The magnetic field due to a current in a straight wire segment of length L at a point on its perpendicular bisector at a distance `r (rgtgtL)`.

Options

A. decreases as `1/r`
B. decreases as `1/r^3`
C. decreases as `1/r^2`
D. increases as `r^2`

Explanation

To find the magnetic field due to a current in a straight wire segment of length \( L \) at a point on its perpendicular bisector at a distance \( r \) (where \( r \gg L \)), we can use the Biot-Savart Law. Here’s the step-by-step solution:

### Step 1: Understand the Biot-Savart Law
The Biot-Savart Law states that the magnetic field \( dB \) at a point in space due to a small segment of current-carrying wire is given by:

\[ dB = \frac{\mu_0}{4\pi} \frac{I \, dL \times \hat{r}}{r^2} \]

where:
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current,
- \( dL \) is the length of the wire segment,
- \( \hat{r} \) is the unit vector pointing from the wire segment to the point where the magnetic field is being calculated,
- \( r \) is the distance from the wire segment to the point.

### Step 2: Set Up the Geometry
For a straight wire segment of length \( L \), we consider a point on its perpendicular bisector at a distance \( r \). The distance from the midpoint of the wire to the point where we are calculating the magnetic field is \( r \).

### Step 3: Calculate the Magnetic Field Contribution
Since the wire segment is straight and the point is on the perpendicular bisector, the contributions to the magnetic field from both ends of the wire will add up. The angle \( \theta \) between \( dL \) and \( \hat{r} \) is \( 90^\circ \), so \( \sin(\theta) = 1 \).

The total magnetic field \( B \) at the point is given by integrating \( dB \) over the length of the wire:

\[ B = \int dB = \int \frac{\mu_0}{4\pi} \frac{I \, dL}{r^2} \]

### Step 4: Evaluate the Integral
Since \( r \) is constant for the entire length of the wire segment, we can take it out of the integral:

\[ B = \frac{\mu_0 I}{4\pi r^2} \int dL \]

The integral \( \int dL \) over the length \( L \) gives us \( L \):

\[ B = \frac{\mu_0 I L}{4\pi r^2} \]

### Step 5: Analyze the Result
From the equation \( B = \frac{\mu_0 I L}{4\pi r^2} \), we can see that the magnetic field \( B \) is inversely proportional to the square of the distance \( r \).

### Conclusion
Thus, the magnetic field due to a current in a straight wire segment of length \( L \) at a point on its perpendicular bisector at a distance \( r \) decreases as \( \frac{1}{r^2} \).