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Chapter 11: Problem 88

A circular coil with 200 turns has a radius of 2.0 cm. (a) What current through the coil results in a magnetic dipole moment of \(3.0 \mathrm{Am}^{2}\) ? (b) What is the maximum torque that the coil will experience in a uniform field of strength \(5.0 \times 10^{-2} \mathrm{T} ?\) (c) If the angle between \(\mu\) and \(B\) is \(45^{\circ},\) what is the magnitude of the torque on the coil? (d) What is the magnetic potential energy of coil for this orientation?

Short Answer

Expert verified
The short answer to the given question is as follows: a) The current needed for a magnetic dipole moment of 3.0 Am虏 is 11.9 A. b) The maximum torque experienced by the coil in a uniform magnetic field of strength 5.0 脳 10鈦宦� T is 0.15 Nm. c) The magnitude of the torque when the angle between the magnetic moment and the magnetic field is 45 degrees is 0.106 Nm. d) The magnetic potential energy of the coil for this orientation is -0.106 J.

Step by step solution

01

a)

Find the amount of current needed such that the magnetic dipole moment of the coil is 3.0 Am虏. We are given the magnetic dipole moment, \(\mu = 3.0\: Am^{2}\), the number of turns, \(n = 200\), and the radius of the coil, \(r = 2.0\: cm = 0.02\: m\). First, let's find the area of the coil: \(A = \pi{r}^{2} = \pi{(0.02)}^{2} = 1.26 \times 10^{-3} m^{2}\) Now, we can use the formula for the magnetic dipole moment to find the current: \(\mu = nIA\) \(I = \frac{\mu}{nA}\) \(I = \frac{3.0}{200 \times 1.26 \times 10^{-3}} = \frac{3.0}{0.252} = 11.9 A\) Thus, the current needed is 11.9 A.
02

b)

Find the maximum torque experienced by the coil in a uniform magnetic field of strength 5.0 脳 10鈦宦� T. We are given the magnetic field strength, \(B = 5.0 \times 10^{-2}T\). Now, to find the maximum torque, the angle between the magnetic moment and the magnetic field should be \(90^{\circ}\), so \(\sin{\theta} = 1\). Using the formula for torque: \(蟿_{max} = \mu{B}\sin{\theta}\) \(蟿_{max} = (3.0)(5.0 \times 10^{-2})(1) = 0.15 Nm\) Thus, the maximum torque experienced by the coil is 0.15 Nm.
03

c)

Determine the magnitude of the torque when the angle between the magnetic moment and the magnetic field is 45 degrees. We are given the angle \(\theta = 45^{\circ}\). Using the torque formula: \(蟿 = \mu{B}\sin{\theta}\) \(蟿 = (3.0)(5.0 \times 10^{-2})\sin{45^{\circ}}\) \(蟿 = 0.15 \times \frac{\sqrt{2}}{2} = 0.106 Nm\) Thus, the magnitude of the torque on the coil when the angle between the magnetic moment and the magnetic field is 45 degrees is 0.106 Nm.
04

d)

Calculate the magnetic potential energy of the coil for this orientation. Using the magnetic potential energy formula, given \(\theta = 45^{\circ}\): \(U = -\mu{B}\cos{\theta}\) \(U = -(3.0)(5.0 \times 10^{-2})\cos{45^{\circ}}\) \(U = -0.15 \times \frac{\sqrt{2}}{2} = -0.106 J\) Thus, the magnetic potential energy of the coil for this orientation is -0.106 J.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Circular Coil
A circular coil is a fundamental component in various electromagnetic applications. It's essentially a wire wound in the shape of a circle, and the number of loops or 'turns' significantly affects its magnetic properties. The magnetic dipole moment of such a coil is a vector quantity pivotal for understanding its interaction with external magnetic fields. It depends on the coil's current (I), the number of turns (n), and the area (A) it encloses.

To compute the area of our coil, we use the area formula of a circle, which is \( A = \pi r^2 \). Once we know the area and we're given the number of turns and the desired magnetic dipole moment, we can solve for the current using the equation \(\mu = nIA\). In our exercise, with a targeted magnetic dipole moment of \(3.0\mathrm{Am}^2\), and given values for n and r, we deduced the current needed to be 11.9 A.
Torque in a Magnetic Field
When a circular coil like ours is placed in a magnetic field, it experiences a torque due to the interaction between the magnetic field and the magnetic dipole moment of the coil. This torque tries to align the dipole moment with the magnetic field. The torque, \(\tau\), can be calculated using the equation \(\tau = \mu B \sin(\theta)\), where \(\theta\) is the angle between the magnetic dipole moment and the magnetic field vector, B.

For maximum torque, this angle is \(90^\circ\), which makes \(\theta\) at its maximum since \(\theta\) equals 1. Conversely, as in our step 3 of the solution, when the angle is \(45^\circ\), we insert the corresponding sine value to calculate the actual torque experienced by the coil.
Magnetic Potential Energy
Magnetic potential energy represents the potential of a magnetic object to do work due to its position or orientation within an external magnetic field. For a coil like the one in our exercise, the magnetic potential energy is given by the formula \(U = -\mu B \cos(\theta)\). The negative sign indicates the work done is towards minimizing the system's energy, trying to align the dipole with the field.

In the given scenario, when the angle is \(45^\circ\), we observe that the potential energy expression involves the cosine of \(\theta\), contrasting with the sine in the torque calculation. By placing the values into our formula, we derive the potential energy of the coil for this particular orientation. This expresses the energy state of our system and how much work could be exerted as the coil tends to align with the magnetic field.

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Most popular questions from this chapter

A mass spectrometer is being used to separate common oxygen-16 from the much rarer oxygen-18, taken from a sample of old glacial ice. (The relative abundance of these oxygen isotopes is related to climatic temperature at the time the ice was deposited.) The ratio of the masses of these two ions is 16 to \(18,\) the mass of oxygen- 16 is \(2.66 \times 10^{-26} \mathrm{kg}, \quad\) and they are singly charged and travel at \(5.00 \times 10^{6} \mathrm{m} / \mathrm{s}\) in a 1.20 -T magnetic field. What is the separation between their paths when they hit a target after traversing a semicircle?

(a) What voltage will accelerate electrons to a speed of \(6.00 \times 10^{-7} \mathrm{m} / \mathrm{s} ?\) (b) Find the radius of curvature of the path of a proton accelerated through this potential in a \(0.500-\mathrm{T}\) field and compare this with the radius of curvature of an electron accelerated through the same potential.

A copper rod of mass \(m\) and length \(L\) is hung from the ceiling using two springs of spring constant \(k\). A uniform magnetic field of magnitude \(B_{0}\) pointing perpendicular to the rod and spring (coming out of the page in the figure) exists in a region of space covering a length \(w\) of the copper rod. The ends of the rod are then connected by flexible copper wire across the terminals of a battery of voltage \(V\) Determine the change in the length of the springs when \(a\) current I runs through the copper rod in the direction shown in figure. (Ignore any force by the flexible wire.)

Describe the primary function of the electric field and the magnetic field in a cyclotron.

The Hall effect is to be used to find the density of charge carriers in an unknown material. A Hall voltage 40 \(\mu \mathrm{V}\) for 3 -A current is observed in a 3 -T magnetic field for a rectangular sample with length \(2 \mathrm{cm},\) width \(1.5 \mathrm{cm},\) and height 0.4 \(\mathrm{cm} .\) Determine the density of the charge carriers.
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