2: Find the Time for Quarter Wave Period

Change in magnetic field occurs over one-quarter of the wave's period. Calculate this time period:\[ \Delta t = \frac{T}{4} \]Using \( T = 8.33 \times 10^{-7} \text{ s} \), we have:\[ \Delta t = \frac{8.33 \times 10^{-7}}{4} \approx 2.08 \times 10^{-7} \text{ s} \]

Step 3: Calculate the Area of the Coil

The coil has a radius \( r = 0.25 \text{ m} \). Calculate the area \( A \) of one loop of the coil using the formula for the area of a circle:\[ A = \pi r^2 \]Substituting the given radius:\[ A = \pi (0.25)^2 \approx 0.196 \text{ m}^2 \]

Step 4: Calculate the Change in Magnetic Flux

Use the formula for magnetic flux \( \Phi \):\[ \Phi = B \cdot A \]The maximum magnetic field \( B = 2.0 \times 10^{-13} \text{ T} \) and the area \( A \approx 0.196 \text{ m}^2 \):\[ \Delta \Phi = B \cdot A = 2.0 \times 10^{-13} \times 0.196 \approx 3.92 \times 10^{-14} \text{ Tm}^2 \]

Step 5: Calculate the Induced EMF Using Faraday's Law

For a coil with \( N = 450 \) turns, the average induced emf is given by Faraday's law of induction:\[ \text{emf} = -N \frac{\Delta \Phi}{\Delta t} \]Substitute \( N = 450 \), \( \Delta \Phi \approx 3.92 \times 10^{-14} \text{ Tm}^2 \), and \( \Delta t \approx 2.08 \times 10^{-7} \text{ s} \):\[ \text{emf} \approx 450 \cdot \frac{3.92 \times 10^{-14}}{2.08 \times 10^{-7}} \approx 8.48 \times 10^{-5} \text{ V} \]

Step 6: Modification for Magnitude and Sign

The negative sign in Faraday's law indicates direction, but we're asked for the magnitude. Thus, the magnitude of the average induced emf is:\[ \approx 8.48 \times 10^{-5} \text{ V} \]

Key Concepts

Faraday's Law

Faraday's Law of Electromagnetic Induction is a fundamental principle used to describe how a change in magnetic field can induce an electric current in a closed circuit. This concept is crucial in understanding how electric generators and transformers work.

The law states that the induced electromotive force (emf) in a closed loop equals the negative change in magnetic flux over time. Mathematically, it can be expressed as:

• \( \text{emf} = - \frac{d\Phi}{dt} \)

Here, \( \Phi \) represents the magnetic flux, which depends on the magnetic field (B), the area of the loop (A), and the orientation of the field concerning the loop.

In practical applications such as the radio wave problem, we see how Faraday's Law relates changing magnetic fields to induced emf, which is critical in antenna technology used for receiving signals.

LC Circuit

An LC circuit consists of a capacitor (C) and an inductor (L) that can store energy in both electric and magnetic fields. These circuits are important elements in electronics, particularly in tuning and resonance applications.

The LC circuit resonates at a natural frequency, which is given by:

• \( f = \frac{1}{2\pi\sqrt{LC}} \)

In radio technology, LC circuits are used to select specific frequencies from a range of radio waves, making them essential in transmitters and receivers.

Understanding how LC circuits work helps explain their role in filtering and tuning signals, key for efficiently receiving and processing radio frequencies in devices such as radios, TVs, and mobile phones.

Radio Waves

Radio waves are electromagnetic waves with long wavelengths that are used in a wide variety of communication systems. They can travel long distances and penetrate through obstacles, which makes them ideal for transmitting signals over vast areas.

These waves have frequencies ranging from about 30 Hz to 300 GHz. Typically, different devices or applications use specific frequency ranges of radio waves, such as AM/FM radio, TV broadcasts, and mobile communications.
An example of radio waves in action is in radio broadcasting, where the wave carries audio signals to receivers like radios in homes or cars. The frequency of the carrier wave, such as the 1.2 MHz wave in the exercise, is used to tune into specific stations, a process that relies on principles like resonance in LC circuits.

Induced EMF

Induced electromotive force (emf) is the energy that pushes charges through a circuit as a result of changing magnetic fields. The concept is primarily derived from Faraday's Law, which explains how magnetic interactions can create electrical energy.

In practical terms, induced emf is what allows antennas to receive energy from radio waves. When a radio wave, which is a type of electromagnetic wave, interacts with the coil in an antenna, it changes the magnetic field through the coil. This change induces an emf, allowing the radio device to convert the wave into an electrical signal that can be processed.

Understanding induced emf is important not just for radio waves but also for all forms of electromagnetic induction, including in devices like electric transformers and induction cooktops, where magnetic fields are key to their operation.