What is Biot Savart Law class 12?
2 Answers
The Biot-Savart Law is a fundamental principle in electromagnetism that describes how a magnetic field is generated by a moving charge or a current-carrying conductor. It is particularly useful for calculating the magnetic field produced by a current element (a small segment of a current-carrying wire). This law is essential for understanding the magnetic effects of electric currents, especially in situations where the current distribution is not uniform or the geometry of the conductor is complex.
### Statement of Biot-Savart Law:
The Biot-Savart Law states that the magnetic field \( d\mathbf{B} \) at a point due to a small current element \( I \, d\mathbf{l} \) is directly proportional to the current \( I \), the length of the current element \( d\mathbf{l} \), the sine of the angle \( \theta \) between the current element and the line joining the element to the point where the field is measured, and inversely proportional to the square of the distance \( r \) between the current element and the point.
### Mathematical Expression:
The mathematical expression for the Biot-Savart Law is given by:
\[
d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3}
\]
Where:
- \( d\mathbf{B} \) is the infinitesimal magnetic field produced by the current element.
- \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A} \)).
- \( I \) is the current flowing through the conductor.
- \( d\mathbf{l} \) is a vector representing the length and direction of the current element.
- \( \mathbf{r} \) is the vector from the current element to the point where the magnetic field is being calculated.
- \( r \) is the magnitude of \( \mathbf{r} \) (the distance between the current element and the point).
- \( \times \) represents the cross-product, indicating that the magnetic field is perpendicular to the plane formed by \( d\mathbf{l} \) and \( \mathbf{r} \).
### Explanation:
1. **Direction of \( d\mathbf{B} \)**: The direction of the magnetic field \( d\mathbf{B} \) is determined by the right-hand rule. If you point the thumb of your right hand in the direction of the current element \( d\mathbf{l} \) and curl your fingers toward the position vector \( \mathbf{r} \), your thumb will point in the direction of the magnetic field.
2. **Magnitude of \( d\mathbf{B} \)**: The magnitude of the magnetic field created by the current element is proportional to the current \( I \), the length of the element \( d\mathbf{l} \), and \( \sin\theta \) (where \( \theta \) is the angle between \( d\mathbf{l} \) and \( \mathbf{r} \)), and inversely proportional to the square of the distance \( r^2 \).
3. **Integration for Total Magnetic Field**: To find the total magnetic field \( \mathbf{B} \) at a point due to an entire current-carrying conductor, you need to integrate the Biot-Savart Law over the entire length of the conductor:
\[
\mathbf{B} = \frac{\mu_0}{4\pi} \int \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3}
\]
### Applications of Biot-Savart Law:
1. **Magnetic Field of a Straight Wire**: The law can be used to derive the magnetic field around a long, straight, current-carrying wire.
2. **Magnetic Field of a Circular Loop**: It helps calculate the magnetic field at the center or on the axis of a circular current loop.
3. **Magnetic Field of a Solenoid**: The Biot-Savart Law can also be applied to derive the magnetic field inside and outside a solenoid.
### Summary:
The Biot-Savart Law is a cornerstone of classical electromagnetism that provides a way to calculate the magnetic field produced by a moving charge or a current element. It demonstrates that the magnetic field is a vector field that depends on the geometry of the current path and the position of the point where the field is being measured.
### Statement of Biot-Savart Law:
The Biot-Savart Law states that the magnetic field \( d\mathbf{B} \) at a point due to a small current element \( I \, d\mathbf{l} \) is directly proportional to the current \( I \), the length of the current element \( d\mathbf{l} \), the sine of the angle \( \theta \) between the current element and the line joining the element to the point where the field is measured, and inversely proportional to the square of the distance \( r \) between the current element and the point.
### Mathematical Expression:
The mathematical expression for the Biot-Savart Law is given by:
\[
d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3}
\]
Where:
- \( d\mathbf{B} \) is the infinitesimal magnetic field produced by the current element.
- \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A} \)).
- \( I \) is the current flowing through the conductor.
- \( d\mathbf{l} \) is a vector representing the length and direction of the current element.
- \( \mathbf{r} \) is the vector from the current element to the point where the magnetic field is being calculated.
- \( r \) is the magnitude of \( \mathbf{r} \) (the distance between the current element and the point).
- \( \times \) represents the cross-product, indicating that the magnetic field is perpendicular to the plane formed by \( d\mathbf{l} \) and \( \mathbf{r} \).
### Explanation:
1. **Direction of \( d\mathbf{B} \)**: The direction of the magnetic field \( d\mathbf{B} \) is determined by the right-hand rule. If you point the thumb of your right hand in the direction of the current element \( d\mathbf{l} \) and curl your fingers toward the position vector \( \mathbf{r} \), your thumb will point in the direction of the magnetic field.
2. **Magnitude of \( d\mathbf{B} \)**: The magnitude of the magnetic field created by the current element is proportional to the current \( I \), the length of the element \( d\mathbf{l} \), and \( \sin\theta \) (where \( \theta \) is the angle between \( d\mathbf{l} \) and \( \mathbf{r} \)), and inversely proportional to the square of the distance \( r^2 \).
3. **Integration for Total Magnetic Field**: To find the total magnetic field \( \mathbf{B} \) at a point due to an entire current-carrying conductor, you need to integrate the Biot-Savart Law over the entire length of the conductor:
\[
\mathbf{B} = \frac{\mu_0}{4\pi} \int \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3}
\]
### Applications of Biot-Savart Law:
1. **Magnetic Field of a Straight Wire**: The law can be used to derive the magnetic field around a long, straight, current-carrying wire.
2. **Magnetic Field of a Circular Loop**: It helps calculate the magnetic field at the center or on the axis of a circular current loop.
3. **Magnetic Field of a Solenoid**: The Biot-Savart Law can also be applied to derive the magnetic field inside and outside a solenoid.
### Summary:
The Biot-Savart Law is a cornerstone of classical electromagnetism that provides a way to calculate the magnetic field produced by a moving charge or a current element. It demonstrates that the magnetic field is a vector field that depends on the geometry of the current path and the position of the point where the field is being measured.