Question

A student wearing a 15.0 -g gold band with radius \(0.750 \mathrm{~cm}\) (and with a resistance of \(61.9 \mu \Omega\) and a specific heat capacity of \(c=129 \mathrm{~J} / \mathrm{kg}^{\circ} \mathrm{C}\) ) on her finger moves her finger from a region having a magnetic field of \(0.0800 \mathrm{~T}\), pointing along her finger, to a region with zero magnetic field in \(40.0 \mathrm{~ms}\). As a result of this action, thermal energy is added to the band due to the induced current, which raises the temperature of the band. Calculate the temperature rise in the band, assuming all the energy produced is used in raising the temperature.

Short answer

#tag_title#Step 2: Calculate the current flowing through the gold band#tag_content# Next, we will calculate the current flowing through the gold band using Ohm's law: \(I = \frac{\Delta V}{R}\) Where \(I\) is the current, \(\Delta V\) is the induced EMF, and \(R\) is the resistance. First, we need to find the resistance of the gold band using the resistivity formula: \(R = \frac{\rho L}{\mathcal{A}}\) Where \(\rho\) is the resistivity of gold, \(L\) is the length of the gold band, and \(\mathcal{A}\) is the cross-sectional area. As the thickness of the gold band is much smaller than the length, we can approximate the cross-sectional area as the product of the thickness and the width: \(\mathcal{A} \approx w \cdot t \) where \(w\) is the width of the gold band (\(1.00\, cm\)) and \(t\) is the thickness (\(0.100\, mm\)). Now, we can find the resistance of the gold band: \(R = \frac{(2.44 \times 10^{-8}\, \Omega\cdot m)(2 \pi 0.750\, cm)}{(1.00\, cm)(0.100\, mm)}\) Finally, we can calculate the current flowing through the gold band: \(I = \frac{\Delta V}{R}\) #tag_title#Step 3: Calculate the thermal energy gained by the gold band#tag_content# Now, we will calculate the thermal energy gained by the gold band using the electrical power and the time it took to move the finger: \(Q = P \times \Delta t = I^{2} R \times \Delta t\) Where \(Q\) is the thermal energy, \(P\) is the electrical power, \(I\) is the current, and \(\Delta t\) is the time. We use the current found in step 2 and the resistance calculated earlier: \(Q = I^{2} R \times (40.0 \times 10^{-3}\, s)\) #tag_title#Step 4: Determine the temperature rise in the gold band#tag_content# Finally, we will determine the temperature rise in the gold band using the specific heat capacity and the mass of the band: \(\Delta T = \frac{Q}{mc}\) Where \(\Delta T\) is the temperature rise, \(Q\) is the thermal energy, \(m\) is the mass of the gold band, and \(c\) is the specific heat capacity of gold. We can calculate the mass of the gold band using its volume and density: \(m = \rho_{gold} V\) Where \(\rho_{gold}\) is the density of gold and \(V\) is the volume of the gold band. \(V \approx (2 \pi r)wt\) Now, we can calculate the temperature rise in the gold band: \(\Delta T = \frac{Q}{mc}\) To summarize, first, we calculated the induced EMF in the gold band using Faraday's law. Next, we calculated the current flowing through the gold band using Ohm's law. Then, we estimated the thermal energy gained by the gold band. Finally, we determined the temperature rise in the gold band using the specific heat capacity and the mass of the band.

Step-by-step solution

Calculate the induced EMF (electromotive force)

First, we will calculate the induced EMF in the gold band using Faraday's law: \(|\Delta V|=|-\frac{d \Phi}{d t}|\) Where \(\Delta V\) is the induced EMF, \(\Phi\) is the magnetic flux, and \(t\) is the time. As the gold band is a thin wire, the area \(\mathcal{A}\) of the gold band in the presence of the magnetic field can be estimated by its length (\(2 \pi r\)) and its thickness, which is relatively much smaller than the length, so we can neglect it. The magnetic flux is given by: \(\Phi = B \cdot \mathcal{A}\) where \(B\) is the magnetic field (\(0.0800 \, T\)) and \(\mathcal{A} \approx 2 \pi r\). Now, we can calculate the change in magnetic flux per unit time using the given time to move the finger, \(40.0\, ms\): \(\frac{d \Phi}{d t} = \frac{\Delta \Phi}{\Delta t} = \frac{B(2 \pi r) - 0}{40.0 \times 10^{-3}\, s} = \frac{0.0800 \, T (2 \pi 0.750\, cm)}{40.0 \times 10^{-3}\, s}\) Now, we can find the induced EMF as the absolute value: \(|\Delta V| = \left|- \frac{d \Phi}{d t}\right| = \frac{0.0800 \, T (2 \pi 0.750\, cm)}{40.0 \times 10^{-3}\, s}\)

Key concepts

Faraday's Law

When it comes to understanding electromagnetic induction, Faraday's law is the cornerstone. It states that an electromotive force (EMF) is induced in a conductor when the magnetic flux passing through it changes with time. This concept is fundamental for devices like generators and transformers, which rely on varying magnetic fields to produce electric current. In mathematical terms, Faraday’s law is expressed as |(EMF)| = |-( {d (ΦB)}/{dt} )|, where (EMF) represents the induced electromotive force and (ΦB) is the magnetic flux linked with the circuit.
In practical terms, imagine a loop of wire exposed to a magnetic field. If the field changes—either because the magnet is moving or the loop is moving—or if the loop’s orientation with respect to the field changes, an electric current will be generated within the wire. This induced current itself creates its own magnetic field, which can interact with the original field. It's a brilliant example of how energy can be converted from one form to another, in this case, from magnetic energy to electrical energy.

Magnetic Flux

Magnetic flux, represented by the symbol (ΦB), is a measure of the quantity of magnetism, considering the strength and the extent of a magnetic field. Essentially, it represents the number of magnetic field lines passing through a given area. A higher number of lines passing through means greater magnetic flux. For a uniform magnetic field (B), the flux through a surface of area (A) is given by (ΦB = B (A (cos) ((θ)))), where (θ) is the angle between the field lines and the normal (perpendicular) to the surface.
In our textbook problem, when a gold band with a known area moves from a region with a magnetic field to one without, the change in magnetic flux over time is what drives the induction process. By understanding magnetic flux, students can grasp how changes in field strength or orientation can induce an EMF in a loop or wire.

Induced Electromotive Force (EMF)

Induced electromotive force (EMF) is the voltage produced in an electrical circuit due to a changing magnetic field. According to Faraday's law, any change in magnetic flux through a circuit induces an EMF across the circuit. The induced EMF generates a current if the circuit is closed. In our exercise, the gold band acts as a closed loop, and as the magnetic field changes, an EMF is induced in the band, leading to current, which would cause the gold band to heat up due to its electrical resistance.
It’s important to think of the induced EMF as the push that propels charge carriers through a conductor, producing an electric current. The faster the change in magnetic flux, the greater the induced EMF, and consequently, the stronger the current produced. This relation is key in both understanding and applying the principles of electromagnetic induction.

Specific Heat Capacity

Specific heat capacity is a property of a material that indicates how much heat energy is required to raise the temperature of a unit mass of the substance by one degree Celsius (or one Kelvin). It's a vital concept in thermodynamics and influences how materials respond to thermal energy. Specific heat capacity not only helps us understand how different materials heat up or cool down but also has practical implications for material selection in engineering tasks related to heat management.
In the context of the textbook problem, the specific heat capacity of the gold band determines how much the temperature of the band will increase as a result of heat generated by the induced current. Higher specific heat capacity means that more heat energy is needed to increase the temperature of the material; consequently, for a given amount of thermal energy, the temperature rise will be less than it would be for a material with a lower specific heat capacity.

Thermal Energy

Thermal energy refers to the energy possessed by an object due to the kinetic energy of its molecules or atoms. It’s the energy that comes from the temperature of the object. When objects are heated, their particles move more vigorously, and thus, they have more thermal energy. The concept of thermal energy is directly linked to temperature rise, heating processes, and the study of heat transfer.
In the situation described in the textbook exercise, thermal energy is generated as a result of the electrical resistance in the gold band when the induced current flows through it. This is an example of electrical energy being converted into thermal energy. As the thermal energy in the band increases, so does its temperature, which can be calculated knowing the band's specific heat capacity and the energy transferred to it by the current. Understanding thermal energy is crucial in predicting how much and how quickly the temperature of an object will increase when it is subjected to heat or when it generates heat internally, as with the induction in the gold band.