Magnetic field due to straight current carrying wire

Biot-Savart Law is the basic formula used to calculate magnetic field due to different kinds of current configurations. Consider an arbitrary current carrying wire. Take a small portion of the wire having length `\vec(dL)` and a point of interest where you need to calculate the magnetic field due to this small section of wire. Assume the radius vector to the point of interest be `\vecr`.

Then, according to Biot-Savart Law, the magnetic field at that point will be given as:

If you want to calculate the magnetic field due to complete wire, then integrate this expression over whole length.

Let's consider a very common case of finite wire as shown below and try to apply B-S Law to calculate the field at point P:

Consider a small piece of wire `\vec(dL)` at a distance `\vecr` from the point of interest. According to the geometry shown above, the small magnetic field created by this small section of wire will be:

Using the geometry of the problem, we can write:

Thus initial magnetic field expression become:

Integrating this expression, we get:

Length of the wire can be calculated as:

Now, consider a special case when the wire is of infinite length. In that case: