Which of the following gives the value of magnetic field according to, Biot-Savart’s law

Options

A. `I Delta lsin theta/r^(2)`
B. `mu_(0)/4pi iDelta l sin theta/r`
C. `mu_(0)/4pi iDeltal sin theta/r^(2)`
D. `mu_(0)/4pi iDelta sin theta`

Answer

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

### Step 1: Understand Biot-Savart's Law
Biot-Savart's law describes the magnetic field generated by a current-carrying conductor. It states that the magnetic field (dB) at a point in space due to a small segment of current (I) is directly proportional to the current, the length of the segment (dl), and the sine of the angle (θ) between the segment and the line connecting the segment to the point where the field is measured.

### Step 2: Write the 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 infinitesimal magnetic field produced by the current element.
- \( \mu_0 \) is the permeability of free space.
- \( I \) is the current flowing through the conductor.
- \( dl \) is the infinitesimal length of the conductor.
- \( \theta \) is the angle between the current element and the line connecting the element to the point where the field is measured.
- \( r \) is the distance from the current element to the point of interest.

### Step 3: Simplify the Expression
From the formula, we can see that the magnetic field \( B \) can be derived by integrating \( dB \) over the entire length of the conductor. However, for a small segment, we can directly use:
\[ B = \frac{\mu_0}{4\pi} \frac{I \, dl \, \sin \theta}{r^2} \]

### Step 4: Identify the Correct Option
When looking at the options provided in the question, we need to match the derived formula with the options. The correct expression according to Biot-Savart's law is:
\[ B = \frac{\mu_0 I \, dl \, \sin \theta}{4\pi r^2} \]
If this matches one of the options given in the question, that option is the correct answer.

### Conclusion
After analyzing the formula and the options, we conclude that the correct expression for the magnetic field according to Biot-Savart's law is:
\[ B = \frac{\mu_0 I \, dl \, \sin \theta}{4\pi r^2} \]