Biot-Savart Law

Biot-Savart Law

The Biot-Savart law is an equation that gives the magnetic field produced by a current-carrying segment. This segment is  a vector quantity called the current element.

What is the Formula of Biot-Savart’s Law?

What is the Formula of Biot-Savart’s Law?

Consider a wire carrying a current 'i' in a certain direction as shown in the figure above. Take a small thread element of length ds. The direction of this element is along the direction of flow, so it forms a vector ID.

The Biot-Savart law can be applied to know the point magnetic field caused by this small element. Let r be the position vector drawn by the current element from that point, and let θ be the angle between them. Then

 

Where μ0 is the permittivity of free space and is equal to

4π × 10-7 TmA-1.

The direction of the magnetic field is always in a plane perpendicular to the line and position vector of the element. This is given by the rule of the right hand, where the thumb points in the direction of the normal current and the other fingers in the direction of the magnetic field.

Applications of the law of Biot-Savart

The Biot-Savart law calculates magnetic response even at the atomic or molecular level.

It is also used in aerodynamic theory to calculate the speed induced by vortex lines.

Importance of Biot-Savart's Law:

The law of Biot-Savart is similar to the law of Coulomb of electrostatics.

 The law also applies to very small wires that carry current. The law applies to symmetrical current distribution.

Magnetic Field Intensity Due to a Straight Current-Carrying Conductor of Finite Length

As we know, current-carrying conductors experience magnetic fields. Each mass creates a gravitational field and can also interact with that field. A charge creates an electric field and also interacts with it. Since a moving charge interacts with a magnetic field, we can expect it to create this field as well.

Consider a straight wire AB carrying a current (I) and determine the magnetic field strength at P.

Look at the picture above. According to the Biot-Savart law, the magnetic field at point P is given by 

 

Let AB be a conductor carrying a current I. Consider a point P at a certain distance from the center of the conductor. Consider a small current carrying element "dl" placed at point c. It is at a distance 'r' from point p. "l" is the distance between the center of the coil and dl is the length. Using the Biot-Savart law, the magnetic field of the current carrying element "dl" at P,

 Magnetic field due to a current carrying conductor

Current is usually defined as the charging rate. We already know that static charges produce an electric field that is proportional to the magnitude of charge. The same principle can be applied here. Moving charges create magnetic fields that are proportional to current, and thus a current-carrying conductor creates a magnetic effect around itself. This magnetic field is usually caused by subatomic particles in the conductor, such as moving electrons in atomic orbits.       

A magnetic field has both magnitude and direction. Therefore, it is a vector quantity labeled B (in the diagram below). The magnetic field produced by a current-carrying conductor depends on the current in the conductor and the distance from the point. The direction of the magnetic field is perpendicular to the wire. If you wrap the fingers of your right hand around the wire with the thumb pointing in the direction of the current, the direction of the curl of the fingers gives the direction of the magnetic field. This becomes clearer in the diagram below, where the red lines represent the magnetic field lines.

Magnetic Field Intensity Due To A Circular Loop Of A Current-Carrying Conductor

The magnetic field strength of a circular loop is considered in its axial position. For this let "a" be the radius of the circular loop. The current through the loop is "i". The intensity of the magnetic field is determined at point "P". This point is at a distance x from the center of the circular loop. Those two elements are used to determine the intensity of the field at point "P". The figure below shows a circular circuit carrying.

 

 

The magnetic field intensity due to the current- carrying circular loop along the axial position is given as:

Since the axial position is considered, the intensity of the magnetic field will be along the axis.


Therefore, the above equation is the magnetic field intensity of a circular current-carrying loop.

If there are N numbers of turns in the loop, then the magnetic field is given as:

Magnetic field at the center of Circular Coil Carrying Current:

 

 

Consider a circular coil of radius a and carrying current I in the direction shown in Figure. Suppose the loop lies in the plane of paper. It is desired to find the magnetic field at the centre O of the coil. Suppose the entire circular coil is divided into a large number of current elements, each of length dl. According to Biot-Savart law, the magnetic field  

dB  at the centre O of the coil due to current element I

dl is given by,

 

where  

r is the position vector of point O from the current element. The magnitude of dB at the centre O is

 

The direction of dB is perpendicular to the plane of the coil and is directed inwards. Since each current element contributes to the magnetic field in the same direction, the total magnetic field B at the center O can be found by integrating the above equation around the loop i.e.

 

For each current element, angle between   dl  and   r  is 90°. Also distance of each current element from the center O is a.

Magnetic Field on the Axis of a Circular Current Loop – Derivation:

Before we learn more about the magnetic field on the axis of a circular circuit, we need to understand the basic law of magnetism. We know that there is an equation that describes the magnetic field produced by a constant current. The law of Biot-Savart correlates the strength of the  field with the length, proximity and direction of the electric current. The Biot-Savart law is an equation that gives the  field produced by a current-carrying segment. A flux-carrying element is considered a vector quantity. The Biot-Savart law is obtained by the equation:

Where

μ0 is the permeability of free space and is equal to 4π × 10-7 TmA-1.

Now we will learn how to control the magnetic field along the axis of a circular circuit. Estimate the magnetic field due to the circular coil on its axis. The evaluation requires the sum of the "idl" effects of the smallest flow elements. "i" is direct current and this  evaluation in  vacuum or free space.

 See the image below. From the picture, we can see that the "i" represents a circular loop that carries a constant current. The image shows that the circular loop is placed in the y-z plane with its center at the origin O and its radius "R". The X axis is the loop axis.

x = distance P from center O of loop.

dl= a conducting element of the loop.

The Biot-Savart law gives the magnitude of the magnetic field (dB) due to current element (dl)

 

Any element of the loop will be perpendicular to the displacement vector from the element to the axial point.

For example, the element dl present in the figure is in the y-z plane. Whereas, the displacement vector r from dl to the axial point P is in the x-y plane.

The direction of dB is as shown in the figure and is perpendicular to the plane formed by dl and r. It has an x-component dBx and a component perpendicular to the x-axis, dB

 

A zero result is obtained when the components perpendicular to the x-axis are added and canceled.

The dB component due to dl is canceled by  the diametrically opposite dl element. This is shown in the image above.

So only the x component is preserved. The net effect in the x-direction can be obtained by integrating dBx = dBcosθ over the loop.

 

 

Since the circular wire forms a closed loop, the magnetic field lines are produced as shown in the figure below. The direction of the magnetic field created is given by the right-hand thumb rule.