https://file.aiquickdraw.com/mathbot/file/1728159143504jxfon1f0.jpg

Please give me another easy method I can use to derivate the above formula

Thank you for sharing the image. From the content provided, it looks like the text is discussing the magnetic field produced by a straight current-carrying conductor, and there is an integral involved to derive the magnitude of the magnetic field at a point, labeled as \( B \).

The original derivation utilizes an integral approach to derive the magnetic field \( B \) due to a finite conductor of length \( 2a \), with some limiting considerations for a long conductor. Let me guide you through an easier alternative approach for deriving the magnetic field, particularly if you're interested in simplifying the derivation for an infinitely long conductor.

### Using Ampère's Law to Derive the Formula
Instead of using the direct integration method shown in the image, you can apply **Ampère's Law**, which is much more straightforward in this case if you assume the conductor is infinitely long.

Ampère's Law states:

\[
\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}
\]

Where:
- \(\oint \vec{B} \cdot d\vec{l}\) is the line integral of the magnetic field around a closed loop (an Amperian loop).
- \(\mu_0\) is the permeability of free space.
- \(I_{\text{enc}}\) is the current enclosed by the loop.

#### Steps to Derive the Magnetic Field Using Ampère's Law:

1. **Choose an Amperian Loop**: 
   - For a straight, infinitely long current-carrying conductor, the magnetic field forms concentric circles around the wire. 
   - To apply Ampère's Law, choose a circular Amperian loop of radius \( r \) centered on the wire.

2. **Symmetry Considerations**:
   - By symmetry, the magnetic field \( B \) at any point on the loop is constant and tangential to the loop. Thus, the integral simplifies to:

   \[
   \oint \vec{B} \cdot d\vec{l} = B \oint d l = B (2 \pi r)
   \]

3. **Ampère's Law Application**:
   - According to Ampère’s Law:

   \[
   B (2 \pi r) = \mu_0 I
   \]

4. **Solve for \( B \)**:
   - Rearranging to solve for \( B \):

   \[
   B = \frac{\mu_0 I}{2 \pi r}
   \]

This formula gives the magnetic field at a distance \( r \) from a long, straight conductor carrying current \( I \). Notice how much simpler it is compared to performing the direct integration, especially in the case of an infinitely long conductor.

### Summary of Differences
- **Direct Integration**: The original method integrates contributions from each segment of the conductor to find the total magnetic field. It involves trigonometric substitution and can be complex, especially for a finite length conductor.
- **Ampère’s Law**: This approach is easier and more direct for an infinitely long conductor due to the high degree of symmetry, reducing the problem to a simple algebraic equation.

Would you like more details about using Ampère's Law or anything else about this topic?

### Follow-up Questions for Deeper Understanding:
1. How would the derived magnetic field change if the conductor were not infinitely long?
2. What assumptions are made in using Ampère's Law, and why are they valid here?
3. How can Biot-Savart law be used to derive the magnetic field for a finite length of conductor?
4. What is the physical significance of the magnetic field direction around the conductor?
5. How would the magnetic field expression change if the current were alternating instead of direct?

### Tip:
Always remember that Ampère's Law is most effective when there's a high degree of symmetry in the problem (such as infinite wires, solenoids, or toroids). For more complex geometries, Biot-Savart is often more appropriate.

Please give me another easy method I can use to derive the magnetic field formula for a straight current-carrying conductor.

Learn how to easily derive the magnetic field formula for a straight current-carrying conductor using Ampère's Law. This method simplifies the process, bypassing the complexity of integration, and is ideal for physics students familiar with integral calculus and electromagnetism.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation