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Chapter 28: Problem 77

A horizontally oriented coil of wire of radius \(5.00 \mathrm{~cm}\) and carrying a current, \(i\), is being levitated by the south pole of a vertically oriented bar magnet suspended above the center of the coil. If the magnetic field on all parts of the coil makes an angle \(\theta\) of \(45.0^{\circ}\) with the vertical, determine the magnitude and the direction of the current needed to keep the coil floating in midair. The magnitude of the magnetic field is \(B=0.0100 \mathrm{~T}\), the number of turns in the coil is \(N=10.0\), and the total coil mass is \(10.0 \mathrm{~g}\).

Short Answer

Expert verified
Answer: The magnitude of the current needed is \(6.34 \mathrm{~A}\), and the current should flow in a clockwise direction when viewed from above.

Step by step solution

01

Calculate the Gravitational Force

To calculate the force of gravity acting on the coil, we can use the formula: $$F_g = m \cdot g$$ where \(F_g\) is the gravitational force, \(m\) is the mass of the coil, and \(g\) is the acceleration due to gravity (approximately \(9.81 \mathrm{~m/s^2}\)). Given the mass \(m = 10.0 \mathrm{~g} = 0.010 \mathrm{~kg}\), we can calculate the gravitational force: $$F_g = 0.010 \mathrm{~kg} \times 9.81 \mathrm{~m/s^2} = 9.81 \times 10^{-2} \mathrm{N}$$
02

Calculate the Magnetic Force

The magnetic force acting on the coil can be calculated using the formula: $$F_m = B \cdot I \cdot L \cdot \sin(\theta)$$ where \(F_m\) is the magnetic force, \(B\) is the magnetic field, \(I\) is the current, \(L\) is the length of the wire, and \(\theta\) is the angle between the magnetic field and the vertical direction. In our case, \(B = 0.0100 \mathrm{~T}\), \(\theta = 45.0^{\circ}\), and since there are \(N = 10.0\) turns in the coil, the total length of the wire in the field is $$L = 2\pi r \cdot N = 2\pi \cdot 0.05 \mathrm{~m} \cdot 10.0 = \pi \mathrm{~m}$$
03

Equate Magnetic Force to Gravitational Force

To keep the coil levitated, we need the magnetic force to be equal to the gravitational force: $$F_m = F_g$$ Using the magnetic force formula, we have: $$B \cdot I \cdot L \cdot \sin(\theta) = 9.81 \times 10^{-2} \mathrm{N}$$ Now, we can solve for the current \(I\): $$I = \frac{9.81 \times 10^{-2} \mathrm{N}}{0.0100 \mathrm{~T} \cdot \pi \mathrm{~m} \cdot \sin(45.0^{\circ})}$$
04

Calculate the Current

Now, let's plug in the values and find the magnitude of the current needed: $$I = \frac{9.81 \times 10^{-2} \mathrm{N}}{0.0100 \mathrm{~T} \cdot \pi \mathrm{~m} \cdot \frac{\sqrt{2}}{2}} = \frac{9.81 \times 10^{-2} \mathrm{N} \cdot 2 \sqrt{2}}{0.0100 \mathrm{~T} \cdot \pi \mathrm{~m}}$$ $$I = 6.34 \mathrm{~A}$$
05

Determine the Direction of the Current

To determine the direction of the current, we need to consider the orientation of the magnetic field and the levitating force. Since the south pole of the magnet is above the coil, the magnetic field lines are directed downwards. By using the Right Hand Rule, we can determine that the current needs to flow in a clockwise direction when viewed from above for the magnetic force to act upwards and to counteract the gravitational force. So, to keep the coil floating in midair, the required current is \(6.34 \mathrm{~A}\) flowing in a clockwise direction.

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Most popular questions from this chapter

Many electrical applications use twisted-pair cables in which the ground and signal wires spiral about each other. Why?

Two particles, each with charge \(q\) and mass \(m\), are traveling in a vacuum on parallel trajectories a distance \(d\) apart, both at speed \(v\) (much less than the speed of light). Calculate the ratio of the magnitude of the magnetic force that each exerts on the other to the magnitude of the electric force that each exerts on the other: \(F_{\mathrm{m}} / \mathrm{F}_{\mathrm{e}}\)

A current element produces a magnetic field in the region surrounding it. At any point in space, the magnetic field produced by this current element points in a direction that is a) radial from the current element to the point in space. b) parallel to the current element. c) perpendicular to the current element and to the radial direction.

Consider a model of the hydrogen atom in which an electron orbits a proton in the plane perpendicular to the proton's spin angular momentum (and magnetic dipole moment) at a distance equal to the Bohr radius, \(a_{0}=5.292 \cdot 10^{-11} \mathrm{~m} .\) (This is an oversimplified classical model.) The spin of the electron is allowed to be either parallel to the proton's spin or antiparallel to it; the orbit is the same in either case. But since the proton produces a magnetic field at the electron's location, and the electron has its own intrinsic magnetic dipole moment, the energy of the electron differs depending on its spin. The magnetic field produced by the proton's spin may be modeled as a dipole field, like the electric field due to an electric dipole discussed in Chapter 22 Calculate the energy difference between the two electronspin configurations. Consider only the interaction between the magnetic dipole moment associated with the electron's spin and the field produced by the proton's spin.

Discuss how the accuracy of a compass needle in showing the true direction of north can be affected by the magnetic field due to currents in wires and appliances in a residential building.
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