Inductance is the property of electrical circuits containing coils in which a change in the electrical current induces an electromotive force (emf). This value of induced emf opposes the change in current in electrical circuits and electric current ‘I’ produces a magnetic field which generates magnetic flux acting on the circuit containing coils . This magnetic flux, due to Lenz’s law, tends to act to oppose changes in the flux by generating a voltage (a back EMF) that counters or tends to reduce the rate of change in the current. The ratio of the magnetic flux to the current is called the self-inductance. The term ‘inductance’ was coined by Oliver Heaviside in February 1886. It is customary to use the symbol ‘L’ for inductance, possibly in honour of the physicist Heinrich Lenz. In honour of Joseph Henry, the unit of inductance has been given the name Henry (H): 1H=1Wb/A.
The phenomenon of inducing an emf in a coil whenever a current linked with coil changes is called induction. The quantitative definition of the inductance of a wire loop in SI units is (1)
Here units of L are Weber per ampere which is equivalent to Henry. denotes the magnetic flux through the area spanned by one loop, and N is the number of loops in the coil. The flux so linked with the loop is,
N= LI (2)
Self and mutual inductances also occur in the expression for the energy of the magnetic field generated by K electrical circuits where In is the current in the nth circuit. This equation is an alternative definition of inductance that also applies when the currents are not confined to thin wires so that it is not immediately clear what area is encompassed by the circuit nor how the magnetic flux through the circuit is to be defined. The definition L = N / I, in contrast, is more direct and more intuitive. It may be shown that the two definitions are equivalent by equating the time derivative of W and the electric power transferred to the system. It should be noted that this analysis assumes linearity, not nonlinearity.
1.1.2 LAWS RELATED WITH INDUCTION
1.1.2a FARADAYS FIRST LAW:
Whenever the magnetic flux linked with a closed circuit changes, an e.m.f. is induced in the circuit. The induced e.m.f. last long as the change in magnetic flux continues.
1.1.2b FARADAYS SECOND LAW:
The magnitude of induced e.m.f. is directly proportional to time rate of change of magnetic flux linked with the circuit.
Faraday made his discovery of electromagnetic induction with an experiment using two coils of wire wound around opposite sides of a ring of soft iron similar to the experiment shown in Figure 1 below.
The first coil on the right is attach to a battery. The second coil contains a compass, which acts as a galvanometer to detect current flow. When the switch is closed, a current passes through the first coil and the iron ring becomes magnetized. When the switch is first closed, the compass in the second coil deflects momentarily and returns immediately to its original position. The deflection of the compass is an indication that an electromotive force was induced causing current to flow momentarily in the second coil. Faraday also observed that when the switch is opened, the compass again deflects momentarily, but in the opposite direction. Faraday was aware that that a coil of wire with an electric current flowing through it generates a magnetic field. Therefore, he hypothesized that a changing magnetic field induces a current in the second coil. The closing and opening of the switch cause a magnetic field to change: to expand and collapse respectively.
Lenz’s Law: According to this law: – “The direction of any magnetic induction effect is such as to oppose the cause of the effect”
To determine the direction of the current produced when electric potential is induced, we use Lenz’s Law: the induced current flows in a direction that opposes the change that induced the current. This is more easily understood through an example[a]. In the following example the permanent magnet moves to the left.
What is the direction of the current through the resistor?
The movement of the north end of the permanent magnet away from the solenoid induces electric potential in the solenoid. To oppose the motion of the magnet, the left end of the solenoid becomes south, attracting the magnet. The attraction is not strong enough to prevent the movement; it just offers resistance to the movement.
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Using the right hand rule for solenoids, we point the thumb of the right hand along the direction of the field through the solenoid (ie. to the right). When we “grab” the solenoid with our right hand, the fingers curl upward behind the solenoid and come over top the solenoid and down in front of the solenoid. This is the direction of conventional current flow through the solenoid. (For electron flow use the left hand.) Since the current flows downwards in front of the solenoid, it must travel to the right through the resistor.
1.1.3 Properties of inductance
Taking the time derivative of both sides of the equation N = Li yields:
In most physical cases, the inductance is constant with time and so
By Faraday’s Law of Induction we have:
Where is the Electromotive force (emf) and v is the induced voltage. Note that the emf is opposite to the induced voltage. Thus:
These equations together state that, for a steady applied voltage v, the current changes in a linear manner, at a rate proportional to the applied voltage, but inversely proportional to the inductance. Conversely, if the current through the inductor is changing at a constant rate, the induced voltage is constant.
The effect of inductance can be understood using a single loop of wire as an example. If a voltage is suddenly applied between the ends of the loop of wire, the current must change from zero to non-zero. However, a non-zero current induces a magnetic field by Ampere’s law. This change in the magnetic field induces an emf that is in the opposite direction of the change in current. The strength of this emf is proportional to the change in current and the inductance. When these opposing forces are in balance, the result is a current that increases linearly with time where the rate of this change is determined by the applied voltage and the inductance.
An alternative explanation of this behaviour is possible in terms of energy conservation. Multiplying the equation for di / dt above with Li leads to
Since is the energy transferred to the system per time it follows that is the energy of the magnetic field generated by the current. A change in current thus implies a change in magnetic field energy, and this only is possible if there also is a voltage.A mechanical analogy is a body with mass M, velocity v and kinetic energy (M / 2) v2. A change in velocity (current) requires or generates a force (an electrical voltage) proportional to mass (inductance).
18.104.22.168 Phasor circuit analysis and impedance
Using phasors, the equivalent impedance of an inductance is given by:
where j is the arbitrary unit,
L is the inductance,
is the angular frequency,
f is the frequency and
is the inductive reactance.
22.214.171.124 Relation between inductance and capacitance
Inductance per length L’ and capacitance per length C’ are related to each other in the special case of transmission lines consisting of two parallel perfect conductors of arbitrary but constant cross section, 
Here and μ denote dielectric constant and magnetic permeability of the medium the conductors are embedded in. There is no electric and no magnetic field inside the conductors (complete skin effect, high frequency). Current flows down on one line and returns on the other. The signal propagation speed coincides with the propagation speed of electromagnetic waves in the bulk.
The flux through the i-th circuit in a set is given by:
so that the induced emf, , of a specific circuit, i, in any given set can be given directly by
1.1.4 Applications of Inductance
Inductance is typified by the behavior of a coil of wire in resisting any change of electric current through the coil. Arising from Faraday’s law the inductance L may be defined in terms of the e.m.f. generated to oppose a given change in current:
The properties of inductors make them very useful in various applications. For example, inductors oppose any changes in current. Therefore, inductors can be used to protect circuits from surges of current. Inductors are also used to stabilize direct current and to control or eliminate alternating current. Inductors used to eliminate alternating current above a certain frequency are called chokes.
One of the most common uses of electromagnetic inductance is in the generation of electric current.
Inductors can be used in circuits with capacitors to generate and isolate high-frequency currents. For example, inductor coils are used with capacitors in tuning circuits of radios. In Figure 4, a variable capacitor is connected to an antenna-transformer circuit. Transmitted radio waves cause an induced current to flow in the antenna through the primary inductor coil to ground.
A secondary current in the opposite direction is induced in the secondary inductor coil. This current flows to the capacitor. The surge of current to the capacitor induces a counter electromotive force. This counter electromotive force is call capacitive reactance. The induced flow of current through the coil also induces a counter electromotive force. This is called inductive reactance. So we have both capacitive and inductive reactances in the circuit.
At higher frequencies, inductive reactance is greater and capacitive reactance is smaller. At lower frequencies the opposite is true. A variable capacitor is used to equalize the inductive and capacitive reactances. The condition in which the reactances are equalized is called resonance. The particular frequency that is isolated by the equalized reactances is called the resonant frequency. A radio circuit is tuned by adjusting the capacitance of a variable capacitor to equalize the inductive and capacitive reactance of the circuit for the desired resonant frequency, or in other words, to tune in the desired radio station. Inductor coils and a variable capacitor are used to tune in radio frequencies.
(iii) Metal Detectors:
The operation of a metal detector is based upon the principle of electromagnetic induction. Metal detectors contain one or more inductor coils. When metal passes through the magnetic field generated by the coil or coils, the field induces electric currents in the metal. These currents are called eddy currents. These eddy currents in turn induce their own magnetic field, which generates current in the detector that powers a signal indicating the presence of the metal. Observe the magnetic fields and eddy currents generated by a metal detector.
Chapter – 2
2.1 Mutual Induction
There may, however, be contributions from other circuits for induction. Consider for example two circuits C1, C2, carrying the currents i1, i2. The flux linkages of C1 and C2 are given by
According to the definition, L11 and L22 are the self-inductances of C1 and C2, respectively. It can be shown that the other two coefficients are equal: L12 = L21 = M, where M is called the mutual inductance of the pair of circuits. The number of turns N1 and N2 occur somewhat asymmetrically in the definition above. But actually Lmn always is proportional to the product NmNn, and thus the total currents Nmim contribute to the flux.
Mutual induction is the property of two coils by virtue of which each opposes any change in the strength of current flowing through the other by developing an induced emf. 
If the current i in one circuit changes with time, the flux through the area bounded by the second circuit also changes. This phenomenon is called mutual induction. 
Suppose that one circuit (the primary) employs a changing current to create a magnetic field changes with time – inducing a current in another (secondary) circuit.  In other words, Mutual inductance tells us how large a change in a circuit (primary) is needed to produce a given secondary current (voltage)
2.1.2 Coefficient of mutual induction
It is a measure of the induction between two circuits; the ratio of the electromotive force in a circuit to the corresponding change of current in a neighbouring circuit; usually measured in henries.
Coefficient of mutual induction of two coils is numerically equal to the amount of magnetic flux linked with one coil when unit current flows through the neighbouring coil.
Now, the emf induced in the coil is given by
If dI/dt = 1, then =-M*1 or M =-
Hence coefficient of mutual induction of two coil is equal to the e.m.f. induced in one coil when rate of change of current through the other coil is unity.
Units S.I Unit of L=1Volt/1Amp/sec=1Henry
The SI unit of M is Henry, when a current change at the rate of one ampere/sec in one coil induces an e.m.f. of one volt in the other coil.
Note: 1 Volt / Amp = 1 Ohm ; 1 Henry = 1 Ohm / sec=1Weber/ampere = 1volt-sec/ampere
(i) The mutual inductance of two coils depends on the geometry of the two coils, distance between the coils and orientation of the two coils.
(ii) Distance between two coils,
(iii) Relative placement of two coil i.e. orientation of the two coils.
The circuit diagram representation of mutually inducting inductors. The two vertical lines between the inductors indicate a solid core that the wires of the inductor are wrapped around.  “n:m” shows the ratio between the number of windings of the left inductor to windings of the right inductor. This picture also shows the dot convention.
Mutual inductance occurs when the change in current in one inductor induces a voltage in another nearby inductor. It is important as the mechanism by which transformers work, but it can also cause unwanted coupling between conductors in a circuit. The mutual inductance, M, is also a measure of the coupling between two inductors. The mutual inductance by circuit i on circuit j is given by the double integral Neumann formula i.e.in 2.1.2
The mutual inductance also has the relationship:
Where, M21 is the mutual inductance, and the subscript specifies the relationship of the voltage induced in coil 2 to the current in coil 1.
N1 is the number of turns in coil 1,
N2 is the number of turns in coil 2,
P21 is the permeance of the space occupied by the flux.
The mutual inductance also has a relationship with the coupling coefficient. The coupling coefficient is always between 1 and 0, and is a convenient way to specify the relationship between a certain orientations of inductor with arbitrary inductance:
Where, k is the coupling coefficient and 0 ≤ k ≤ 1,
L1 is the inductance of the first coil, and
L2 is the inductance of the second coil.
Once the mutual inductance, M, is determined from this factor, it can be used to predict the behavior of a circuit:
Where, V is the voltage across the inductor of interest,
L1 is the inductance of the inductor of interest,
dI1 / dt is the derivative, with respect to time, of the current through the inductor of interest,
dI2 / dt is the derivative, with respect to time, of the current through the inductor that is coupled to the first inductor, and M is the mutual inductance. The minus sign arises because of the sense the current has been defined in the diagram. With both currents defined going into the dots the sign of M will be positive.
When one inductor is closely coupled to another inductor through mutual inductance, such as in a transformer, the voltages, currents, and number of turns can be related in the following way:
Where, Vs is the voltage across the secondary inductor, Vp is the voltage across the primary inductor (the one connected to a power source), Ns is the number of turns in the secondary inductor, and Np is the number of turns in the primary inductor.
Conversely the current:
Where, Is is the current through the secondary inductor, Ip is the current through the primary inductor (the one connected to a power source), Ns is the number of turns in the secondary inductor, and Np is the number of turns in the primary inductor.
Note that the power through one inductor is the same as the power through the other. Also note that these equations don’t work if both transformers are forced (with power sources).
When either side of the transformer is a tuned circuit, the amount of mutual inductance between the two windings determines the shape of the frequency response curve. Although no boundaries are defined, this is often referred to as loose-, critical-, and over-coupling. When two tuned circuits are loosely coupled through mutual inductance, the bandwidth will be narrow. As the amount of mutual inductance increases, the bandwidth continues to grow. When the mutual inductance is increased beyond a critical point, the peak in the response curve begins to drop, and the center frequency will be attenuated more strongly than its direct sidebands. This is known as over coupling.
2.1.3 Calculation techniques
The mutual inductance by a filamentary circuit i on a filamentary circuit j is given by the double integral Neumann formula
The symbol μ0 denotes the magnetic constant (4π – 10-7 H/m), Ci and Cj are the curves spanned by the wires, Rij is the distance between two points.
2.1.4 Application of Mutual Induction
(i) Electric toothbrush
A transformer is an example of a device that uses circuits with maximum mutual inductance. The device shown in the below photograph is a kind of transformer, with two concentric wire coils. It is actually intended as a precision standard unit for mutual inductance, but for the purposes of illustrating what the essence of a transformer is, it will suffice. The two wire coils can be distinguished from each other by colour: the bulk of the tube’s length is wrapped in green-insulated wire (the first coil) while the second coil (wire with bronze-coloured insulation) stands in the middle of the tube’s length. The wire ends run down to connection terminals at the bottom of the unit. Most transformer units are not built with their wire coils exposed like this.
Current flow in a conductor produces a magnetic field around the conductor. When the current is increasing, decreasing, or changing direction, the magnetic field changes. The magnetic field expands, contracts, or changes direction in response to the changes in current flow. A changing magnetic field induces an additional electromotive force, or voltage in the conductor. The induction of this additional voltage is called self-induction, because it is induced within the conductor itself. The direction of the self-induced electromotive force, or voltage, is in the opposite direction of the current flow that generated it. This is consistent with Lenz’s law, which can be expressed as follows:”An induced electromotive force (voltage) in any circuit is always in a direction in opposition to the current that produced it.” The effect of self-induction in a circuit is to oppose any change in current flow in the circuit. For example, when voltage is applied to a circuit, current begins to flow in all parts of the circuit. This current induces a magnetic field around it. As the field is expanding, a counter voltage, sometimes called back voltage, is generated in the circuit. This back voltage causes a current flow in the opposite direction of the main current flow. Inductance at this stage acts to oppose the buildup of current. When the induced magnetic field becomes steady, it ceases to induce back voltage.
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When a current is established in a closed conducting loop, it produces a magnetic field. This magnetic field has its flux through the area bounded by the loop. If the current changes with time, the flux through the loop changes and hence an emf is induced in the loop. This process is called self induction.  Self induction is the property of a coil opposes any change in the strength of current flowing through it by inducing an emf in itself. 
We need to distinguish carefully between emfs and currents that are caused by batteries or other sources and those that are induced by changing magnetic fields. [6a]
3.1.2 Coefficient of self induction
Coefficient of self induction of a coil is numerically equal to the amount of magnetic flux linked with the coil when unit current flows through the coil.
Now, the emf induced in the coil is given by
If dI/dt = 1, then =-L*1 or L= –
Hence coefficient of self induction of a coil is equal to the e.m.f. induced in the coil when rate of change of current through the coil is unity.
S.I Unit of L = 1 Volt / 1 Amp / sec = 1 Henry
Note: 1 Volt / Amp = 1 Ohm
1 Henry = 1 Ohm / sec
1 Henry=1Weber/ampere = 1volt-sec/ampere
3.1.2 Discription with example
We use the adjective source (as in the terms source emf and source current) to describe the parameters associated with a physical source, and we use the adjective induced to describe those emfs and currents caused by a changing magnetic field.
Consider a circuit consisting of a switch, a resistor, and a source of emf. When the switch is thrown to its closed position, the source current does not immediately jump from zero to its maximum value Faraday’s law of electromagnetic induction can be used to describe this effect as follows: As the source current increases with time, the magnetic flux through the circuit loop due to this current also increases with time. This increasing flux creates an induced emf in the circuit. The direction of the induced emf is such that it would cause an induced current in the loop (if a current were not already flowing in the loop), which would establish a magnetic field that would oppose the change in the source magnetic field. Thus, the direction of the induced emf is opposite the direction of the source emf; this results in a gradual rather than instantaneous increase in the source current to its final equilibrium value.
This effect is called self-induction because the changing flux through the circuit and the resultant induced emf arise from the circuit itself. The emf set up in this case is called a self-induced emf. It is also often called a back emf. As a second example of self-induction, which shows, a coil wound on a cylindrical iron core. Assume that the source current in the coil either increases or decreases with time. When the source current is in the direction shown, a magnetic field directed from right to left is set up inside the coil, as the source current changes with time, the magnetic flux through the coil also changes and induces an emf in the coil. From Lenz’s law, the polarity of this induced emf must be such that it opposes the change in the magnetic field from the source current. If the source current is increasing, the polarity of the induced emf is as pictured in and if the source current is decreasing, the polarity of the induced emf.
To obtain a quantitative description of self-induction, we recall from Faraday’s law that the induced emf is equal to the negative time rate of change of the magnetic flux. The magnetic flux is proportional to the magnetic field due to the source current, which in turn is proportional to the source current in the circuit. Therefore, a self-induced emf (EL) is always proportional to the time rate of change of the source current. For a closely spaced coil of N turns (a toroid or an ideal solenoid) carrying a source current I, we find that
Self-induced emf: EL=-LdI/dt
Formally the self-inductance of a wire loop would be given by the above equation with i =j. However, 1 / R becomes infinite and thus the finite radius a along with the distribution of the current in the wire must be taken into account.  There remain the contribution from the integral over all points where and a correction term,
Here ‘a’ and ‘l’ denote radius and length of the wire, and Y is a constant that depends on the distribution of the current in the wire: Y = 0 when the current flows in the surface of the wire (skin effect), Y = 1 / 4 when the current is homogeneous across the wire. This approximation is accurate when the wires are long compared to their cross-sectional dimensions. Here is a derivation of this equation.
Pradeep’s Fundamentals Physics pg.4/13
 Verma, H.C., Concepts of Physics, Bharati Bhawan, 2008, B.B. Printers, Patna, Fifth Electromagnetic induction, pg 295
 Verma, H.C., Concepts of Physics, Bharati Bhawan, 2008, B.B. Printers, Patna, Fifth Electromagnetic induction, pg 295
 Pradeep’s Fundamentals Physics pg.4/11
[6a] Haliday-Resnick-Walker Fundamentals of Physics Pg.1016
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