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Biot Savart Law

To understand completely Biot Savart Law some important rules or laws should be known are Maxwell cork screw Law, Fleming’s right hand rule.

If current passes through a conductor it creates a magnetic field around it. The direction of the magnetic field for passing current through conductor explained by Maxwell cork screw Law & Fleming’s right hand rule.

Maxwell cork screw law:

Maxwell explained the direction of magnetic field with cork screw. If we turn the screw with right hand the turn of thumb indicates the direction of magnetic field.

According to Maxwell corkscrew rule if current goes through a wire, the direction of magnetic field due to flow of current is turning the screw approaching direction. The figure shows the direction of magnetic field.

Maxwell cork screw

Photo by brainshare

Fleming right hand rule:

Fleming explained the direction of magnetic field with right hand due to flow of current via conductor. The thumb indicates the flow of current and all the other fingers indicates the direction of magnetic field due to flowing current shown in figure.

Fleming's right hand rule

Photo by Jfmelero

Flowing of current through conductor produces magnetic field, the direction of magnetic field explained Maxwell corkscrew rule and Fleming right hand rule.

If charges pass through magnetic field what happens?

The straight answer is that the charges experience a force due to magnetic field. The magnitude of the force depends upon the quantity of charge, moving velocity of the charge, strength of magnetic field and angle between magnetic field and charge moving direction.

Mathematical expression is

Where,

q = moving charge quantity

V = velocity of the charge

B = magnitude of magnetic field

θ = angle between magnetic field and moving charge

Figure 1 is showing angle between magnetic field direction and moving charge is 90 degree or perpendicular each other. Figure 2 shows any angle of them. When they are in 90 degree it is the peak value of force and if the moving charge goes parallel to magnetic field direction then the force is zero. For any other angle we will get a force on moving charge from above equation.

In aforesaid topic all are important to understand Biot Savart Law. We know if current passes through a conductor it produces magnetic field and we can determine the direction of magnetic field using Fleming right hand rule and Maxwell corkscrew rule also. But what is the magnitude of the magnetic field arising due to flow of current. That is explained by Biot Savart Law.

Biot Savart’s Law

In a constant medium if current passes through a small conductor it creates magnetic field, the magnitude of the magnetic field is proportional to the flowing current, length of the conductor, angle between the middle point of conductor and the certain position straight line and inversely square to the distance between the certain point and the middle point of the conductor.

If the length of the conductor is dl, I current passing through the conductor, P is a certain point where we will find the magnitude of magnetic field due to flow of current dB, angle between the certain point and the middle point is theta (θ). Here we take small length of conductor so produce magnetic field is small. As we take dl for length and dB for magnetic field.

Mathematically

Where K is proportionality constant. The value of K depends upon the magnetic effect in medium and other parameters value.

 

In free space the value of K for Biot Savart Law is found

Application of Biot Savart’s Law:

Applying Biot Savart Law we can measure the amount of flowing current passing a conductor.

Magnetic field due to current carrying straight wire of infinite length:

biot savart formula straight wire

biot savart's law straight wire

 

Let a straight conductor along XY at infinite length carrying I current. dx is a small portion of the length. We determine the magnetic field magnitude B for dx length I carrying current. Suppose P is a point from the midpoint of dx. The distance between P and midpoint is r.

According to Biot Savart Law magnetic field in P point due to flow of current I via dx,

We have to find magnetic field B for total length.

As the length can be any value so we have to integrate x = -∞  to x = +∞ .

Total magnetic field in P point,

the biot savart's formula

This is the equation of magnetic field due to infinite long current carrying straight conductor.

Magnetic field at the center of circular loop:

biot savart law circular loop caryying current

Let a circle shape conductor carrying I current have r radius. Center of the circle is O. we have to determine magnetic field B in the center O.

Consider, XY = dl is small length of the total coil.

According to Biot Savart law we get magnetic field at center O of the coil for small length dl,

Here Ѳ is the angle between dl and r, it is 90. We will get total magnetic field at center O integration above equation. Limit of the integration will be l = 0 to l = 2πr because length of the coil is circumference of the circle 2πr.

This is the equation of magnetic field at the center of a current carrying circular coil.

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