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Chapter 27: Problem 16

The magnetic field inside a 20 -cm-diameter solenoid is increasing at 2.4 T/s. How many turns should a coil wrapped around the outside of the solenoid have so that the emf induced in the coil is \(15 \mathrm{V} ?\)

Short Answer

Expert verified
The solenoid should have 200 turns for the emf induced in the coil to be 15V.

Step by step solution

01

Identify the given values

The given values are: Diameter of the solenoid, \(d\)=20 cm, rate of increase of magnetic field, \( \frac{dB}{dt}\) =2.4 T/s, and the emf, \(E\) = 15 V
02

Calculate the cross-sectional area of the solenoid

The cross-sectional area, \(A\), can be calculated using the formula \(A= \frac{\pi d^2}{4}\), where \(d\) is the diameter of the solenoid. Convert \(d\) into meter first by dividing by 100, so \(d\)=0.2 m. So, \(A =\frac{\pi (0.2 m)^2}{4} = 0.0314 m^2\)
03

Use Faraday’s Law

Faraday’s law states that the emf, \(E\), induced in a coil is equal to the negative rate of change of magnetic flux \(\phi\), through the coil. This can be expressed as follows: \(E = -N \frac{d\phi}{dt}\), where \(N\) is the number of turns of the coil, \( \frac{d\phi}{dt}\) is the rate of change of magnetic flux. The magnetic flux \(\phi\) is equal to the product of the magnetic field \(B\) and the cross-sectional area \(A\) of the coil: \(\phi = B A\). So, \( \frac{d\phi}{dt} = A \frac{dB}{dt}\). Substitute this into the previous equation: \(E = -N A \frac{dB}{dt}\). From this, we can express N as \(N = -\frac{E}{A \frac{dB}{dt}}\)
04

Calculate the number of turns

Plugging the given values into the formula, we find \(N = \frac{-15V}{0.0314 m^2 * 2.4 T/s} = -200\) turns. The number of turns cannot be negative, so we take the absolute value to find that the solenoid should have 200 turns.

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Most popular questions from this chapter

A stent is a cylindrical tube, often made of metal mesh, that's inserted into a blood vessel to overcome a constriction. It's sometimes necessary to heat the stent after insertion to prevent cell growth that could cause the constriction to recur. One method is to place the patient in a changing magnetic field, so that induced currents heat the stent. Consider a stainless- steel stent 12 mm long by 4.5 mm diameter, with total resistance \(41 \mathrm{m} \Omega\). Treating the stent as a wire loop in the optimum orientation, find the rate of change of magnetic field needed for a heating power of \(250 \mathrm{mW}.\)

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A generator consists of a rectangular coil \(75 \mathrm{cm}\) by \(1.3 \mathrm{m},\) spinning in a 0.14 -T magnetic field. If it's to produce a \(60-\mathrm{Hz}\) alternating emf with peak value \(6.7 \mathrm{kV},\) how many turns must it have?

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