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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 Two circular loops of wire, each containing a single turn, have the same radius of 4.0 \(\mathrm{cm}\) and a common center. The planes of the loops are perpendicular. Each carries a current of 1.7 \(\mathrm{A}\) . What is the magnitude of the net magnetic field at the common center?

A proton is projected perpendicularly into a magnetic field that has a magnitude of 0.50 T. The ficld is then adjusted so that an electron will follow a circular path of the same radius when it is projected perpendicularly into the field with the same velocity that the proton had. What is the magnitude of the field used for the electron?

Two infinitely long, straight wires are parallel and separated by a distance of one meter. They carry currents in the same direction. Wire 1 carries four times the current that wire 2 carries. On a line drawn perpendicular to both wires, locate the spot (relative to wire 1) where the net magnetic field is zero. Assume that wire 1 lies to the left of wire 2 and note that there are three regions to consider on this line: to the left of wire 1, between wire 1 and wire 2, and to the right of wire 2.

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