0.3 Magnetic field due to current in straight wire

The Biot-Savart law allows us to calculate magnetic field due to steady current through a small element of wire. Since direction of magnetic field due to different current elements of an extended wire carrying current is not unique, we need to add individual magnetic vectors to obtain resultant or net magnetic field at a point. This method of determining the net magnetic field follows superposition principle, which says that magnetic fields due to individual small current element are independent of each other and that the net magnetic field at a point is obtained by vector sum of individual magnetic field vectors :

$$\mathbf{B}=\sum {\mathbf{B}}_i={\mathbf{B}}_1+{\mathbf{B}}_2+{\mathbf{B}}_3+\dots \dots$$

We calculate magnetic field due to individual current element (I d l ) using Biot-Savart law :

$$d\mathbf{B}=\frac{{\mu }_0}{4\pi }\frac{Iđ\mathbf{l}X\mathbf{r}}{r^3}$$

where "d l " is referred as “current length element” and "I d l " as “current element”.

In the case of a straight wire, the task of vector addition is simplified to a great extent because direction of magnetic field at a point due to all current elements comprising the straight wire is same.

Direction of magnetic field (right hand thumb rule)

A straight line and a point constitute an unique plane. This is true for all points in three dimensional rectangular space (x,y,z). For convenience, let us consider that the point of observation (P) lies in xy plane as shown in the figure below. We can say that the straight wire along y-axis also lie in xy plane. Clearly, this plane is the plane of current length element d l and displacement vector r , which appear in the Biot-Savart expression. The direction of magnetic field is vector cross product d l X r , which is clearly perpendicular to the plane xy. This means that the magnetic field is along z-axis. This conclusion is independent of the relative positions of current length elements of the wire with respect to observation point P.

In a nutshell, we conclude that the directions of magnetic fields due to all current elements constituting straight wire at a point P are same. Though, magnitudes of magnetic fields are different as different current elements are located at different linear distance from the point i.e. displacement vectors ( r ) are different for different current length elements (d l ).

Magnetic field due to current in straight wire

See in the figure how magnetic fields due to three current elements in positive y-direction are acting in negative z-direction. The magnetic fields due to different current elements are ${\mathbf{B}}_1$ , ${\mathbf{B}}_2$ and ${\mathbf{B}}_3$ acting along PZ’ as shown in the figure. Note that magnitudes of magnetic fields are not equal as current elements are positioned at different linear distance.

The magnetic field is along z-axis either in positive or negative z direction depending on the direction of current and whether observation point is on right or left of the current carrying straight wire. By convention, magnetic field vector into the plane of drawing is denoted by a cross (X) and magnetic field vector out of the plane of drawing is denoted by a dot (.). Following this convention, magnetic field depicted on either side of a current carrying straight wire is as shown here :