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Chapter 21: Problem 47

Two pieces of the same wire have the same length. From one piece, a square coil containing a single loop is made. From the other, a circular coil containing a single loop is made. The coils carry different currents. When placed in the same magnetic field with the same orientation, they experience the same torque. What is the ratio \(I_{\text { square }} / I_{\text { circle }}\) the current in the square coil to the current in the circular coil?

Short Answer

Expert verified
The ratio of the currents is \( \frac{\pi}{4} \).

Step by step solution

01

Relate Torque with Magnetic Moment

The torque \( \tau \) experienced by a current-carrying loop in a magnetic field is given by \( \tau = nIAB\sin(\theta) \). Here, \( n \) is the number of turns (which is 1 for both loops), \( I \) is the current, \( A \) is the area, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the plane of the coil and the magnetic field. Since \( \theta \) is the same for both coils and \( \sin(\theta) = 1 \), we have \( \tau = IAB \).
02

Compare Coil Areas

The total length of the wire for each coil is \( L \). For the square coil, each side is \( \frac{L}{4} \), giving an area of \( (\frac{L}{4})^2 = \frac{L^2}{16} \). For the circular coil, the circumference is \( L = 2\pi r \), giving a radius of \( r = \frac{L}{2\pi} \) and an area of \( \pi (\frac{L}{2\pi})^2 = \frac{L^2}{4\pi} \).
03

Set Torque Equations Equal

Since the torque experienced by both coils is the same, equate torques: \( I_{\text{square}}\frac{L^2}{16}B = I_{\text{circle}}\frac{L^2}{4\pi}B \). Simplifying, \( I_{\text{square}}\frac{1}{16} = I_{\text{circle}}\frac{1}{4\pi} \).
04

Solve for the Current Ratio

Re-arrange the equation from Step 3 to find the ratio of currents: \( \frac{I_{\text{square}}}{I_{\text{circle}}} = \frac{4\pi}{16} \). Simplifying gives \( \frac{I_{\text{square}}}{I_{\text{circle}}} = \frac{\pi}{4} \).

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Key Concepts

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

Magnetic Moment
The magnetic moment is a fundamental concept in understanding how objects interact with magnetic fields. It quantifies the strength and orientation of a magnet or current-carrying coil's reaction to an external magnetic field. In simple terms, the magnetic moment is a measure of the coil's ability to align with the magnetic field lines.
For a loop of current, the magnetic moment (\( ext{m}\( ext{\)\)) is defined as:
  • The product of the current (\(I\( ext{\)\)) flowing through the coil
  • And the area (\(A\( ext{\)\)) of the loop.
Mathematically, it can be expressed as:\[\text{m} = IA\]This concept helps explain how the same torque can be exerted on two different shaped coils, as the magnetic moment directly affects the torque experienced in a magnetic field.
Loop of Current
A loop of current refers to any continuous path of an electrical current that forms a closed circle or shape. It is essential in creating a magnetic field inside and around the loop.
A couple of key factors define a loop of current:
  • The shape of the loop which can be square, circular, or any other closed form.
  • The amount of current flowing through the loop which affects the strength of the magnetic field it produces.
In the context of our original exercise, the loop of current is depicted as either a square or circular wire shape. Both loops interact with the magnetic field to produce the same torque, illustrating the importance of understanding the loop of current and its characteristics.
Area of Coil
The area of a coil is crucial in determining its magnetic properties, especially its magnetic moment. It refers to the surface area enclosed by the wire loop:
For different shapes, the area calculations vary:
  • For a square coil, the area \(A\) is given by: \[A = \left(\frac{L}{4}\right)^2 = \frac{L^2}{16}\]where \(L\) is the total length of wire forming the square.
  • For a circular coil, the area is calculated as: \[A = \pi \left(\frac{L}{2\pi}\right)^2 = \frac{L^2}{4\pi}\]
Understanding these calculations allows us to comprehend how different shaped coils can affect the magnetic interaction, even when they have the same wire length but different areas enclosed.
Current in Coil
The current in a coil is an essential factor in determining how a coil reacts to a magnetic field. It relates to how much electrical charge flows through the coil per unit time.
For our exercise, the current in the square and circular coils is different but must be understood in the context of the torque they experience. When evaluating the torque equation:
  • The current (\(I\)) is pivotal along with the area (\(A\)) since torque depends on their product.
  • The ratio of currents between different coils is important in balance, which determines their respective torques.
The relationship is crisply given by the equation:\[\frac{I_{\text{square}}}{I_{\text{circle}}} = \frac{\pi}{4}\]This means balancing the equation of torque and understanding these currents' relationship is vital to solving problems involving different wire shapes.

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Most popular questions from this chapter

ssm A long, cylindrical conductor is solid throughout and has a radius \(R\) . Electric charges flow parallel to the axis of the cylinder and pass uniformly through the entire cross section. The arrangement is, in effect, a solid tube of current \(I_{0} .\) The current per unit cross-sectional area (i.e., the current density ) is \(I_{0} /\left(\pi R^{2}\right) .\) Use Ampere's law to show that the magnetic field inside the conductor at a distance \(r\) from the axis is \(\mu_{0} I_{0} r /\left(2 \pi R^{2}\right)\) . Hint: For a closed path, use \(a\) circle of radius \(r\) perpendicular to and centered on the axis. Note that the current through any surface is the area of the surface times the current density.)

Particle 1 and particle 2 have masses of \(m_{1}=2.3 \times 10^{-8} \mathrm{kg}\) and \(m_{2}=5.9 \times 10^{-8} \mathrm{kg},\) but they carry the same charge \(q .\) The two particles accelerate from rest through the same electric potential difference \(V\) and enter the same magnetic field, which has a magnitude \(B\). The particles travel perpendicular to the magnetic field on circular paths. The radius of the circular path for particle 1 is \(r_{1}=12 \mathrm{cm} .\) What is the radius (in cm) of the circular path for particle 2\(?\)

You have a wire of length \(L=1.00 \mathrm{m}\) from which to make the square coil of a dc motor. The current in the coil is \(I=1.7 \mathrm{A},\) and the magnetic field of the motor has a magnitude of \(B=0.34 \mathrm{T}\) . Find the maximum torque exerted on the coil when the wire is used to make a single-turn square coil and a two-turn square coil.

A square coil and a rectangular coil are each made from the same length of wire. Each contains a single turn. The long sides of the rectangle are twice as long as the short sides. Find the ratio \(\tau_{\text { square }} / \tau_{\text { rectangle }}\) of the maximum torques that these coils experience in the same magnetic field when they contain the same current.

A long solenoid has a length of 0.65 \(\mathrm{m}\) and contains 1400 turns of wire. There is a current of 4.7 \(\mathrm{A}\) in the wire. What is the magnitude of the magnetic field within the solenoid?
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