Biot–Savart law

In 1820, it was discovered that electricity and magnetism are intimately related to each other. Besides this, Danish physicist Hans Christian Oersted observed that the current in a straight wire produces a slight deflection in a nearby magnetic compass needle. The deflection increases in the compass when it is brought closer to the current wire or on increasing the current. Oersted deduced from this that moving charges (also known as current) generate a magnetic field in the environment. All magnetic fields are because of current or moving charges. Here in this article, we are going to study the relation between current and magnetic fields as per Biot-Savart’s law. It is a useful law that gives a method to calculate the magnetic field produced by the current distribution. This law was discovered by Jean-Baptiste Biot and Felix Savart in the beginning of the 19th century.  

We are already familiar with the idea that current-carrying conductors generate magnetic fields around themselves. Biot–Savart law describes this situation mathematically and states the intensity of the magnetic field at a point. 

Explanation of Biot–Savart law 

 Consider a small element of a conductor ‘dl’. The magnetic field is produced at distance ‘r’ due to this element. Consider angle current element and position vector ‘r’.  

Biot–Savart law states the following  points:

1. It states that the magnitude of the magnetic field dB is directly proportional to the current I.
2. Also directly proportional to the length of the element dl. 
3. And the square of the distance r is inversely proportional to it.
4. It has a perpendicular direction to the plane containing dl and r.

dB =04Idlr2

Or dB =04I dl r2r 

Where, 04  is a constant of proportionality. The above expression is applied when the medium is a vacuum.  

Here we use the cross-product property. This equation constitutes our basic equation for the magnetic field, where 0 is the permeability of free space.

The relation between Biot–Savart law and Coulomb’s law 

The Biot–Savart law has certain similarities as well as differences with Coulomb’s law for the electrostatic field. 

1. They both are inversely proportional to the square of distance ‘r’ from the current element to the point of the magnetic field.
2. Both fields have the principle of superposition. Note:  This is applied only when the magnetic field is linear in source I. 
3.  The magnetic field is produced due to vector source I dl whereas the electrostatic field is produced by the scalar source known as electric charge.
4. The source and the field point are connected by the electrostatic field and the displacement vector.
5.  The magnetic field is perpendicular to the displacement vector r and the current element I dl.
6.  The angle dependence is not present in the electrostatic field.
7.  The magnetic field at any point in the direction dl is zero. Thus, = 0, sin = 0 and dB = 0.

Now here is a quite interesting relation between the permittivity of free space that is 0, the permeability of free space, and c is the speed of light in a vacuum. 

c=100

Definition of Biot–Savart law

In simple words, it is the mathematical expression in physics that describes the magnetic field generated by a constant electric current.  The magnetic field is affected by the electric current’s size, direction, length, and proximity.

It states that a magnetic field produced due to a small element of current-carrying conductor source at any point is directly proportional to the length of the current element,  the current itself,  the sine of the angle between the direction of current and the line joining the current element and the point. Besides this, it is inversely proportional to the square of the distance of the point r. 

Examples of Biot–Savart law

1) Magnetic field due to finite straight wire: 

 Using Biot-Savart law, we can find out magnetic fields due to finite straight wire. The value of magnetic field will be equal to 

dB =04I dl r2r

Where I is the current flowing in the wire

dl is the small length element,

r is the distance from the wire where the magnetic field is calculated.

1. II)  Magnetic field due to a circular loop: 

Biot-Savart law is used to calculate the magnetic field due to a circular loop. For a circular loop, magnetic field at the center is equal to

B= 0I2R

Where I is the current is passing in the circular loop,

R is the radius of the circular loop.

Importance of Biot–Savart law 

1. It is very important to calculate the magnetic field around the current-carrying conductors.
2. It is used to determine the magnetic field intensity and provides a relation between the magnetic field and the current source.
3. It is also used to determine the magnetic field in Ampere’s law and Coulomb’s force.

Conclusion

From all of the above, we conclude that Biot–Savart law is very useful in physics as it helps to determine the magnetic field around the current-carrying conductor. It states that the magnetic field is produced at a certain distance r from the source of current. This magnetic field is directly proportional to the current, length of the small element of the source and inversely proportional to the square of the distance r. Also, its direction is perpendicular to the plane containing dl and r. It is used to calculate the magnetic field in atomic and molecular theory.  We also conclude that if the current increases, the strength of the magnetic field also increases.