This theorem is just alternative ofThevenin theorem. In Norton theorem, we just replace the circuit connected to a particular branch by equivalentcurrent source. In this theorem, the circuit network is reduced into a single constant current source in which, the equivalent internalresistance is connected in parallel with it. Everyvoltage source can be converted into equivalent current source.
Suppose, in complex network we have to find out thecurrent through a particular branch. If the network has one of more active sources, then it will supplycurrent through the said branch. As in the said branch current comes from the network, it can be considered that the network itself is a current source. So in Norton theorem the network with different active sources is reduced to singlecurrent source that's internalresistance is nothing but the looking back resistance, connected in parallel to the derived source.
Suppose, in complex network we have to find out thecurrent through a particular branch. If the network has one of more active sources, then it will supplycurrent through the said branch. As in the said branch current comes from the network, it can be considered that the network itself is a current source. So in Norton theorem the network with different active sources is reduced to singlecurrent source that's internalresistance is nothing but the looking back resistance, connected in parallel to the derived source.
The looking back resistance of a network is the equivalentelectrical resistance of the network when someone looks back into the network from the terminals where said branch is connected. During calculating this equivalent resistance, all sources are removed leaving their internal resistances in the network. Actually in Norton theorem, the branch of the network through which we have to find out thecurrent, is removed from the network. After removing the branch, we short circuit the terminals where the said branch was connected. Then we calculate the short circuit current that flows between the terminals. This current is nothing but Norton equivalent current INof the source. The equivalentresistance between the said terminals with all sources removed leaving their internal resistances in the circuit is calculated and said it is RN. Now we will form acurrent source that's current is INA and internal shunt resistance is RN Ω.
For getting clearer concept of this theorem, we have explained it by the following example,
In the example tworesistors R1 and R2 are connected in series and this series combination is connected across onevoltage source of emf E with internal resistance Ri as shown. Series combination of one resistive branch of RL and another resistance R3 is connected across the resistance R2 as shown. Now we have to find out thecurrent through RL by applying Norton theorem.
First, we have to remove the resistor RL from terminals A and B and make the terminals A and B short circuited by zero resistance.
Second, we have to calculate the short circuit current or Norton equivalent current IN through the points A and B.
The equivalent resistance of the network,
To determine internalresistance or Norton equivalent resistance RNof the network under consideration, remove the branch between A and B and also replace thevoltage source by its internal resistance. Now the equivalent resistance as viewed from open terminals A and B is RN,
As per Norton theorem, whenresistance RL is reconnected across terminals A and B, the network behaves as a source of constantcurrent IN with shunt connected internal resistance RN and this is Norton equivalent circuit.
For getting clearer concept of this theorem, we have explained it by the following example,
In the example tworesistors R1 and R2 are connected in series and this series combination is connected across onevoltage source of emf E with internal resistance Ri as shown. Series combination of one resistive branch of RL and another resistance R3 is connected across the resistance R2 as shown. Now we have to find out thecurrent through RL by applying Norton theorem.
First, we have to remove the resistor RL from terminals A and B and make the terminals A and B short circuited by zero resistance.
Second, we have to calculate the short circuit current or Norton equivalent current IN through the points A and B.
The equivalent resistance of the network,
To determine internalresistance or Norton equivalent resistance RNof the network under consideration, remove the branch between A and B and also replace thevoltage source by its internal resistance. Now the equivalent resistance as viewed from open terminals A and B is RN,
As per Norton theorem, whenresistance RL is reconnected across terminals A and B, the network behaves as a source of constantcurrent IN with shunt connected internal resistance RN and this is Norton equivalent circuit.
Norton Equivalent Circuit
Biot Savart Law
The mathematical expression formagnetic flux density was derived by Jean Baptiste Biot and Felix Savart. Talking the deflection of a compass needle as a measure of the intensity of a current, varying in magnitude and shape, the two scientists concluded that anycurrent element projects into space amagnetic field, themagnetic flux density of which dB, is directly proportional to the length of the element dl, the current I, the sine of the angle and θ between direction of the current and the vector joining a given point of the field and the current element and is inversely proportional to the square of the distance of the given point from the current element, r. This is Biot Savart law statement. 
Where, K is a constant, depends upon the magnetic properties of the medium and system of the units employed. In SI system of unit,
Therefore, final Biot Savart lawderivation is,
Let us consider a long wire carrying an current I and also consider a point p. The wire is presented in the below picture by red color. Let us also consider an infinitely small length of the wire dl at a distance r from the point P as shown. Here, r is a distance vector which makes an angle θ with the direction of current in the infinitesimal portion of the wire.
Where, K is a constant, depends upon the magnetic properties of the medium and system of the units employed. In SI system of unit,
Therefore, final Biot Savart lawderivation is,
Let us consider a long wire carrying an current I and also consider a point p. The wire is presented in the below picture by red color. Let us also consider an infinitely small length of the wire dl at a distance r from the point P as shown. Here, r is a distance vector which makes an angle θ with the direction of current in the infinitesimal portion of the wire.
If you try to visualize the condition, you can easily understand the magnetic field density at that point P due to that infinitesimal length dl of wire is directly proportional to current carried by this portion of the wire. That means current through this infinitesimal portion of the wire is increased the magnetic field density due to this infinitesimal length of wire, at point P increases proportionally and if the current through this portion of wire is decreased the magnetic field density at point P due to this infinitesimal length of wire decreases proportionally.
As the current through that infinitesimal length of wire is same as the current carried by the wire itself.
It is also very natural to think that the magnetic field density at that point P due to that infinitesimal length dl of wire is inversely proportional to the square of the straight distance from point P to center of dl. That means distance r of this infinitesimal portion of the wire is increased the magnetic field density due to this infinitesimal length of wire, at point P decreases and if the distance of this portion of wire from point P, is decreased, the magnetic field density at point P due to this infinitesimal length of wire increases accordingly.
Lastly, field density at that point P due to that infinitesimal portion of wire is also directly proportional to the actual length of the infinitesimal length dl of wire. As θ be the angle between distance vector r and direction of current through this infinitesimal portion of the wire. The component of dl directly facing perpendicular to the point P is dlsinθ,
Now combining these three statements, we can write,
This is the basic form of Biot Savart's Law
Now putting the value of constant k (which we have already introduced at the beginning of this article) in the above expression, we get
Here, μ0 used in the expression of constant k is absolute permeability of air or vacuum and it's value is 4π10-7 Wb/ A-m in SI system of units. μr of the expression of constant k is relative permeability of the medium.
Now, flux density(B) at the point P due to total length of the current carrying conductor or wire can be represented as,
If D is the perpendicular distance of the point P form the wire, then
Now, the expression of flux density B at point P can be rewritten as,
As per the figure above,
Finally the expression of B comes as,
This angle θ depends upon the length of the wire and the position of the point P. Say for certain limited length of the wire, angle θ as indicated in the figure above varies from θ1 to θ2. Hence, flux density at point P due to total length of theconductor is,
Let's imagine the wire is infinitely long, then θ will vary from 0 to π that is θ1 = 0 to θ2 = π. Putting these two values in the above final expression of Biot Savart law, we get,
This is nothing but the expression of Ampere's Law.
As the current through that infinitesimal length of wire is same as the current carried by the wire itself.
It is also very natural to think that the magnetic field density at that point P due to that infinitesimal length dl of wire is inversely proportional to the square of the straight distance from point P to center of dl. That means distance r of this infinitesimal portion of the wire is increased the magnetic field density due to this infinitesimal length of wire, at point P decreases and if the distance of this portion of wire from point P, is decreased, the magnetic field density at point P due to this infinitesimal length of wire increases accordingly.
Lastly, field density at that point P due to that infinitesimal portion of wire is also directly proportional to the actual length of the infinitesimal length dl of wire. As θ be the angle between distance vector r and direction of current through this infinitesimal portion of the wire. The component of dl directly facing perpendicular to the point P is dlsinθ,
Now combining these three statements, we can write,
This is the basic form of Biot Savart's Law
Now putting the value of constant k (which we have already introduced at the beginning of this article) in the above expression, we get
Here, μ0 used in the expression of constant k is absolute permeability of air or vacuum and it's value is 4π10-7 Wb/ A-m in SI system of units. μr of the expression of constant k is relative permeability of the medium.
Now, flux density(B) at the point P due to total length of the current carrying conductor or wire can be represented as,
If D is the perpendicular distance of the point P form the wire, then
Now, the expression of flux density B at point P can be rewritten as,
As per the figure above,
Finally the expression of B comes as,
This angle θ depends upon the length of the wire and the position of the point P. Say for certain limited length of the wire, angle θ as indicated in the figure above varies from θ1 to θ2. Hence, flux density at point P due to total length of theconductor is,
Let's imagine the wire is infinitely long, then θ will vary from 0 to π that is θ1 = 0 to θ2 = π. Putting these two values in the above final expression of Biot Savart law, we get,
This is nothing but the expression of Ampere's Law.
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