Ch 30: Electromagnetic Induction

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 30: Electromagnetic Induction

Problem 55c

Chapter 30, Problem 55c

INT A 20-cm-long, zero-resistance slide wire moves outward, on zero-resistance rails, at a steady speed of 10 m/s in a 0.10 T magnetic field. (See Figure 30.26.) On the opposite side, a 1.0 Ω carbon resistor completes the circuit by connecting the two rails. The mass of the resistor is 50 mg. If the wire is pulled for 10 s, what is the temperature increase of the carbon? The specific heat of carbon is 710 J/kg K.

Verified step by step guidance

Step 1: Calculate the electromotive force (EMF) induced in the slide wire using Faraday's law of induction. The formula for EMF is \( \text{EMF} = B \cdot v \cdot L \), where \( B \) is the magnetic field strength (0.10 T), \( v \) is the velocity of the wire (10 m/s), and \( L \) is the length of the wire (0.20 m).

Step 2: Determine the current \( I \) flowing through the circuit using Ohm's law. The formula is \( I = \frac{\text{EMF}}{R} \), where \( R \) is the resistance of the carbon resistor (1.0 Ω).

Step 3: Calculate the power dissipated in the resistor using the formula \( P = I^2 \cdot R \). This represents the rate at which electrical energy is converted into heat in the resistor.

Step 4: Find the total energy dissipated over the 10-second interval. Use the formula \( E = P \cdot t \), where \( t \) is the time duration (10 s). This gives the total heat energy transferred to the resistor.

Step 5: Calculate the temperature increase of the carbon resistor using the formula \( \Delta T = \frac{E}{m \cdot c} \), where \( m \) is the mass of the resistor (50 mg or 0.00005 kg) and \( c \) is the specific heat capacity of carbon (710 J/kg K).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field within a closed loop induces an electromotive force (EMF) in the wire. In this scenario, as the slide wire moves through the magnetic field, it experiences a change in magnetic flux, leading to the generation of an induced current that flows through the circuit, including the resistor.

Joule Heating

Joule heating, also known as resistive heating, occurs when an electric current passes through a resistor, converting electrical energy into thermal energy. The amount of heat generated can be calculated using the formula Q = I²Rt, where Q is the heat produced, I is the current, R is the resistance, and t is the time. This principle is crucial for determining the temperature increase of the carbon resistor.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). It is a material-specific property, and for carbon, it is given as 710 J/kg K. This concept is essential for calculating the temperature change of the carbon resistor after it absorbs the heat generated from the current flowing through it.

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Related Practice

Textbook Question

CALC Your camping buddy has an idea for a light to go inside your tent. He happens to have a powerful (and heavy!) horseshoe magnet that he bought at a surplus store. This magnet creates a 0.20 T field between two pole tips 10 cm apart. His idea is to build the hand-cranked generator shown in FIGURE P30.57. He thinks you can make enough current to fully light a 1.0 Ω lightbulb rated at 4.0 W. That's not super bright, but it should be plenty of light for routine activities in the tent. Find an expression for the induced current as a function of time if you turn the crank at frequency f. Assume that the semicircle is at its highest point at t = 0 s.

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Textbook Question

INT A 20-cm-long, zero-resistance slide wire moves outward, on zero-resistance rails, at a steady speed of 10 m/s in a 0.10 T magnetic field. (See Figure 30.26.) On the opposite side, a 1.0 Ω carbon resistor completes the circuit by connecting the two rails. The mass of the resistor is 50 mg. How much force is needed to pull the wire at this speed?

236

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Textbook Question

CALC The L-shaped conductor in FIGURE P30.54 moves at 10 m/s across and touches a stationary L-shaped conductor in a 0.10 T magnetic field. The two vertices overlap, so that the enclosed area is zero, at t = 0 s. The conductor has a resistance of 0.010 ohms per meter. a. What is the direction of the induced current?

INT You've decided to make the magnetic projectile launcher shown in FIGURE P30.58 for your science project. An aluminum bar slides along metal rails through a magnetic field B. The switch closes at t = 0 s, while the bar is at rest, and a battery of emf ε_bat starts a current flowing around the loop. The battery has internal resistance r. The resistances of the rails, which are separated by distance l, and the bar are effectively zero. Show that the bar reaches a terminal speed v_term, and find an expression for v_term.

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Textbook Question

CALC Your camping buddy has an idea for a light to go inside your tent. He happens to have a powerful (and heavy!) horseshoe magnet that he bought at a surplus store. This magnet creates a 0.20 T field between two pole tips 10 cm apart. His idea is to build the hand-cranked generator shown in FIGURE P30.57. He thinks you can make enough current to fully light a 1.0 Ω lightbulb rated at 4.0 W. That's not super bright, but it should be plenty of light for routine activities in the tent. With what frequency will you have to turn the crank for the maximum current to fully light the bulb? Is this feasible?

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