The magnetic induction due to an infinitely long straight wire carrying a current `i` at a distance `r` from wire is given by

Options

A. `|B|=((mu_(0))/(4pi))(2i)/r`
B. `|B|=((mu_(0))/(4pi))r/(2i)`
C. `|B|=((4pi)/(mu_(0)))(2i)/r`
D. `|B|=((4pi)/(mu_(0)))r/(2i)`

Answer

To find the magnetic induction (magnetic field) due to an infinitely long straight wire carrying a current \( i \) at a distance \( r \) from the wire, we can use Ampère's Law or the Biot-Savart Law. Here’s a step-by-step solution:

### Step 1: Understanding the Configuration
We have an infinitely long straight wire carrying a current \( i \). We want to determine the magnetic field \( B \) at a point \( P \) located at a distance \( r \) from the wire.

**Hint:** Visualize the wire and the point where you want to find the magnetic field.

### Step 2: Using the Biot-Savart Law
The Biot-Savart Law states that the magnetic field \( dB \) at a point due to a small segment of current \( dl \) is given by:
\[ dB = \frac{\mu_0}{4\pi} \frac{i \, dl \times \hat{r}}{r^2} \]
where \( \hat{r} \) is the unit vector pointing from the current element to the point where the field is being calculated.

**Hint:** Remember that \( dl \) is a small length of the wire and \( \hat{r} \) is the direction from the wire to the point \( P \).

### Step 3: Setting Up the Geometry
For an infinitely long wire, we can consider the contributions from all segments of the wire. The angle \( \theta \) between \( dl \) and \( \hat{r} \) will be important. The sine of the angle \( \theta \) can be expressed in terms of the geometry of the situation.

**Hint:** The angle \( \theta \) helps in determining how much of the current contributes to the magnetic field at point \( P \).

### Step 4: Integrating Over the Length of the Wire
Since the wire is infinitely long, we need to integrate \( dB \) over the entire length of the wire. The total magnetic field \( B \) can be calculated as:
\[ B = \int dB = \int \frac{\mu_0}{4\pi} \frac{i \, dl \sin \theta}{r^2} \]
For an infinitely long wire, the contributions from both sides of the wire will be symmetrical.

**Hint:** Consider the symmetry of the problem when integrating.

### Step 5: Final Expression
After performing the integration, we find that the magnetic field \( B \) at a distance \( r \) from an infinitely long straight wire carrying current \( i \) is given by:
\[ B = \frac{\mu_0 i}{2 \pi r} \]

**Hint:** This formula shows that the magnetic field decreases inversely with distance from the wire.

### Conclusion
The magnetic induction (magnetic field) due to an infinitely long straight wire carrying a current \( i \) at a distance \( r \) is:
\[ B = \frac{\mu_0 i}{2 \pi r} \]