Problem 19 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A 25 -cm-diameter circular loop of wire has a resistance of \(150 \Omega\). It is initially in a 0.40-T magnetic field, with its plane perpendicular to \(\overrightarrow{\mathbf{B}},\) but is removed from the field in \(120 \mathrm{~ms}\). Calculate the electric energy dissipated in this process.

Step-by-Step Solution

Answer

The electric energy dissipated is approximately \(2.19 \times 10^{-5} \, \text{J}\).

Step 1: Determine the Initial Magnetic Flux

The initial magnetic flux through the loop is given by the formula \( \Phi = B \cdot A \cdot \cos(\theta) \). In this scenario, the magnetic field \( B = 0.40 \, \text{T} \), the area \( A = \pi (r^2) \), where \( r = \frac{25}{2} \times 10^{-2} \text{ m} \), and \( \theta = 0 \) because the plane is perpendicular. So, \( \Phi = 0.40 \cdot \pi \left( \frac{25}{2} \times 10^{-2} \right)^2 \cdot \cos(0) \).

Step 2: Calculate the Area of the Loop

First, calculate the radius: \( r = \frac{25}{2} = 12.5 \, \text{cm} = 0.125 \, \text{m} \). The area \( A = \pi r^2 = \pi (0.125)^2 \approx 0.0491 \, \text{m}^2 \).

Step 3: Find the Initial Magnetic Flux

Substitute the area into the flux equation: \( \Phi = 0.40 \cdot 0.0491 = 0.01964 \, \text{Wb} \).

Step 4: Determine the Change in Magnetic Flux

Initially, \( \Phi_{initial} = 0.01964 \, \text{Wb} \) and when the loop is removed, \( \Phi_{final} = 0 \, \text{Wb} \). Hence, change in flux \( \Delta \Phi = \Phi_{final} - \Phi_{initial} = 0 - 0.01964 = -0.01964 \, \text{Wb} \).

Step 5: Calculate the Induced EMF

Using Faraday's Law of Induction, induced EMF \( \mathcal{E} = - \frac{\Delta \Phi}{\Delta t} \). Here, \( \Delta t = 120 \, \text{ms} = 0.12 \, \text{s} \), so \( \mathcal{E} = - \frac{-0.01964}{0.12} \approx 0.16367 \, \text{V} \).

Step 6: Calculate the Electric Energy Dissipated

The energy dissipated \( W \) in a resistor is given by \( W = \frac{\mathcal{E}^2 \cdot t}{R} \). Substitute the values: \( W = \frac{(0.16367)^2 \times 0.12}{150} \approx 2.19 \times 10^{-5} \, \text{J} \).

Key Concepts

Magnetic Flux

Magnetic flux is a measure of the amount of magnetic field passing through a given area. It is linked to the concept of magnetic field lines, which are used to represent the strength and direction of magnetic fields. Think of magnetic flux as the number of magnetic field lines passing through a loop. The formula to calculate magnetic flux \( \Phi \) is:

\[\Phi = B \cdot A \cdot \cos(\theta)\]

where:

• \(B\) is the magnetic field (in teslas).
• \(A\) is the area of the loop (in square meters).
• \(\theta\) is the angle between the magnetic field and the normal to the surface.

In our exercise, the magnetic field was perpendicular to the loop, meaning \(\theta = 0\). This makes \(\cos(0) = 1\), simplifying the formula. Think of a perpendicular magnetic field as being at its most effective strength through the loop, maximizing the flux.

The unit for magnetic flux is the weber (Wb). Understanding magnetic flux is critical when studying electromagnetic induction and analyzing circuits in changing magnetic fields.

Faraday's Law of Induction

Faraday's Law of Induction is essential to understanding how electric currents can be generated in a conductor by changing magnetic flux. It states that a change in magnetic flux through a loop induces an electromotive force (EMF) in the loop. The law can be expressed mathematically as:

\[\mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \]

where:

• \(\mathcal{E}\) is the induced EMF (in volts).
• \(\Delta \Phi\) is the change in magnetic flux (in webers).
• \(\Delta t\) is the time over which the change occurs (in seconds).

The negative sign is a consequence of Lenz's Law, indicating that the induced EMF will generate a current in a direction to oppose the change in magnetic flux.

For instance, imagine removing our loop from a magnetic field: initially, there's flux through it, but it reduces to zero as it leaves the field. The reduction in flux induces an EMF. Calculating this EMF allows us to determine the current generated by the change. Faraday’s Law illustrates the interplay between electricity and magnetism, forming the foundation of many electromagnetic applications, like transformers and electric generators.

Electric Energy

Electric energy in circuits can be calculated with the knowledge of voltage, current, and resistance. In our scenario, the energy dissipated in the wire loop, similar to a resistor, is crucial to compute. The formula for energy dissipated \(W\) in a resistor through a time \(t\) is:

\[W = \frac{\mathcal{E}^2 \cdot t}{R}\]

where:

• \(\mathcal{E}\) is the induced EMF (in volts).
• \(t\) is the time during which it acts (in seconds).
• \(R\) is the resistance (in ohms).

In the exercise, this method calculates the energy lost as heat due to the induced current (Ohmic loss). This energy is a direct conversion of the change in magnetic environment to thermal energy due to resistive forces. The principle could be simplified as converting magnetic energy (induced due to a changing field) into electric energy, then into thermal form. By understanding the transition of energies here, one can better comprehend how electric devices consume energy and the factors influencing efficiency.