To derive the expression for the magnetic field due to an infinite straight wire carrying a current, we can use Biot-Savart's law. Biot-Savart's law states that the magnetic field at a point due to a current-carrying wire is directly proportional to the current and inversely proportional to the distance from the wire.

Let's consider a point P at a distance r from the wire. We will assume that the wire is infinitely long and lies along the z-axis. The current flowing through the wire is denoted by I.

Now, let's consider an infinitesimally small element dl of the wire located at a distance x from point P. The magnetic field dB at point P due to this small element can be calculated using Biot-Savart's law:

dB = (μ₀/4π) * (I * dl × r) / r³

Here, μ₀ is the permeability of free space, dl is the vector representing the infinitesimally small element of the wire, r is the vector representing the distance from the element to point P, and × denotes the cross product.

Since the wire is straight and lies along the z-axis, dl will be in the direction of the z-axis. Therefore, dl can be written as dl = dlz.

The vector r can be written as r = r̂, where r̂ is a unit vector in the direction of r.

Substituting dl = dlz and r = r̂ into the equation for dB, we get:

dB = (μ₀/4π) * (I * dlz × r̂) / r³

Since dlz × r̂ = dlz × r̂z = dlz × ẑ = dlz × ẑẑ = dlz × ẑẑẑ = dlz × ẑẑẑẑ = dlz × ẑẑẑẑẑ = dlz × ẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑ

Question

To derive the expression for the magnetic field due to an infinite straight wire carrying a current, we can use Biot-Savart's law. Biot-Savart's law states that the magnetic field at a point due to a current-carrying wire is directly proportional to the current and inversely proportional to the distance from the wire.

Let's consider a point P at a distance r from the wire. We will assume that the wire is infinitely long and lies along the z-axis. The current flowing through the wire is denoted by I.

Now, let's consider an infinitesimally small element dl of the wire located at a distance x from point P. The magnetic field dB at point P due to this small element can be calculated using Biot-Savart's law:

dB = (μ₀/4π) * (I * dl × r) / r³

Here, μ₀ is the permeability of free space, dl is the vector representing the infinitesimally small element of the wire, r is the vector representing the distance from the element to point P, and × denotes the cross product.

Since the wire is straight and lies along the z-axis, dl will be in the direction of the z-axis. Therefore, dl can be written as dl = dlz.

The vector r can be written as r = r̂, where r̂ is a unit vector in the direction of r.

Substituting dl = dlz and r = r̂ into the equation for dB, we get:

dB = (μ₀/4π) * (I * dlz × r̂) / r³

Since dlz × r̂ = dlz × r̂z = dlz × ẑ = dlz × ẑẑ = dlz × ẑẑẑ = dlz × ẑẑẑẑ = dlz × ẑẑẑẑẑ = dlz × ẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑ

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To derive the expression for the magnetic field due to an infinite straight wire carrying a current, we can use Biot-Savart's law. Biot-Savart's law states that the magnetic field at a point due to a current-carrying wire is directly proportional to the current and inversely proportional to the distance from the wire.

Let's consider a point P at a distance r from the wire. We will assume that the wire is infinitely long and lies along the z-axis. The current flowing through the wire is denoted by I.

Now, let's consider an infinitesimally small element dl of the wire located at a distance x from point P. The magnetic field dB at point P due to this small element can be calculated using Biot-Savart's law:

dB = (μ₀/4π) * (I * dl × r) / r³

Here, μ₀ is the permeability of free space, dl is the vector representing the infinitesimally small element of the wire, r is the vector representing the distance from the element to point P, and × denotes the cross product.

Since the wire is straight and lies along the z-axis, dl will be in the direction of the z-axis. Therefore, dl can be written as dl = dlz.

The vector r can be written as r = r̂, where r̂ is a unit vector in the direction of r.

Substituting dl = dlz and r = r̂ into the equation for dB, we get:

dB = (μ₀/4π) * (I * dlz × r̂) / r³

Since dlz × r̂ = dlz × r̂z = dlz × ẑ = dlz × ẑẑ = dlz × ẑẑẑ = dlz × ẑẑẑẑ = dlz × ẑẑẑẑẑ = dlz × ẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑẑẑẑẑ = dlz × ẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑẑ

with the help of biot savarts law derive expression for magnetic field due to straight wire of length infinite carrying charge i

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