biosavert's law

The mathematical expression for magnetic 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 any current element projects into space a magnetic field, the magnetic 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 law derivation 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 W_b/ A-m in S.I system of units. μ_r 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 the conductor 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 .

Ohms Law | Equation Formula and Limitation of Ohms Law

Joules Law of Heating

 This theorem has been introduced in the year of 1952 by Dutch Electrical Engineer Bernard D.H. Tellegen. This is a very useful theorem in network analysis. According to Tellegen theorem, the summation of instantaneous powers for the n number of branches in an electrical network is zero. Are you confused? Let's explain. Suppose n number of branches in an electrical network have i₁, i₂, i₃, .............i_n respective instantaneous currents through them. These currents satisfy Kirchhoff's current law. Again, suppose these branches have instantaneous voltages across them are v₁, v₂, v₃, ........... v_n respectively. If these voltages across these elements satisfy Kirchhoff Voltage law then,

v_k is the instantaneous voltage across the k^th branch and i_k is the instantaneous current flowing through this branch. Tellegen theorem is applicable mainly in general class of lumped networks that consist of linear, non-linear, active, passive, time variant and time variant elements. This theorem can easily be explained by the following example.

In the network shown, arbitrary reference directions have been selected for all of the branch currents, and the corresponding branch voltages have been indicated, with positive reference direction at the tail of the current arrow. For this network, we will assume a set of branch voltages satisfy the Kirchhoff voltage law and a set of branch current satisfy Kirchhoff current law at each node. We will then show that these arbitrary assumed voltages and currents satisfy the equation.

And it is the condition of Tellegen theorem.

In the network shown in the figure, let v₁, v₂ and v₃ be 7, 2 and 3 volts respectively. Applying Kirchhoff voltage law around loop ABCDEA. We see that v₄ = 2 volt is required. Around loop CDFC, v₅ is required to be 3 volt and around loop DFED, v₆ is required to be 2. We next apply Kirchhoff current law successively to nodes B, C and D.

At node B let i_i = 5 A, then it is required that i₂ = − 5 A. At node C let i₃ = 3 A and then i₅ is required to be − 8. At node D assume i₄ to be 4 then i₆ is required to be − 9. Carrying out the operation of equation,

we get, 7 X 5 + 2 X ( − 5) + 3 X 3 + 2 X 4 + 3 X ( − 8) + 2 X ( − 9) = 0 Hence Tellegen theorem is verified.

Objective Questions on Circuit Theory (MCQs)

Superposition Theorem

Thevenin Theorem and Thevenin Equivalent Voltage and Resistance

Norton Theorem | Norton Equivalent Current and Resistance

Maximum Power Transfer Theorem

Millman Theorem

Reciprocity Theorem

Substitution Theorem

Compensation Theorem

Please add your con

What is Reciprocal Property?

In many electrical networks it is found that if the positions of voltage source and ammeter are interchanged, the reading of ammeter remains the same. It is not clear to you. Let's explain it in details. Suppose a voltage source is connected to a passive network and an ammeter is connected to other part of the network to indicate the response. Now any one interchanges the positions of ammeter and voltage source that means he or she connects the voltage source at the part of the network where the ammeter was connected and connects ammeter to that part of the network where the voltage source was connected. The response of the ammeter means current through the ammeter would be the same in both the cases. This is where the property of reciprocity comes in the circuit. The particular circuit that has this reciprocal property, is called reciprocal circuit. This type of circuit perfectly obeys reciprocity theorem.

Explanation of Reciprocity Theorem

The voltage source and the ammeter used in this theorem must be ideal. That means the internal resistance of both the voltage source and ammeter must be zero. The reciprocal circuit may be a simple or complex network. But every complex reciprocal passive network can be simplified into a simple network. As per reciprocity theorem, in a linear passive network, supply voltage V and output current I are mutually transferable. The ratio of V and I is called the transfer resistance. The theorem can easily be understood by this following example.

substitution theorem

Statement

Substitution theorem states that "if an element in a network is replaced by a voltage source whose voltage at any instant of time is equals to the voltage across the element in the previous network then the initial condition in the rest of the network will be unaltered". Or alternately "if an element in a network is replaced by a current source whose current at any instant of time is equal to the current through the element in the previous network then the initial condition in the rest of the network will be unaltered".

Explanation

Let us take a circuit as shown in fig - a, Let, V is supplied voltage and Z₁, Z₂ & Z₃ is different circuit impedances. V₁, V₂ and V₃ are the voltages across Z₁, Z₂ & Z₃ impedance respectively and I is the supplied current whose I₁ part is flowing through the Z₁ impedance whereas I₂ part is flowing through the Z₂ and Z₃ impedance.

Now if we replace Z₃ impedance with V₃ voltage source as shown in fig-b or with I₂ current source as shown in fig-c then according to Substitution Theorem all initial condition through other impedances and source will remain unchanged. i.e. - current through source will be I, voltage across Z₁ impedance will be V₁, current through Z₂ will be I₂ etc.

Example of Substitution Theorem

For more efficient and clear understanding let us go through a simple practical example:

Let us take a circuit as shown in fig - d As per voltage division rule voltage across 3Ω and 2Ω resistance are If we replace the 3Ω resistance with a voltage source of 6V as shown in fig - e, then According to Ohm’s law the voltage across 2Ω resistance and current through the circuit is Alternately if we replace 3Ω resistance with a current source of 2A as shown in fig - f, then Voltage across 2Ω is V_2Ω = 10 - 3× 2 = 4 V and voltage across 2A current source is V_2A = 10 - 4 = 6 V We can see the voltage across 2Ω resistance and current through the circuit is unaltered i.e all initial condition of the circuit is intact.

Superposition Theorem

Thevenin Theorem and Thevenin Equivalent Voltage and Resistance

Norton Theorem | Norton Equivalent Current and Resistance

Maximum Power Transfer Theorem

Millman Theorem

Reciprocity Theorem

Maximum Power Transfer Theorem

Suppose we have a voltage source or battery that's internal resistance is R_i and a load resistance R_L is connected across this battery . Maximum power transfer theorem determines the value of resistance R_L for which, the maximum power will be transferred from source to it. Actually the maximum power, drawn from the source, depends upon the value of the load resistance. There may be some confusion let us clear it. Power delivered to the load resistance,

To find the maximum power, differentiate the above expression with respect to resistance R_L and equate it to zero. Thus,

Thus in this case, the maximum power will be transferred to the load when load resistance is just equal to internal resistance of the battery . Maximum power transfer theorem can be applicable in complex network as follows-

A resistive load in a resistive network will abstract maximum power when the load resistance is equal to the resistance viewed by the load as it looks back to the network. Actually this is nothing but the resistance presented to the output terminals of the network. This is actually Thevenin equivalent resistance as we explained in Thevenin's theorem if we consider the whole network as a voltage source. Similarly, if we consider the network as current source, this resistance will be Norton equivalent resistance as we explained in Norton theorem.

Thevenin Theorem

This theorem is very conceptual. If we think deeply about an electrical circuit, we can visualize the statements made in Thevenin theorem. Suppose we have to calculate the current through any particular branch in a circuit. This branch is connected with rest of the circuits at its two terminal. Due to active sources in the circuit, there is one electric potential difference between the points where the said branch is connected. The current through the said branch is caused by this electric potential difference that appears across the terminals. So rest of the circuit can be considered as a single voltage source, that's voltage is nothing but the open circuit voltage between the terminals where the said branch is connected and the internal resistance of the source is nothing but the equivalent resistance of the circuit looking back into the terminals where, the branch is connected. So the Thevenin theorem can be stated as follows,

1. When a particular branch is removed from a circuit, the open circuit voltage appears across the terminals of the circuit, is Thevenin equivalent voltage and,
2. The equivalent resistance of the circuit network looking back into the terminals, is Thevenin equivalent resistance.
3. If we replace the rest of the circuit network by a single voltage source , then the voltage of the source would be Thevenin equivalent voltage and internal resistance of the voltage source would be Thevenin equivalent resistance which would be connected in series with the source as shown in the figure below.

To make Thevenin theorem easy to understand, we have shown the circuit below, Here two resistors R₁ and R₂ are connected in series and this series combination is connected across one voltage source of emf E with internal resistance R_i as shown. One resistive branch of R_L is connected across the resistance R₂ as shown. Now we have to calculate the current through R_L. First, we have to remove the resistor R_L from the terminals A and B. Second, we have to calculate the open circuit voltage or Thevenin equivalent voltage V_T across the terminals A and B.

The current through resistance R₂, Hence voltage appears across the terminals A and B i.e. Third, for applying Thevenin theorem, we have to determine the Thevenin equivalent electrical resistance of the circuit, and for that; first we have to replace the voltage source from the circuit, leaving behind only its internal resistance R_i. Now view the circuit inwards from the open terminals A and B. It is found the circuits now consist of two parallel paths - one consisting of resistance R₂ only and the other consisting of resistance R₁ and R_i in series. Thus the Thevenin equivalent resistance R_T is viewed from the open terminals A and B is given as. As per Thevenin theorem, when resistance R_L is connected across terminals A and B, the network behaves as a source of voltage V_T and internal resistance R_T and this is called Thevenin equivalent circuit. The current through R_L is given as,

Thevenin Equivalent Circuit