You know a charge has an electric field around it. The interesting thing is when the charge moves, it also has another type of field called magnetic field. So, if a charge is moving, it now has two fields one is electric field which was already there and another is magnetic field.
Just like electric field \(\vec E\) is a vector field, the magnetic field \(\vec B\) is also a vector field. The electric field exerts force on a charge \(q\), that is \(\vec F = q\vec E\). Similarly the magnetic field exerts force on another moving charge. Note that magnetic field does not exert force on stationary charge. You know moving charge is current, which means a current produces magnetic field and exerts force on other currents in its influence.
Here we focus on the magnetic field of an isolated moving charge to understand how the magnetic field due to an isolated moving charge is calculated even if no such isolated moving charge is possible (explained later). The definition of magnetic field of an isolated moving charge allows us to understand how the magnetic field is determined for other moving charge distributions, that is current or collection of currents.
Now let's determine the magnetic field of a moving charge at a field point \(p\) at a particular instant of the motion. The charge is moving so we have to determine the field an instant. The magnetic field is based on the experimental evidences as we did to determine the magnetic force.
We defined magnetic force or field of a moving charge before truly defining the magnetic field but all of these definitions are in accordance with the real experiments conducted by Biot and Savart which is explained in Biot-Savart law.
Here we are going to do something similar to what we did in Coulomb's law. The experimental evidences suggest that the magnetic field \(\vec B\) is proportional to the square of distance from the source point to the field point, that is \(B \propto 1/r^2\) (see Figure 1). Note that the magnetic field is inversely proportional to the distance.
The magnetic field is also proportional to the speed of moving charge, that is \(B \propto v\) (in this case the magnetic field is directly proportional to the speed). And the magnetic field is directly proportional to the magnitude of the moving charge \(|q|\). The interesting thing here is that the magnetic field is also proportional to the sine of angle \(\theta\) between the charge's velocity vector and the position vector \(\vec r\) of the field point.
The interesting difference between electric field and magnetic field is that in electric field the direction was along the line joining the source point to the field point but in magnetic field the direction is perpendicular to the plane containing the velocity vector \(\vec v\) and position vector \(\vec r\) joining the source point and field point as shown in figure above.
Combing all the relationships imparting the magnetic field we get an expression for the magnetic field, that is
\[B = k \, \frac{|q| \, v \, \sin\theta}{r^2}\]
where \(k\) is the proportionality constant and it's value is \(k = \mu_0/4\pi\). So the full expression of the magnetic field is
\[B = \frac{\mu_0}{4\pi} \frac{|q| \, v \, \sin\theta}{r^2} \tag{1}\label{1}\]
The above equations can not be verified experimentally because it is based on the isolated moving charge and no such charge is possible. You know in electric circuit that a charge can only move if it is part of a complete electric circuit.
The Equation \eqref{1} can be expressed in vector form as the cross product of \(\vec v\) and unit vector \(\hat r\),
\[\vec B = \frac{\mu_0}{4\pi} \frac{q \, \vec v \times \hat r}{r^2} \tag{2}\label{2}\]
The direction of magnetic field can be determined by using the right hand rule. In this case you can curl your fingers around \(\vec v\) pointing your thumb in the direction of \(\vec v\) and the curled fingers give the direction of magnetic field for a positive moving charge. If the moving charge is negative, the direction of magnetic field is opposite to the direction of curled fingers for the positive charge case. Or you can simply curl your fingers in the sense of \(\vec v\) rotating into \(\hat r\) keeping thumb straight and the thumb gives the direction of magnetic field for positive charge. If the charge is negative the direction is opposite. In this case in Figure 1 at the point \(p\) the direction is outward, that is towards you.
The magnetic field has maximum magnitude when the angle between \(\vec v\) and \(\vec r\) is \(90^\circ\) and zero when the angle is \(0^\circ\). The SI unit of magnetic field is Tesla, \(\text{T}\). Note that the coulomb (C) per second is ampere (A).
\[1\text{T} = 1\frac{\text{N}}{\text{C}\cdot \text{m/s}} = 1\frac{\text{N}}{\text{A}\cdot \text{m}}\]
The real actual experiments were done by Biot and Savart for current carrying conductor called and the summarized version of their experiments is called BIot-Savart law. Here the magnetic field of moving charge is determined for an isolated moving charge and this is truly valid in terms of Biot-Savart law even if no isolated charge is possible.
