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Chapter 22: Problem 16

A square loop of wire consisting of a single turn is perpendicular to a uniform magnetic field. The square loop is then re-formed into a circular loop, which also consists of a single turn and is also perpendicular to the same magnetic field. The magnetic flux that passes through the square loop is \(7.0 \times 10^{-3}\) Wb. What is the flux that passes through the circular loop?

Short Answer

Expert verified
The magnetic flux through the circular loop is approximately 8.9 x 10^-3 Wb.

Step by step solution

01

Understand Magnetic Flux

Magnetic flux (\(\Phi\)) through a loop is defined as the product of the magnetic field (\(B\)) and the area (\(A\)) through which the field lines pass, perpendicular to both the field and the surface of the area. It can be expressed as \(\Phi = B \cdot A\).
02

Recall Area Formulas

The area of a square with side length \(s\) is calculated as \(s^2\), while the area of a circle with radius \(r\) is given by \(\pi r^2\). Since both loops are made from the same length of wire, we equate the perimeter of the square and the circumference of the circle.
03

Calculate Side Length of Square

Knowing that flux through the square loop is \(7.0 \times 10^{-3}\) Wb, and the formula \(\Phi = B \cdot s^2\), where \(s\) is the side of the square. The perimeter of the square is \(4s\).
04

Calculate Radius of Circle

Using the same wire length, the circumference of the circle is equal to the perimeter of the square, \(4s\). This gives the circumference as \(2\pi r = 4s\), which simplifies to \(r = \frac{2s}{\pi}\). Then, find the area \(A = \pi r^2 = \pi \left(\frac{2s}{\pi}\right)^2 = \frac{4s^2}{\pi}\).
05

Calculate Magnetic Flux Through the Circle

Because the magnetic field is unchanged, we express the flux through the circular loop as \(\Phi_{circle} = B \cdot \frac{4s^2}{\pi}\). Substitute the initial magnetic flux for \(B \cdot s^2\) in the formula, giving us \(\Phi_{circle} = \frac{4}{\pi} \Phi_{square}\).
06

Calculate Flux Through Circular Loop

Substitute \(\Phi_{square} = 7.0 \times 10^{-3}\) Wb into the equation to find the flux through the circular loop: \(\Phi_{circle} = \frac{4}{\pi} \times 7.0 \times 10^{-3}\) Wb, which simplifies to approximately \(8.9 \times 10^{-3}\) Wb.

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Key Concepts

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

Magnetic Field
The magnetic field is a fundamental concept in physics. It represents a region around a magnetic material or moving electric charge where forces of magnetism act.
Magnetic fields are often depicted using lines of force that show the direction and strength of the field. These lines are denser where the magnetic field is stronger.
  • Magnetic fields are created by magnets or moving charges.
  • The field's strength and direction can influence other magnets or charged particles within its reach.
  • Field lines enter the south pole and exit the north pole of a magnet.
These fields play a vital role in many technological and natural phenomena, affecting everything from electric motors to the Earth's own geomagnetic field.
Square Loop
A square loop is simply a loop of wire shaped as a square. When placed in a magnetic field, it interacts with the field lines.
This interaction can help us understand electromagnetic effects such as magnetic flux. The side length of the square determines its area, calculated as: \[ A_{square} = s^2 \] where \(s\) is the length of one side of the square.
  • The total length of wire forms the perimeter of the square, calculated by \(4s\).
  • When perpendicular to the magnetic field, the square loop captures maximum flux.
In practical applications, square loops are used in devices like transformers and sensors, where precise geometry affects functionality.
Circular Loop
When the same wire is reshaped into a circular loop, the geometry changes, but the wire's length remains the same. This transformation gives rise to different calculations for area and flux.
  • The circumference of the circle is equivalent to the perimeter of the square: \(2\pi r = 4s\).
  • From this, we find the radius: \(r = \frac{2s}{\pi}\).
The area of a circular loop is:\[ A_{circle} = \pi r^2 = \frac{4s^2}{\pi} \] This new area influences the magnetic flux that passes through the loop, illustrating how shape impacts physical properties.
Magnetic Flux Calculation
Magnetic flux, denoted by \( \Phi \), quantifies the total magnetic field passing through a given area. It combines both the field strength and the area through which it acts.
  • The basic formula for magnetic flux is: \( \Phi = B \cdot A \), where \(B\) is the magnetic field strength.
  • For the given problem, the flux through a square loop is known: \(7.0 \times 10^{-3}\) Wb.
For the circular loop, we use the relationship:\[ \Phi_{circle} = \frac{4}{\pi} \Phi_{square} \] This equation stems from the change in area due to the new circular shape, leading to a calculated flux of approximately \(8.9 \times 10^{-3}\) Wb. This exercise highlights how the same physical material can produce different effects based on its shape and orientation.

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Most popular questions from this chapter

A uniform magnetic field is perpendicular to the plane of a single-turn circular coil. The magnitude of the field is changing, so that an emf of \(0.80 \mathrm{V}\) and a current of \(3.2 \mathrm{A}\) are induced in the coil. The wire is then re-formed into a single-turn square coil, which is used in the same magnetic field (again perpendicular to the plane of the coil and with a magnitude changing at the same rate). What emf and current are induced in the square coil?

The resistances of the primary and secondary coils of a transformer are 56 and \(14 \Omega,\) respectively. Both coils are made from lengths of the same copper wire. The circular turns of each coil have the same diameter. Find the turns ratio \(N_{s} / N_{\mathrm{p}}\)

Suppose there are two transformers between your house and the high-voltage transmission line that distributes the power. In addition, assume that your house is the only one using electric power. At a substation the primary coil of a step-down transformer (turns ratio \(=1: 29\) ) receives the voltage from the high-voltage transmission line. Because of your usage, a current of \(48 \mathrm{mA}\) exists in the primary coil of this transformer. The secondary coil is connected to the primary of another step-down transformer (turns ratio \(=1: 32\) ) somewhere near your house, perhaps up on a telephone pole. The secondary coil of this transformer delivers a \(240-\mathrm{V}\) emf to your house. How much power is your house using? Remember that the current and voltage given in this problem are rms values.

A flat coil of wire has an area \(A, N\) turns, and a resistance \(R\). It is situated in a magnetic field, such that the normal to the coil is parallel to the magnetic field. The coil is then rotated through an angle of \(90^{\circ},\) so that the normal becomes perpendicular to the magnetic field. The coil has an area of \(1.5 \times 10^{-3} \mathrm{m}^{2}, 50\) turns, and a resistance of \(140 \Omega .\) During the time while it is rotating, a charge of \(8.5 \times 10^{-5} \mathrm{C}\) flows in the coil. What is the magnitude of the magnetic field?

A magnetic field is passing through a loop of wire whose area is \(0.018 \mathrm{m}^{2} .\) The direction of the magnetic field is parallel to the normal to the loop, and the magnitude of the field is increasing at the rate of \(0.20 \mathrm{T} / \mathrm{s}\) (a) Determine the magnitude of the emf induced in the loop. (b) Suppose that the area of the loop can be enlarged or shrunk. If the magnetic field is increasing as in part (a), at what rate (in \(\mathrm{m}^{2} / \mathrm{s}\) ) should the area be changed at the instant when \(B=1.8 \mathrm{T}\) if the induced emf is to be zero? Explain whether the area is to be enlarged or shrunk.
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