Question

A student wearing a \(15.0-\mathrm{g}\) 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 that all the energy produced is used in raising the temperature.

Short answer

Answer: Approximately 0.0341 °C.

Step-by-step solution

Calculate the initial magnetic flux inside the band

We are given the magnetic field strength B and the radius of the band r. The surface area of a circle can be calculated using the formula A = πr². Then, we can find the magnetic flux (Φ₁) using the formula Φ = BA, where B is the magnetic field strength and A is the area. B = 0.0800 T r = 0.750 cm = 0.00750 m A = πr² = π(0.00750)² ≈ 1.7671 × 10⁻⁴ m² Φ₁ = BA = (0.0800)(1.7671 × 10⁻⁴) = 1.4137 × 10⁻⁵ Wb

Calculate the final magnetic flux inside the band

Since the band is moved to a region with zero magnetic field, the final magnetic flux (Φ₂) becomes 0. Φ₂ = 0 Wb

Find the change in magnetic flux (∆Φ)

The change in magnetic flux is the difference between the final and initial magnetic fluxes: ∆Φ = Φ₂ - Φ₁ = 0 - 1.4137 × 10⁻⁵ = -1.4137 × 10⁻⁵ Wb

Calculate the induced EMF using Faraday's law of induction

Faraday's law states that the induced EMF (ε) equals the negative rate of change of magnetic flux: ε = -∆Φ/∆t. We are given the time (∆t) as 40.0 ms. ∆t = 40.0 ms = 4.0 × 10⁻² s ε = -(-1.4137 × 10⁻⁵)/(4.0 × 10⁻²) = 3.5343 × 10⁻⁴ V

Find the induced current using Ohm's law

Now, we can use Ohm's law to find the induced current (I). Ohm's law states that I = ε/R, where R is the resistance of the band, which is given as 61.9 μΩ. R = 61.9 μΩ = 61.9 × 10⁻⁶ Ω I = (3.5343 × 10⁻⁴)/(61.9 × 10⁻⁶) = 5.7080 A

Calculate the energy generated by the induced current

The energy generated by the induced current is given by the formula E = I²R∆t. E = (5.7080)²(61.9 × 10⁻⁶)(4.0 × 10⁻²) ≈ 0.0806 J

Apply the specific heat capacity formula to find the temperature rise

We use the formula Q = mc∆T, where Q is the energy, m is the mass of the band, c is the specific heat capacity, and ∆T is the temperature rise. We are given the mass (m) of the gold band as 15.0 g and the specific heat capacity (c) as 129 J/kg°C. We need to find ∆T. m = 15.0 g = 0.0150 kg c = 129 J/kg°C ∆T = Q/(mc) = (0.0806)/((0.0150)(129)) ≈ 0.0341 °C So, the temperature rise in the gold band is approximately 0.0341 °C.

Key concepts

Understanding Faraday's Law of Induction

Faraday's law of induction is a fundamental principle in electromagnetism, which plays a crucial role whenever the magnetic environment around a conductor changes. It describes how an electromotive force (EMF) is induced in a loop when the magnetic flux through the loop changes over time.

In our textbook exercise, a gold band experiences such a change when it moves from an area of a magnetic field to one without. The initial magnetic flux is calculated using the equation \( \Phi_1 = B A \) with \( B \) representing the magnetic field strength and \( A \) being the area enclosed by the loop. After the movement, the final flux, \( \Phi_2 \) is zero since the magnetic field it is in becomes null.

The change in magnetic flux, denoted as \( \Delta\Phi \) is simply the difference between \( \Phi_1 \) and \( \Phi_2 \) which, according to Faraday's law, induces an EMF. The induced EMF (\(\epsilon\)) is calculated using the formula \( \epsilon = -\frac{\Delta\Phi}{\Delta t} \), where \( \Delta t \) is the time taken for the change. The negative sign indicates that the induced EMF opposes the change in flux, adhering to Lenz's law which states that induced current directions always act to oppose the change in flux that produced them.

Specific Heat Capacity and Temperature Change

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. This quantity, often denoted \( c \) and measured in \( \text{J}/(\text{kg}\cdot \text{°C}) \), tells us about the thermal inertia of a material — how much it 'resists' temperature change when absorbing or releasing heat.

In the exercise, the specific heat capacity of the gold band is critical to determining how much the temperature of the band rises due to the thermal energy generated by the induced current. We calculate the actual rise in temperature (\( \Delta T \) ) using the formula \( Q = mc\Delta T \) where \( Q \) is the heat energy transferred, \( m \) is the mass of the object, and \( c \) is the specific heat capacity.

Since all of the energy produced by the induced current goes into heating the band, equating the energy generated to \( Q \) lets us isolate and calculate \( \Delta T \) , giving us the temperature rise due to the band’s movement through the magnetic fields.

Ohm's Law and Induced Current Calculations

Ohm's law is a cornerstone of electrical circuits, encapsulating the relationship between current, voltage, and resistance. It postulates that the current (\( I \) ) flowing through most conductors is directly proportional to the voltage (\( V \) ) across it, provided the temperature remains constant. The law is succinctly expressed as \( V = IR \) where \( R \) is the resistance.

In our student’s scenario with the gold band, once Faraday's law has given us the induced EMF (\( \epsilon \) ), Ohm's law can be used to find the current (\( I \) ) induced in the band. The band's resistance (\( R \)) is a known quantity, allowing us to rearrange Ohm’s law to \( I = \epsilon / R \).

With this current, we can further investigate the amount of energy that translates into thermal energy by utilizing the formula \( E = I^2R\Delta t \) . This relation indicates that the energy generated is proportional to the square of the current, the resistance, and the time over which current flows. The calculated energy helps us in understanding how much heat is generated, which ultimately raises the temperature of the gold band as described in the exercise.