To derive the Biot-Savart Law, we start with the fundamental concepts of electromagnetism, specifically focusing on how a moving charge creates a magnetic field. Here’s a step-by-step outline of the derivation:

### Step 1: Understand the Magnetic Field due to a Current Element

1. **Current Element Definition**: Consider a small current element \( I \, d\mathbf{l} \) at position \( \mathbf{r}' \) in space, where \( \mathbf{l} \) is the direction of current flow.
2. **Electric Current**: A current \( I \) is defined as the flow of charge per unit time, \( I = \frac{dq}{dt} \).

### Step 2: Determine the Magnetic Field Contribution

3. **Magnetic Field from a Current Element**: The contribution \( d\mathbf{B} \) to the magnetic field \( \mathbf{B} \) at a point \( \mathbf{r} \) due to the current element is given by:
   \[
   d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{r}}{|\mathbf{r}|^3}
   \]
   where \( \mathbf{r} = \mathbf{r} - \mathbf{r}' \) is the vector pointing from the current element to the point of observation.

### Step 3: Vector Cross Product

4. **Cross Product Interpretation**: The cross product \( d\mathbf{l} \times \mathbf{r} \) gives a vector that is perpendicular to both the direction of the current and the vector pointing from the current element to the observation point, which is consistent with the right-hand rule.

### Step 4: Integrate Over the Current Distribution

5. **Total Magnetic Field**: To find the total magnetic field \( \mathbf{B} \) at point \( \mathbf{r} \), integrate the contributions from all current elements along the current path \( C \):
   \[
   \mathbf{B}(\mathbf{r}) = \int_C d\mathbf{B} = \frac{\mu_0}{4\pi} \int_C \frac{I \, d\mathbf{l} \times \mathbf{r}}{|\mathbf{r}|^3}
   \]

### Step 5: Final Form of the Biot-Savart Law

6. **Biot-Savart Law**: The law states that the magnetic field \( \mathbf{B} \) at a point in space due to a current-carrying wire is given by:
   \[
   \mathbf{B}(\mathbf{r}) = \frac{\mu_0 I}{4\pi} \int_C \frac{d\mathbf{l} \times \mathbf{r}}{|\mathbf{r}|^3}
   \]

### Conclusion

The Biot-Savart Law encapsulates how current distributions generate magnetic fields in their vicinity. This law is fundamental in electromagnetism and is used extensively in both theoretical and practical applications.

### Do you want more details or have any questions?

Here are some related questions you might consider:
1. What are the applications of the Biot-Savart Law in real-world scenarios?
2. How does the Biot-Savart Law compare to Ampère's Law?
3. What is the significance of the constant \( \mu_0 \) in the law?
4. Can you derive the magnetic field of a straight current-carrying wire using this law?
5. What assumptions are made in the Biot-Savart Law? 

**Tip**: To better understand the Biot-Savart Law, visualize the current-carrying wire and the resulting magnetic field lines, applying the right-hand rule for direction.

Explore the derivation of the Biot-Savart Law, a fundamental principle in electromagnetism, detailing each step from current element definition to final formula. Perfect for advanced high school students.

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