If length of the arc is 20 cm, radius R=8 cm and a current through a loop I= 20 A, what is the magnitude and direction of the magnetic field at the center O (fig.3)? (show step by step solution)
If length of the arc is 20 cm, radius R=8 cm and a current through a loop I= 20 A, what is the magnitude and direction of the magnetic field at the center O (fig.3)? (show step by step solution)
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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![**Problem Statement:**
If the length of the arc is 20 cm, radius \( R = 8 \) cm, and a current through a loop \( I = 20 \) A, what is the magnitude and direction of the magnetic field at the center \( O \) (fig. 3)? (Show step-by-step solution)
**Diagram Explanation (Fig. 3):**
- The diagram shows a segment of a circular wire carrying a current \( I \).
- The arc length of the segment is given as 20 cm, with the radius of the segment \( R \) as 8 cm.
- The diagram illustrates the wire forming part of a circle with its center labeled as \( O \).
- Arrows in the diagram indicate the direction of the current flowing through the wire segment.
**Step-by-Step Solution:**
1. **Parameter Calculation:**
- Given:
- Arc length \( l = 20 \) cm \( = 0.2 \) m
- Radius \( R = 8 \) cm \( = 0.08 \) m
- Current \( I = 20 \) A
2. **Magnetic Field Calculation:**
- The magnetic field at the center \( O \) due to a current-carrying arc can be calculated using the formula:
\[
B = \frac{\mu_0 I \theta}{4\pi R}
\]
where \( \mu_0 \) is the permeability of free space \((4\pi \times 10^{-7} \, \text{T m/A})\), and \( \theta \) is the angle in radians subtended by the arc.
3. **Angle \( \theta \) Calculation:**
- The angle \( \theta \) is given by:
\[
\theta = \frac{l}{R} = \frac{0.2}{0.08} = 2.5 \, \text{radians}
\]
4. **Magnetic Field \( B \):**
- Substituting the values into the equation:
\[
B = \frac{4\pi \times 10^{-7} \times 20 \times 2.5}{4\pi \times 0.08} = \frac{2.5 \times 10^{-6}}{0.08](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a2ce857-5f61-4ff8-8c92-7aabb081b77b%2Fc54f45d2-3fc1-47f2-9c33-62699f076df4%2Fj12mvtl_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
If the length of the arc is 20 cm, radius \( R = 8 \) cm, and a current through a loop \( I = 20 \) A, what is the magnitude and direction of the magnetic field at the center \( O \) (fig. 3)? (Show step-by-step solution)
**Diagram Explanation (Fig. 3):**
- The diagram shows a segment of a circular wire carrying a current \( I \).
- The arc length of the segment is given as 20 cm, with the radius of the segment \( R \) as 8 cm.
- The diagram illustrates the wire forming part of a circle with its center labeled as \( O \).
- Arrows in the diagram indicate the direction of the current flowing through the wire segment.
**Step-by-Step Solution:**
1. **Parameter Calculation:**
- Given:
- Arc length \( l = 20 \) cm \( = 0.2 \) m
- Radius \( R = 8 \) cm \( = 0.08 \) m
- Current \( I = 20 \) A
2. **Magnetic Field Calculation:**
- The magnetic field at the center \( O \) due to a current-carrying arc can be calculated using the formula:
\[
B = \frac{\mu_0 I \theta}{4\pi R}
\]
where \( \mu_0 \) is the permeability of free space \((4\pi \times 10^{-7} \, \text{T m/A})\), and \( \theta \) is the angle in radians subtended by the arc.
3. **Angle \( \theta \) Calculation:**
- The angle \( \theta \) is given by:
\[
\theta = \frac{l}{R} = \frac{0.2}{0.08} = 2.5 \, \text{radians}
\]
4. **Magnetic Field \( B \):**
- Substituting the values into the equation:
\[
B = \frac{4\pi \times 10^{-7} \times 20 \times 2.5}{4\pi \times 0.08} = \frac{2.5 \times 10^{-6}}{0.08
![**Problem Statement:**
If the length of the arc is 20 cm, the radius \( R = 8 \) cm, and a current through a loop \( I = 20 \) A, what is the magnitude and direction of the magnetic field at the center \( O \) (Fig. 3)? Provide a step-by-step solution.
**Diagram Explanation:**
- The diagram shows an arc of a circle with radius \( R \) and current \( I \) flowing through it.
- The arc subtends an angle at the center labeled \( O \).
- Arrows indicate the direction of current flow through the arc.
**Solution Steps:**
To find the magnetic field at the center \( O \), we use the Biot-Savart Law for an arc:
1. **Calculate the Angle Subtended by the Arc:**
\[
\theta = \frac{\text{Length of the arc}}{\text{Radius}} = \frac{20 \, \text{cm}}{8 \, \text{cm}} = 2.5 \, \text{radians}
\]
2. **Using the Biot-Savart Law for an Arc:**
The magnetic field \( B \) at the center due to an arc is given by:
\[
B = \frac{\mu_0 I \theta}{4 \pi R}
\]
Where:
- \( \mu_0 = 4\pi \times 10^{-7} \, \text{T}\cdot \text{m/A} \) (permeability of free space)
- \( I = 20 \, \text{A} \)
- \( \theta = 2.5 \, \text{radians} \)
- \( R = 8 \, \text{cm} = 0.08 \, \text{m} \)
3. **Substitute Values:**
\[
B = \frac{(4\pi \times 10^{-7}) \times 20 \times 2.5}{4 \pi \times 0.08}
\]
Simplifying:
\[
B = \frac{2.5 \times 10^{-7} \times 20}{0.08}
\]
\[
B = \frac{5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a2ce857-5f61-4ff8-8c92-7aabb081b77b%2Fc54f45d2-3fc1-47f2-9c33-62699f076df4%2Frbiyjrn_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
If the length of the arc is 20 cm, the radius \( R = 8 \) cm, and a current through a loop \( I = 20 \) A, what is the magnitude and direction of the magnetic field at the center \( O \) (Fig. 3)? Provide a step-by-step solution.
**Diagram Explanation:**
- The diagram shows an arc of a circle with radius \( R \) and current \( I \) flowing through it.
- The arc subtends an angle at the center labeled \( O \).
- Arrows indicate the direction of current flow through the arc.
**Solution Steps:**
To find the magnetic field at the center \( O \), we use the Biot-Savart Law for an arc:
1. **Calculate the Angle Subtended by the Arc:**
\[
\theta = \frac{\text{Length of the arc}}{\text{Radius}} = \frac{20 \, \text{cm}}{8 \, \text{cm}} = 2.5 \, \text{radians}
\]
2. **Using the Biot-Savart Law for an Arc:**
The magnetic field \( B \) at the center due to an arc is given by:
\[
B = \frac{\mu_0 I \theta}{4 \pi R}
\]
Where:
- \( \mu_0 = 4\pi \times 10^{-7} \, \text{T}\cdot \text{m/A} \) (permeability of free space)
- \( I = 20 \, \text{A} \)
- \( \theta = 2.5 \, \text{radians} \)
- \( R = 8 \, \text{cm} = 0.08 \, \text{m} \)
3. **Substitute Values:**
\[
B = \frac{(4\pi \times 10^{-7}) \times 20 \times 2.5}{4 \pi \times 0.08}
\]
Simplifying:
\[
B = \frac{2.5 \times 10^{-7} \times 20}{0.08}
\]
\[
B = \frac{5
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