A long wire carrying a current `i` is bent to form a plane angle. The magnitude of magnetic field at height `d`, above the point of bending

Options

A. `(sqrt(2) mu_(0)i)/(4pid)`
B. `(3sqrt(2) mu_(0)i)/(4pid)`
C. `(5sqrt(2) mu_(0)i)/(4pid)`
D. `(7sqrt(2) mu_(0)i)/(4pid)`

Explanation

To solve the problem of finding the magnitude of the magnetic field at height `d` above the point of bending of a long wire carrying a current `i` that is bent to form a plane angle, we can follow these steps:

### Step 1: Understand the Configuration
We have a long wire bent at a 90-degree angle. The current `i` flows through the wire, and we need to find the magnetic field at a point located at a height `d` above the point where the wire is bent.

### Step 2: Identify the Sections of the Wire
The wire can be considered as two semi-infinite straight wires:
1. One part of the wire extends vertically (let's call it wire A).
2. The other part extends horizontally (let's call it wire B).

### Step 3: Use the Formula for Magnetic Field Due to a Semi-Infinite Wire
The magnetic field `B` at a distance `r` from a straight wire carrying current `I` is given by the formula:
\[ B = \frac{\mu_0 I}{4 \pi r} \]
where `μ₀` is the permeability of free space.

### Step 4: Calculate the Magnetic Field from Wire A
For wire A (the vertical wire), the distance from the point of interest (height `d`) to the wire is `d`. Thus, the magnetic field `B_A` due to wire A at the point above the bending is:
\[ B_A = \frac{\mu_0 i}{4 \pi d} \]
The direction of the magnetic field due to wire A can be determined using the right-hand rule. If the current flows upwards, the magnetic field will circulate around the wire, pointing towards the right (let's assume this is in the positive x-direction).

### Step 5: Calculate the Magnetic Field from Wire B
For wire B (the horizontal wire), the distance from the point of interest to the wire is also `d`. Thus, the magnetic field `B_B` due to wire B at the same point is:
\[ B_B = \frac{\mu_0 i}{4 \pi d} \]
The direction of the magnetic field due to wire B can also be determined using the right-hand rule. If the current flows to the right, the magnetic field will point upwards (in the positive z-direction).

### Step 6: Combine the Magnetic Fields
Since the magnetic fields from wire A and wire B are perpendicular to each other (one in the x-direction and one in the z-direction), we can use the Pythagorean theorem to find the resultant magnetic field `B_net`:
\[ B_{net} = \sqrt{B_A^2 + B_B^2} \]
Substituting the values of `B_A` and `B_B`:
\[ B_{net} = \sqrt{\left(\frac{\mu_0 i}{4 \pi d}\right)^2 + \left(\frac{\mu_0 i}{4 \pi d}\right)^2} \]
\[ B_{net} = \sqrt{2 \left(\frac{\mu_0 i}{4 \pi d}\right)^2} = \frac{\mu_0 i}{4 \pi d} \sqrt{2} \]

### Step 7: Final Result
Thus, the magnitude of the magnetic field at height `d` above the point of bending is:
\[ B_{net} = \frac{\mu_0 i \sqrt{2}}{4 \pi d} \]