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Chapter 19: Problem 73

Two concentric circular wire loops in the same plane each carry a current. The larger loop has a current of 8.46 A circulating clockwise and has a radius of \(6.20 \mathrm{cm} .\) The smaller loop has a radius of \(4.42 \mathrm{cm} .\) What is the current in the smaller loop if the total magnetic field at the center of the system is zero? [See Eq. \((19-16) .]\)

Short Answer

Expert verified
Answer: The current in the smaller loop is 5.07 A counterclockwise.

Step by step solution

01

Find the magnetic field due to the larger loop at the center

According to Eq. (19-16) if we have I current in a wire loop with radius R, then the magnetic field at the center of the loop is given by: \(B = \frac{\mu _{0} I}{2R}\) Where \(\mu_0 = 4\pi 脳10^{-7} T.m/A\) is the permeability of free space. For the larger loop, we have I = 8.46 A (clockwise) and R = 6.20 cm: \(B_{large} = \frac{4\pi 脳10^{-7} \times 8.46}{2\times0.062} \mathrm{T}\) Note that when the larger loop is clockwise current, its magnetic field will direct inside the screen (Apply right-hand rule).
02

Find the magnetic field due to the smaller loop at the center

In order to make the total magnetic field at the center zero, the magnetic field of the smaller loop should have the opposite direction to the larger loop, so the current in the smaller loop must be counterclockwise. For the smaller loop, we have R = 4.42 cm and we need to find the current I: \(B_{small} = \frac{4\pi 脳10^{-7} \times I}{2\times0.0442} \mathrm{T}\) Since the magnetic field due to the smaller loop should be opposite to the magnetic field due to the larger loop and equal in magnitude: \(B_{small} = -B_{large}\)
03

Calculate the current in the smaller loop

Now we can just equate \(B_{small}\) and \(-B_{large}\), and solve for I (current in the smaller loop): \(\frac{4\pi 脳10^{-7} \times I}{2\times0.0442} = -\frac{4\pi 脳10^{-7} \times 8.46}{2\times0.062}\) We cancel out the \(4\pi\times10^{-7}\) on both sides and solve for I: \(I = -8.46 \times \frac{0.0442}{0.062} \mathrm{A}\) The negative sign indicates that the current flows counterclockwise, as we expected.
04

Find the current in the smaller loop

Calculate the current in the smaller loop: \(I = -8.46 \times \frac{0.0442}{0.062} = -5.07 \mathrm{A}\) So the current in the smaller loop is 5.07 A counterclockwise.

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Most popular questions from this chapter

An electron moves with speed \(2.0 \times 10^{5} \mathrm{m} / \mathrm{s}\) in a uniform magnetic field of \(1.4 \mathrm{T},\) pointing south. At one instant, the electron experiences an upward magnetic force of $1.6 \times 10^{-14} \mathrm{N} .$ In what direction is the electron moving at that instant? Be specific: give the angle(s) with respect to $\mathrm{N}, \mathrm{S}, \mathrm{E}, \mathrm{W},$ up, down. (If there is more than one possible answer, find all the possibilitics.)
The conversion between atomic mass units and kilograms is $$1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{kg}$$ In one type of mass spectrometer, ions having the same velocity move through a uniform magnetic field. The spectrometer is being used to distinguish $^{12} \mathrm{C}^{+}\( and \)^{14} \mathrm{C}^{+}$ ions that have the same charge. The \(^{12} \mathrm{C}^{+}\) ions move in a circle of diameter \(25 \mathrm{cm} .\) (a) What is the diameter of the orbit of \(^{14} \mathrm{C}^{+}\) ions? (b) What is the ratio of the frequencies of revolution for the two types of ion?

An electromagnetic rail gun can fire a projectile using a magnetic field and an electric current. Consider two conducting rails that are \(0.500 \mathrm{m}\) apart with a \(50.0-\mathrm{g}\) conducting rod connecting the two rails as in the figure with Problem \(46 .\) A magnetic field of magnitude \(0.750 \mathrm{T}\) is directed perpendicular to the plane of the rails and rod. A current of \(2.00 \mathrm{A}\) passes through the rod. (a) What direction is the force on the rod? (b) If there is no friction between the rails and the rod, how fast is the rod moving after it has traveled \(8.00 \mathrm{m}\) down the rails?
The magnetic field in a cyclotron is \(0.360 \mathrm{T}\). The dees have radius \(82.0 \mathrm{cm} .\) What maximum speed can a proton achicve in this cyclotron?
The conversion between atomic mass units and kilograms is $$1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{kg}$$ Natural carbon consists of two different isotopes (excluding $^{14} \mathrm{C},$ which is present in only trace amounts). The isotopes have different masses, which is due to different numbers of neutrons in the nucleus; however, the number of protons is the same, and subsequently the chemical properties are the same. The most abundant isotope has an atomic mass of \(12.00 \mathrm{u} .\) When natural carbon is placed in a mass spectrometer, two lines are formed on the photographic plate. The lines show that the more abundant isotope moved in a circle of radius \(15.0 \mathrm{cm},\) while the rarer isotope moved in a circle of radius \(15.6 \mathrm{cm} .\) What is the atomic mass of the rarer isotope? (The ions have the same charge and are accelerated through the same potential difference before entering the magnetic field.)
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