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Ch 29: Electromagnetic Induction
Chapter 29, Problem 29

The magnetic field within a long, straight solenoid with a circular cross section and radius R is increasing at a rate of dB/dt. (e) What is the magnitude of the induced emf in a circular turn of radius R/2 that has its center on the solenoid axis?

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Identify the relevant concepts: The problem involves the concept of electromagnetic induction, specifically Faraday's Law of Electromagnetic Induction, which states that the induced electromotive force (emf) in any closed circuit is equal to the negative rate of change of the magnetic flux through the circuit.
Determine the magnetic flux through the circular turn: The magnetic flux, \( \Phi \), through the circular turn of radius \( R/2 \) is given by the product of the magnetic field \( B \) and the area \( A \) of the circle. The area can be calculated using the formula \( A = \pi (R/2)^2 \).
Apply Faraday's Law: According to Faraday's Law, the magnitude of the induced emf \( \mathcal{E} \) is given by \( \mathcal{E} = |\frac{d\Phi}{dt}| \). Since the magnetic field is changing at a rate of \( \frac{dB}{dt} \), and the area of the circle is constant, the rate of change of the magnetic flux can be expressed as \( \frac{d\Phi}{dt} = A \frac{dB}{dt} \).
Substitute the expressions: Substitute the expression for the area \( A \) into the formula for \( \frac{d\Phi}{dt} \) to find the rate of change of magnetic flux through the circular turn.
Calculate the induced emf: Finally, use the expression for \( \frac{d\Phi}{dt} \) to find the magnitude of the induced emf using Faraday's Law. This will give you the final expression for the induced emf in terms of \( R \), \( \pi \), and \( \frac{dB}{dt} \).

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Key Concepts

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

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a closed loop induces an electromotive force (emf) in that loop. The induced emf is proportional to the rate of change of the magnetic flux, which is the product of the magnetic field strength and the area through which it passes. This principle is fundamental in understanding how changing magnetic fields can generate electric currents.
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Magnetic Flux

Magnetic flux is a measure of the quantity of magnetism, taking into account the strength and the extent of a magnetic field. It is defined as the product of the magnetic field (B) and the area (A) perpendicular to the field through which it passes, expressed as Φ = B · A. In the context of the solenoid, the flux changes as the magnetic field strength increases, which is crucial for calculating the induced emf.
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Induced EMF in a Loop

The induced emf in a circular loop placed in a varying magnetic field can be calculated using the formula ε = -dΦ/dt, where ε is the induced emf and dΦ/dt is the rate of change of magnetic flux through the loop. For a loop of radius R/2 centered on the axis of a solenoid, the area and the magnetic field strength at that radius must be considered to determine the magnitude of the induced emf accurately.
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Related Practice
Textbook Question
A flat, rectangular coil of dimensions l and w is pulled with uni-form speed v through a uniform magnetic field B with the plane of its area perpen-dicular to the field (Fig. E29.14). (a) Find the emf induced in this coil. (b) If the speed and magnetic field are both tripled, what is the induced emf?
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Textbook Question
The conducting rod ab shown in Fig. E29.29 makes contact with metal rails ca and db. The apparatus is in a uniform magnetic field of 0.800 T, perpendicular to the plane of the figure.

(b) In what direction does the current flow in the rod?
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Textbook Question
A metal ring 4.50 cm in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic field. These magnets produce an initial uniform field of 1.12 T between them but are gradually pulled apart, causing this field to remain uniform but decrease steadily at 0.250 T/s. (a) What is the magnitude of the electric field induced in the ring?
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Textbook Question
The magnetic field within a long, straight solenoid with a circular cross section and radius R is increasing at a rate of dB/dt. (g) What is the magnitude of the induced emf if the radius in part (e) is 2R?
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Textbook Question
A long, thin solenoid has 900 turns per meter and radius 2.50 cm. The current in the solenoid is increasing at a uniform rate of 36.0 A/s. What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) 0.500 cm from the axis of the solenoid; (b) 1.00 cm from the axis of the solenoid?
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Textbook Question
A long, thin solenoid has 400 turns per meter and radius 1.10 cm. The current in the solenoid is increasing at a uniform rate di/dt. The induced electric field at a point near the center of the solenoid and 3.50 cm from its axis is 8.00*10^-6 V/m. Calculate di/dt.
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