Electromagnetics deals with Electric and Magnetic fields. Biot-Savart’s Law is the basic governing law for the study of static magnetic fields. In this article lets us discuss it along with its applications in our EMT.
The magnetic field intensity dH at a given point, by the current element is proportional to the current element and sine of the angle between the element and the line joining that element with the given point and inversely proportional to the square of distance between the current element and the given point.
Explanation of the Statement of Biot-Savart’s Law
Consider a wire of any length having current I flowing through it.
Now, let me explain the term “Current element“. It is the product of current and an infinitesimal part/length (dl) of the wire. So current element would be Idl. Moreover current element is treated as a vector quantity with the direction along the direction of current.
So consider a current element present at any location of the wire and let us find infinitesimal magnetic field intensity (dH) due to this current element at a point P.
Remember: Biot-Savart’s Law gives you the magnetic field intensity for the infinitesimal current element only and not for the entire wire.
Let us assume that point P is at a distance of ‘r’ from the current element. In other words, let us say length of the line joining current element and P is ‘r’. And let us say the angle between current element and r is θ then according to Biot-Savart’s Law we have,
Remember: The constant of proportionality is experimentally found to be 1/4π.
Vector form of the Biot-Savart’s Law
We know that, magnetic field intensity is a vector quantity. So obviously we are interested in the direction of this infinitesimal H i.e. dH. And for that we could state this law in the vector form.
So talking in terms of the vectors, the dH vector is pointing along the cross product of current element vector and r vector. We already know that current element is a vector quantity having direction along the current. And ‘r’ vector is the vector joining current element and the required point with a direction pointing towards the point.
Remember: If you write ‘r’ vector instead of unit vector along the ‘r’ then there will be r3 in the denominator.
An easy way to get the direction of ‘dH’
Alternatively, we can find the direction of dH using Right Hand Screw Rule.
Right Hand Screw Rule
Outstretch your thumb in the direction of current (Idl) with rest of the fingers curled. Or, think like screw is placed along the wire pointed in the direction of current flow. Then, the direction of rotation of fingers or direction of advancement of the screw is the direction of ‘dH’.
Biot-Savart’s Law for various Current Distributions
We cannot always expect a current in a wire. NO! It can be over the plane surface lamina or through the volume of object. Meaning, we have THREE types of current distributions i.e. line, surface and volume. So, is Biot-Savart’s Law stated different for each current distribution?
The answer is NO. Remember that the law gives you an infinitesimal magnetic field for an infinitesimally small current element. So irrespective of the current distribution, the statement will remain same. Only the change would be the “Current element”. Current element or an infinitesimal part of the current carrying conductor is different for various distributions.
The idea of “Current element” is the infinitesimally small part of the current carrying conductor.
The small portion of current would vary according to the distribution of current over the conductor. And distribution of current can be explained with the term current density rather than current. So, in other words, ultimately the current element depends on the given current density of the conductor. Isn’t it?
Current Density and its types
Like different charge configurations, we can have different current densities. And as I stated in the above paragraph, current density and not current, would sophistically represents the distributed current. What is distributed current? For that just compare it with distributed charge. As we know about the charge distributions, charge can be physically deposited over the line, surface or within the volume. Based on that we say distributed charge can be line charge , surface charge or volume charge and we define respective charge densities as linear charge density, surface charge density and volume charge density.
Likewise, the current carrying conductor can have current distributed over the line, surface or volume viz line current (or current), surface current and volume current. And we can define the respective current densities.
Line Current or just Current
Basically, it would be wrong to say a line current. But I have done so, to compare it with the line charge. Always keep in mind that concepts line charge and line current are totally different. You can define the line charge density i.e. charge per unit length for the line charge. But you can never define the line current density. NO!
Actually we never say a line current. It’s a current only. How do we define current? It is the rate of flow of positive charge. Isn’t it? It is a stream of continuous travelling charge. And that stream or that that line is actually the current.
When we have a current ‘I’ between points A and B that means we have a continuous flow of charge between A and B. The same charge leaving A is reaching to B travelling along the path. And the rate of flow for that charge is the current. So, the path of travelling of the charge or the line of stream of charge is actually through the conductor or wire between A and B. So if we say that there is current I, that means there is current ‘I’ at every point between A and B. Isn’t it?
So, technically, the current is the rate of flow of charge considering the stream of charge to be singular meaning area of cross-section of conductor/wire is just negligible. And as that path or stream appears to be a line, you can say it is a LINE CURRENT.
Surface Current and its Density (K)
Now, think that multiple streams of charge are spread over the surface or lamina. Such current is called as the SURFACE CURRENT.
We can say that multiple line currents are present over the surface to form the current over the plane lamina. And these line currents are distributed along the cross-sectional length of that surface.
So, surface current is made of the line currents stacked along the cross-sectional length of the surface. And you can visualize the total current on the surface is distributed over the cross-sectional length of the surface.
Hence, surface current density (K) is defined as the Current per unit cross-sectional length of the surface. It is represented by letter K. Obviously, surface current density, K is a vector quantity pointing along the direction of current and having unit as Ampere per meter (A/m).
Volume Current and its Density (J)
Now, think that multiple streams of charge are travelling through the volume. Such Current is called as the VOLUME CURRENT.
We can say that multiple line currents are present within the volume to form the volume current. And these line currents are distributed along the cross-sectional area of that volume.
So, volume current is made of the line currents stacked within the volume. And you can visualize the total current within the volume is distributed over the cross-sectional surface of the volume.
Hence, volume current density (J) is defined as the Current per unit cross-sectional surface. It is represented by letter J. Obviously, volume current density, J is a vector quantity pointing along the direction of current and having unit as Ampere per meter square (A/m2).
Current elements for different current distributions
Having known with different current distributions and respective current densities, let us now write current elements for them. Remember that Biot-Savart’s law gives you the infinitesimal magnetic field intensity (dH) for the infinitesimal part of the current carrying conductor i.e. current element.
Remember that the idea of “Current element” in the Biot-Savart’s Law is the product of respective current density and very small or an infinitesimal part of the given current carrying structure. So, we can say the current elements “Idl” in Line current, “Kds” in Surface current and “Jdv” in Volume current are all analogous.
Hence, Biot-Savart’s Law for different current distributions is written as follows:
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