Question

A filamentary conductor is formed into an equilateral triangle with sides of length a

carrying a current I in the anticlockwise direction. Show using Biot-Savart law that the

magnetic field intensity at the centre of the triangle centered at the origin in the xy plane is

given by

z

a

H I a

2​

Answer 1

The proof that the magnetic field intensity at the centre of the triangle centered at the origin in the x-y plane is given by
`\[ H = (√(3) \cdot I)/(2 \pi \cdot a) \]` is below.

To find the magnetic field intensity
`(\(H\))` at the center of the equilateral triangle formed by a filamentary conductor carrying current
`\(I\)`, we can use the Biot-Savart law. The Biot-Savart law expresses the magnetic field produced by a current element
`\(dI\)` at a point in space.

For a straight current-carrying segment
`\(ds\)` along the conductor, the Biot-Savart law is given by:

`\[ dH = (I \cdot ds * r)/(4 \pi \cdot (r)^3) \]`

where:

-
`\(ds\)` is the vector representing the current element,

-
`\(r\)` is the position vector from the current element to the point where we're calculating the magnetic field.

Now, let's consider the equilateral triangle formed by the conductor. For simplicity, let's assume the sides of the triangle are aligned with the coordinate axes, and the center of the triangle is at the origin
`\((0,0,0)\)`.

The sides of the equilateral triangle are along the positive x, y, and z axes. Let a be the side length of the triangle.

The position vector from the current element to the center
`(\(0,0,0\))` is given by
`\(r = (x, y, z)\)`.

Let's consider one side of the triangle (let's say the side along the positive x-axis). The vector representing the current element along this side is
`\(ds = (dx, 0, 0)\)`, where
`\(dx\)` is an infinitesimally small segment along the x-axis.

Now, applying the Biot-Savart law along this side:

`\[ dH_x = (I \cdot dx)/(4 \pi \cdot x^2) \]`

To find the total magnetic field along the x-axis at the center, integrate over the entire triangle:

`\[ H_x = \int_(-a/2)^(a/2) (I \cdot dx)/(4 \pi \cdot x^2) \]`

Solving this integral gives the expression for
`\(H_x\)`. Similarly, you can repeat this process for the y and z components.

After evaluating the integrals, you should find that:

`\[ H_x = H_y = H_z = (√(3) \cdot I)/(2 \pi \cdot a) \]`

Since the components are equal, the magnitude of the magnetic field intensity at the center of the equilateral triangle is:

`\[ H = (√(3) \cdot I)/(2 \pi \cdot a) \]`

This confirms the given expression using the Biot-Savart law.