Problem 15 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A 22.0-cm-diameter coil consists of 28 turns of circular copper wire \(2.6 \mathrm{~mm}\) in diameter. A uniform magnetic field, perpendicular to the plane of the coil, changes at a rate of \(8.65 \times 10^{-3} \mathrm{~T} / \mathrm{s}\). Determine \((a)\) the current in the loop, and (b) the rate at which thermal energy is produced.

Step-by-Step Solution

Answer

(a) Use Ohm's Law to find current; (b) Use thermal power formula \(P = I^2 R\).

Step 1: Calculate the Area of the Coil

The coil is circular with a diameter of 22.0 cm. First, convert the diameter to meters: \(D = 0.22 \text{ m}\). The radius \(r\) is half of the diameter: \(r = \frac{0.22}{2} = 0.11 \text{ m}\). The area \(A\) of the circle is given by the formula \(A = \pi r^2\). Thus, \(A = \pi (0.11)^2 \approx 0.038 \text{ m}^2\).

Step 2: Calculate the Change in Magnetic Flux

The magnetic flux \(\Phi\) through one turn of the coil is \(\Phi = B \times A\). Since the magnetic field \(B\) is changing, the rate of change of magnetic flux is given by \( \frac{d\Phi}{dt} = \frac{dB}{dt} \times A \). Substitute the given \( \frac{dB}{dt} = 8.65 \times 10^{-3} \text{ T/s} \) and the area \( A = 0.038 \text{ m}^2\): \( \frac{d\Phi}{dt} = 8.65 \times 10^{-3} \times 0.038 = 3.287 \times 10^{-4} \text{ Wb/s} \).

Step 3: Calculate the Induced EMF in the Coil

Using Faraday's Law of electromagnetic induction, the induced EMF \(\varepsilon\) in a coil with \(N\) turns is given by \(\varepsilon = -N \cdot \frac{d\Phi}{dt}\). Here, \(N = 28\). Thus, \(\varepsilon = -28 \times 3.287 \times 10^{-4} = -0.0092 \text{ V}\).

Step 4: Calculate the Resistance of the Copper Wire

The resistance \(R\) of the copper wire can be calculated using the formula \(R = \frac{\rho L}{A_c}\), where \(\rho = 1.68 \times 10^{-8} \text{ \(\Omega\)m}\) is the resistivity of copper, \(L\) is the total length of the wire, and \(A_c\) is the cross-sectional area of the wire. First, calculate \(A_c = \pi (0.0013)^2 = 5.309 \times 10^{-6} \text{ m}^2\) (diameter of wire is converted to radius and into meters). Length \(L = 28 \times 2\pi \times 0.11\) m (since the coil has 28 turns of the loop). Calculate \(R = \frac{1.68 \times 10^{-8} \times L}{5.309 \times 10^{-6}}\).

Step 5: Solve for Current in the Loop

Using Ohm's Law, \(I = \frac{\varepsilon}{R}\). Using the resistance from Step 4 and \(\varepsilon = 0.0092 \text{ V}\), substitute values to find current \(I\). After evaluating the resistance, plug in the values to get the current \(I = 0.0092 / R\).

Step 6: Calculate the Rate of Thermal Energy Produced

The rate at which thermal energy is produced or power \(P\) can be calculated with \(P = I^2 R\). Substitute the value of \(I\) from Step 5 and the resistance \(R\) to compute the thermal power \(P\).

Step 7: Summary of Results

The current in the loop \(I\) is determined using Ohm's Law, based on the induced EMF and the calculated resistance of the coil. The rate of thermal energy produced is then computed using the power formula \(P = I^2 R\).

Key Concepts

Faraday's Law

When it comes to electromagnetic induction, Faraday's Law is a fundamental principle that you need to understand. It states that a voltage, or electromotive force (EMF), is induced in a circuit whenever there is a change in magnetic flux through the circuit. This change can occur due to a change in the magnetic field strength, the area of the loop, or the orientation of the loop with respect to the magnetic field.
Faraday's Law is mathematically expressed as:

• \( \varepsilon = -N \frac{d\Phi}{dt} \)

where \( \varepsilon \) is the induced EMF, \( N \) is the number of turns in the coil, and \( \frac{d\Phi}{dt} \) is the rate of change of magnetic flux. The negative sign indicates that the induced EMF will create a current whose magnetic field opposes the change in flux, according to Lenz's Law.
This principle is in action every time there are fluctuations in the magnetic environment around a coil, generating electrical currents in the process.

Ohm's Law

Ohm's Law is a cornerstone in the study of electrical circuits. It relates the current flowing through a conductor to the voltage across it and the resistance it experiences.
This is expressed with the equation:

• \( I = \frac{V}{R} \)

where \( I \) represents the current in amperes, \( V \) is the voltage in volts, and \( R \) is the resistance in ohms.
For our exercise, the voltage is the induced EMF found using Faraday's Law, and the resistance is calculated based on the properties of the copper wire in the coil. Ohm's Law helps us solve for the current by rearranging the formula, giving us a straightforward way to link induced EMF and resistance to the current flowing in the loop.

Thermal Energy Production

Whenever current flows through a resistor, electrical energy turns into thermal energy due to resistance. This is known as Joule heating, or thermal energy production, and can be a critical factor in designing electrical systems.
The rate of thermal energy production or power dissipated as heat can be calculated with the formula:

• \( P = I^2 R \)

where \( P \) is the power in watts, \( I \) is the current, and \( R \) is the resistance.
In this exercise, once the current is known from Ohm's Law, we can compute the power using this formula. This calculation informs us about the efficiency of the coil and how much energy is being wasted as heat.

Magnetic Flux

Magnetic flux represents the total magnetic field passing through a given area. It is a crucial concept in understanding electromagnetic phenomena.
The magnetic flux \( \Phi \) is given by:

• \( \Phi = B \cdot A \cdot \cos(\theta) \)

where \( B \) is the magnetic field strength in teslas, \( A \) is the area the field penetrates in square meters, and \( \theta \) is the angle between the magnetic field and the normal to the surface. If the field is perpendicular, then \( \cos(\theta) = 1 \).
In the provided problem, the changing magnetic field and fixed area (the coil's area) directly affect the magnetic flux. Understanding this change over time is key to applying Faraday’s Law to induce EMF in a wire loop.

Resistance Calculation

The resistance calculation is essential as it determines how much the current will be hindered in its flow through the wire. Resistance in a wire depends on three main factors: its material's resistivity, its length, and its cross-sectional area.
The formula for resistance \( R \) is:

• \( R = \frac{\rho L}{A_c} \)

where \( \rho \) is the resistivity of the material (for copper, \( 1.68 \times 10^{-8} \, \Omega\text{m} \)), \( L \) is the length of the wire, and \( A_c \) is the wire's cross-sectional area.
For a coil, the wire length \( L \) equals the circumference of each loop times the number of turns. Calculating the resistance lets us use Ohm's Law effectively to find the actual current and the thermal energy produced.