Which of the following given the value of magnetic field due to small current elememt accordong to Biot -Savart's law

Options

A. `(iDeltalsintheta)/(r^(2))`
B. `(mu_(0))/(4pi)(iDeltalsintheta)/(r)`
C. `(mu_(0))/(4pi)(iDeltalsintheta)/(r^(2))`
D. `(mu_(0))/(4pi)(iDeltalsintheta)/(r^(3))`

Answer

To solve the question regarding the value of the magnetic field due to a small current element according to Biot-Savart's law, we can follow these steps:

### Step-by-Step Solution

1. **Understanding Biot-Savart's Law**:
Biot-Savart's law helps us calculate the magnetic field generated by a small segment of current-carrying wire. It states that the magnetic field \( dB \) at a point in space due to a small current element \( I \, dL \) is proportional to the current \( I \), the length of the current element \( dL \), and the sine of the angle \( \theta \) between the current element and the line connecting the element to the point where the field is being calculated.

2. **Biot-Savart's Law Formula**:
The mathematical expression for Biot-Savart's law is given by:
\[ dB = \frac{\mu_0}{4\pi} \frac{I \, dL \, \sin \theta}{r^2} \]
where:
- \( dB \) is the magnetic field due to the small current element,
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current flowing through the wire,
- \( dL \) is the length of the current element,
- \( \theta \) is the angle between the current element and the line to the point of interest,
- \( r \) is the distance from the current element to the point where the magnetic field is being calculated.

3. **Identifying the Components**:
In the formula:
- \( \mu_0 \) is a constant (approximately \( 4\pi \times 10^{-7} \, T \cdot m/A \)),
- \( I \) is the current in amperes,
- \( dL \) is the differential length of the wire in meters,
- \( \sin \theta \) accounts for the angle between the current direction and the line to the point,
- \( r^2 \) is the square of the distance from the current element to the point.

4. **Conclusion**:
The expression derived from Biot-Savart's law gives us the value of the magnetic field due to a small current element. Therefore, the correct option regarding the value of the magnetic field due to a small current element according to Biot-Savart's law is:
\[ dB = \frac{\mu_0}{4\pi} \frac{I \, dL \, \sin \theta}{r^2} \]