The figure below shows a cross section of an "infinitely" long solenoid (circle) that has n=18000 turns/m; the axis of the solenoid (a bold dot) is perpendicular to the plane of cross section. The radius of the solenoid, r=87 cm. A square frame with a side a=8 cm is located in the plane of cross section. The current in the solenoid is I=3.33 A and it does not change with time, the resistance of the frame is R=13 Ω. The frame is rotating at 6000 turns/min with respect to one of its sides.Find the value of the magnetic field inside the solenoid: B= T.Find the maximum value of induced e.m.f. in the frame: ℰind= V.Find the maximum value of induced current in the frame: Iind= A. **Area Analysis of Concentric Shapes**In this diagram, we have two primary geometric shapes: a circle and a square. These shapes are arranged concentrically, meaning they share the same center point.**Key Components:**1. **Circle:** - **Radius (r):** The circle has a radius denoted by 'r', which is the distance from the center of the circle to any point on its perimeter. - The circle is the larger of the two shapes and encompasses the square.2. **Square:** - **Side Length (a):** The square is inscribed within the circle, and each side of the square is denoted by 'a'. - The center of the square coincides with the center of the circle.**Important Relationships:**- **Inscribed Square:** Since the square is inscribed within the circle, the diameter of the circle is equal to the diagonal of the square.- **Pythagorean Theorem Application:** Using the Pythagorean Theorem, the diagonal of the square (which is also equal to 2r, the diameter of the circle) can be calculated as \(\sqrt{2a^2}\), leading to \(2r = a\sqrt{2}\). Therefore, the radius \(r\) can be expressed in terms of the side length 'a' as \(r = \frac{a\sqrt{2}}{2}\).**Exploration Questions:**1. How do changes in the side length of the square affect the radius of the circle?2. What is the area of the circle in terms of the side length 'a' of the square?3. How do these shapes conform to basic geometric principles, such as the properties of inscribed shapes?**Applications:**Understanding the relationship between inscribed shapes and their respective dimensions is fundamental in various fields such as architecture, design, and more advanced mathematical studies. This diagram visually represents the connection between linear dimensions and circular form factors, which is an essential concept in geometry.

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College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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The figure below shows a cross section of an "infinitely" long solenoid (circle) that has n=18000 turns/m; the axis of the solenoid (a bold dot) is perpendicular to the plane of cross section. The radius of the solenoid, r=87 cm. A square frame with a side a=8 cm is located in the plane of cross section. The current in the solenoid is I=3.33 A and it does not change with time, the resistance of the frame is R=13 Ω. The frame is rotating at 6000 turns/min with respect to one of its sides.

Find the value of the magnetic field inside the solenoid: B=   T.

Find the maximum value of induced e.m.f. in the frame: ℰ_ind=   V.

Find the maximum value of induced current in the frame: I_ind=  A.

**Area Analysis of Concentric Shapes**

In this diagram, we have two primary geometric shapes: a circle and a square. These shapes are arranged concentrically, meaning they share the same center point.

**Key Components:**
1. **Circle:**
- **Radius (r):** The circle has a radius denoted by 'r', which is the distance from the center of the circle to any point on its perimeter.
- The circle is the larger of the two shapes and encompasses the square.

2. **Square:**
- **Side Length (a):** The square is inscribed within the circle, and each side of the square is denoted by 'a'.
- The center of the square coincides with the center of the circle.

**Important Relationships:**
- **Inscribed Square:** Since the square is inscribed within the circle, the diameter of the circle is equal to the diagonal of the square.
- **Pythagorean Theorem Application:** Using the Pythagorean Theorem, the diagonal of the square (which is also equal to 2r, the diameter of the circle) can be calculated as \(\sqrt{2a^2}\), leading to \(2r = a\sqrt{2}\). Therefore, the radius \(r\) can be expressed in terms of the side length 'a' as \(r = \frac{a\sqrt{2}}{2}\).

**Exploration Questions:**
1. How do changes in the side length of the square affect the radius of the circle?
2. What is the area of the circle in terms of the side length 'a' of the square?
3. How do these shapes conform to basic geometric principles, such as the properties of inscribed shapes?

**Applications:**
Understanding the relationship between inscribed shapes and their respective dimensions is fundamental in various fields such as architecture, design, and more advanced mathematical studies. This diagram visually represents the connection between linear dimensions and circular form factors, which is an essential concept in geometry.

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