A long conducting wire of radius 5cm, carrying a current of 2A is placed with its axis coinciding with Y axis and lies on x z plane . Calculate the magnetic field induction at point A(3cm,3cm,3cm) and (8cm,8cm,8cm) if relative magnetic permeability of conductor is 300 ?

Question

Answer

Calculation of Magnetic Field Induction at Two Different Points

Given:

Radius of wire, r = 5 cm

Current flowing through the wire, I = 2 A

Relative magnetic permeability of the conductor, μr = 300

Point A(3cm,3cm,3cm)

Let's consider a small element dl of the wire at a distance r from point A. The magnetic field induction dB due to this element can be calculated using the Biot-Savart Law:

dB = μ0/4π * I * dl x r / r^3

where μ0 is the permeability of free space, dl is the length of the element, and r is the distance between the element and the point A.

As the wire is symmetric about the Y-axis and lies on the XZ plane, the magnetic field induction at point A is only in the Y-direction.

Therefore, the magnetic field induction at point A is:

B = ∫dB = μ0/4π * I * 2πr * r / r^3 = μ0/2 * I / r

As μr ≠ 1, we need to multiply the above expression by the relative magnetic permeability of the conductor:

B = μ0 * μr/2 * I / r = 4π * 10^-7 * 300/2 * 2 / 0.03 ≈ 1.26 x 10^-3 T

Point B(8cm,8cm,8cm)

Using the same approach as above, the magnetic field induction at point B can be calculated:

B = μ0 * μr/2 * I / r = 4π * 10^-7 * 300/2 * 2 / 0.08 ≈ 3.97 x 10^-4 T

Explanation:

The magnetic field induction at a point due to a current-carrying wire depends on the distance between the point and the wire, the current flowing through the wire, and the relative magnetic permeability of the conductor. By applying the Biot-Savart Law, we can calculate the magnetic field induction at any point in space due to a small element of the wire and then integrate over the entire wire to obtain the total magnetic field induction at the point. In this case, as the wire is symmetric about the Y-axis and lies on the XZ plane, the magnetic field induction at the two given points is only in the Y-direction.

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