**Problem 5.***In your own words, explain Ampere's Law and its significance (explain both sides of the equation). Determine then, how we find the magnetic field inside and outside of a wire with current \(i\). Next, provide an example with a grouping of wires with an Amperian loop to show total enclosure of current.***Explanation for Educational Website:**1. **Introduction to Ampere's Law:** - Ampere's Law is a fundamental principle of electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop. It is mathematically expressed as: \[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} \] Here, \(\oint \mathbf{B} \cdot d\mathbf{l}\) represents the line integral of the magnetic field \(\mathbf{B}\) around a closed path (Amperian loop), \(\mu_0\) is the permeability of free space, and \(I_{\text{enc}}\) is the total current enclosed by the loop.2. **Significance:** - The left-hand side of the equation (\(\oint \mathbf{B} \cdot d\mathbf{l}\)) represents the total circulation of the magnetic field along the chosen path. - The right-hand side (\(\mu_0 I_{\text{enc}}\)) represents the product of the permeability of free space and the total current enclosed by the path. - Ampere’s Law helps in understanding how currents produce magnetic fields and is used extensively in designing electrical devices like transformers, inductors, and solenoids.3. **Finding the Magnetic Field Inside and Outside a Current-Carrying Wire:** - **Inside the Wire:** Consider a cylindrical wire of radius \(R\) with current \(i\). If you take an Amperian loop of radius \(r\) inside the wire (\(r < R\)), the enclosed current is proportional to the area. We have: \[ I_{\text{enc}} = i \frac{\pi r^2}{\pi R^2} = i \frac{r^2}{R^2} \] Using Ampere’s Law: \[ B(2 \pi r) = \mu_

Every magnetic has its own magnetic field. The Magnetic field is defined as the region of space…

5. In your own words, explain Ampere's Law and its significance (explain both sides of equation). Determine then, how we find the magnetic field inside and outside of a wire with current i. Next, provide an example with a grouping of wires with an Amperian loop to show total enclosure of current.

5. In your own words, explain Ampere's Law and its significance (explain both sides of equation). Determine then, how we find the magnetic field inside and outside of a wire with current i. Next, provide an example with a grouping of wires with an Amperian loop to show total enclosure of current.

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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Question

**Problem 5.**

*In your own words, explain Ampere's Law and its significance (explain both sides of the equation). Determine then, how we find the magnetic field inside and outside of a wire with current \(i\). Next, provide an example with a grouping of wires with an Amperian loop to show total enclosure of current.*

**Explanation for Educational Website:**

1. **Introduction to Ampere's Law:**
- Ampere's Law is a fundamental principle of electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop. It is mathematically expressed as:
\[
\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}
\]
Here, \(\oint \mathbf{B} \cdot d\mathbf{l}\) represents the line integral of the magnetic field \(\mathbf{B}\) around a closed path (Amperian loop), \(\mu_0\) is the permeability of free space, and \(I_{\text{enc}}\) is the total current enclosed by the loop.

2. **Significance:**
- The left-hand side of the equation (\(\oint \mathbf{B} \cdot d\mathbf{l}\)) represents the total circulation of the magnetic field along the chosen path.
- The right-hand side (\(\mu_0 I_{\text{enc}}\)) represents the product of the permeability of free space and the total current enclosed by the path.
- Ampere’s Law helps in understanding how currents produce magnetic fields and is used extensively in designing electrical devices like transformers, inductors, and solenoids.

3. **Finding the Magnetic Field Inside and Outside a Current-Carrying Wire:**
- **Inside the Wire:**
Consider a cylindrical wire of radius \(R\) with current \(i\). If you take an Amperian loop of radius \(r\) inside the wire (\(r < R\)), the enclosed current is proportional to the area. We have:
\[
I_{\text{enc}} = i \frac{\pi r^2}{\pi R^2} = i \frac{r^2}{R^2}
\]
Using Ampere’s Law:
\[
B(2 \pi r) = \mu_

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