Magnetic Field Due to a Current Element: Class 12 Physics Notes, Definition, Working Principle & Formula

The magnetic effect produced in space due to small current carrying element is referred to as magnetic field due to a current element. This is the essential concept of electromagnetism which shows that moving electric current generates magnetic fields. The current element is a small part of the wire carrying current.  

Magnetic field dure to current element is also referred to as Biot-Savart Law. This law is used to determine the magnetic field due to a small segment of current. The formula to determine the magnetic field due to a current element is,

$\begin{array}{l}\left|dB\right|=(\frac{{\mu }_0}{4\pi })(\frac{Idssin\theta }{r^2})\end{array}$

Importance of Determining the Magnetic Field due to a Current Element 

Determining the magnetic field due to a current element is essential in both theoretical and practical aspects of physics and engineering. Below is the importance.  

1. Foundation of Electromagnetism 

1. Design of electrical devices 

1. Magnetic field Mapping 

1. Navigation and Sensing 

1. Medical and Scientific Application 

The working principle of magnetic field due to a current element is based on Biot-Savart Law. As per this law, we can determine the magnetic field in a surrounding space due to the small element of current-carrying conductor. 

Working Principle 

• Electric Current Produces Magnetic Field 

• • The flow of electric current generated magnetic field around the conductor. 

• Direction of Magnetic Field: Direction of magnetic field is given by the right-hand rule 

• • The thumb is the direction of the current 

• • Fingers indicate the direction of the magnetic field 

• Magnitude of Magnetic Field: The magnetic field strength at a point is based on 

• • Magnitude of current 

• • Length and orientation of the current element 

• • Distance between the current element and the observation point 

• • Angle between current element and line connecting to a point. 

• Vector Nature: 

• • Magnetic field is a vector quantity 

• • Magnetic field is calculated using the cross product of current element and position vector.  

Using this fundamental formula, we can derive the expression of $\vec{\mathit{B}}$  due to a long wire.

$\vec{B}$ due to a straight wire :

Due to a straight wire ‘PQ’ carrying a current ‘i’ the $\vec{\mathit{B}}$  at A is given by the formula

B = $\frac{{\mu }_0{\rm I}}{4\pi r}$  (sin Θ₁ + sin Θ₂)   

Shortcut for Direction:

The direction of the magnetic field at a point P due to a straight wire can be found by a slight variation in the right-hand thumb rule. If we stretch the thumb of the right hand along the current and curl our fingers to pass through the point P, the direction of the fingers at P gives the direction of the magnetic field there.

We can draw magnetic field lines on the pattern of electric field lines. A tangent to a magnetic field line is given by the direction of the magnetic field existing at that point. For a straight wire, the field lines are concentric circles with their centres on the wire and in the plane perpendicular to the wire. There will be an infinite number of such lines in the planes parallel to the above-mentioned plane.

Special Case:

If the wire is infinitely long, then the magnetic field at ‘P’ (as shown in the figure) is given by (using
_Θ1 = Θ₂ = 90º and the formula of ‘B’ due to a straight wire)

The direction of $\vec{B}$  at various is as shown in the figure. The magnetic lines of force will be concentric circles

around the wire (as shown earlier)

If the wire is infinitely long but ‘P’ is as shown in the figure. The direction of $\vec{B}$ at various points is as shown in the figure. At ‘P’