Whenever the magnetic flux linked with an electric circuit changes, an emf is induced in the circuit. This phenomenon is called electromagnetic induction.
The magnetic flux (f) linked with a surface held in a magnetic field (B) is
defined as the number of magnetic lines of force crossing that area (A). If q is
the angle between the direction of the field and normal to the area, (area
vector) then
φ = B.A = BA cos θ
Faraday performed various experiments to discover and understand the phenomenon of electromagnetic
induction. Some of them are :
Whenever there is a relative motion between the source of magnetic field (magnet) and the coil, an emf is induced in the coil. When the magnet and coil move towards each other then the flux linked with the coil increases and emf is induced. When the magnet and coil move away from each other the magnetic flux linked with the coil decreases, again an emf is induced. This emf lasts so long the flux is changing. Due to this emf an electric current start to flow and the galvanometer shows deflection. The deflection in galvanometer last as long the relative motion between the magnet and coil continues. Whenever relative motion between coil and magnet takes place an induced emf produced in coil. If coil is in closed circuit then current and charge is also induced in the circuit. This phenomenon is called electro magnetic induction.
E ∝ dφ / dt ⇒ E = – dφ / dt
where constant of proportionality is one and negative sign indicates Lenz’s law.
Here, flux = NBA cos θ, SI unit of φ = weber,
CGS unit of φ = maxwell, 1 weber = 108 maxwell,
Dimensional formula of magnetic flux
[φ] = [ML2T-2A-2]
The negative sign in Faraday's equations of electromagnetic induction describes the direction in
which the induced emf drives current around a circuit. The direction is
determined with the help of Lenz’s law. This law states that
“The direction of any magnetic induction effect is such as to oppose the cause of the effect.”
or
"The induced current produces magnetic fields which tend to oppose the change in magnetic
flux that induces such currents."
If a rod of length l moves perpendicular to a magnetic field B, with a velocity v, then induced emf produced in it given by
E = B * v * I = bvl
If a metallic rod of length 1 rotates about one of its ends in a plane perpendicular to the magnetic field, then the induced emf produced across its ends is given by
E = 1 / 2 bωr2 = BAf
where, ω = angular velocity of rotation, f = frequency of rotation and A = πr2 = area of disc.
The direction of induced current in any conductor can be obtained from Fleming’s right hand rule.
A rectangular coil moves linearly in a field when coil moves with constant velocity in a uniform magnetic field, flux and induced emf will be zero.
A rod moves at an angle θ with the direction of magnetic field, velocity
E = – Blv sin θ.
An emf is induced
If we stretch the thumb, the forefinger and the central finger of right hand in such a way that all three are perpendicular to each other, th. if thumb represent the direction of motion,
the forefinger represent tile direction of magnetic field, then central finger will represent the direction of induced current.
If R is the electrical resistance of the circuit, then induced current in the circuit is given by
I = E / R
If induced current is produced in a coil rotated in uniform magnetic field, then
I = NBA ω sin ωt / R = Io sin ωt
where, Io = NBA ω = peak value of induced current,
N = number of turns in the coil,
B= magnetic induction,
ω = angular velocity of rotation and
A = area of cross-section of the coil.
If a piece of metal is placed in a varying magnetic field or rotated high speed in a uniform magnetic field, then induced current set up the piece are like whire pool of air, called eddy currents.
The magnitude of eddy currents is given by
i = – e / R = dφ / dt / R
where R is the resistance.
Eddy currents are also known as Facault’s current.
The phenomena of production of induced emf in a circuit due to change in current flowing in its own, is called self induction.
Coefficient of Self-Induction
The magnetic flux linked with a coil
φ = LI
where, L = = coefficient of self induction.
The induced emf in the coil
E = – L dl / dt
it unit of self induction is henry (H) and its dimensional formula is [ML2T -2A-2].
Self – inductance of a long solenoid is given by normal text
L = μo N2 A / l = μo n2 Al
where. N = total number of turns in the solenoid,
1 = length of the coil, n = number of turns in the coil and
A = area of cross-section of the coil.
If core of the solenoid is of any other magnetic material, then
L = μo μr N2 A / l
Self – inductance of a toroid L = μo N2 A / 2πr
Where, r = radius of the toroid
Energy stored in an inductor E = 1 / 2 LI2
The phenomena of production of induced emf in a circuit due to the change in magnetic flux in its neighbouring circuit, is called mutual induction.
Coefficient of Mutual Induction
If two coils are coupled with each, other then magnetic flux linked with a Coil (secondary coil)
φ = MI
where M is coefficient of mutual induction and I is current flow in through primary coil.
The induced emf in the secondary coil
E = – M dl / dt
where dl / dt is the rate of change of current through primary coil.
The unit of coefficient of mutual induction is henry (H) and its dimension is [ML2T-2A-2].
The coefficient of mutual induction depends on geometry of two coils, distance between them and orientation of the two coils.
Coefficient of Coupling
Two coils are said to be coupled if full a part of the fuse produced by one links with the other.
K = √M / L1 L2, where L1 and L2 are coefficients of self-induction of the two coils and M is coefficient of mutual induction of the two coils.
Coefficient of coupling is maximum (K = 1) in case (a), when coils are coaxial and minimum in case (b), when coils are placed a right angles.
Mutual inductance of two long coaxial solenoids is given by
M = μ N1 N2 A / l
= μ n1 n2 Al
where N1 and N2 are total number of turns in both coils, n1 n2 are number of turns per unit length in coils, A is area of cross-section of coils and 1 is length of the coils.
(a) When three coils of inductances L1, L2 and L3 are connected in series and the coefficient of coupling K = 0, as in series, then
L = L1 + L2 + L3
(b) When three coils of inductances L1, L2 and L3 are connected in parallel and the coefficient of coupling K= 0 as in parallel, then
L = 1 / L1 + 1 / L2 + 1 / L3
If coefficient of coupling K = 1, then
(i) In series
(a) If current in two coils are in the same direction, then
L = L1 + L2 + 2M
(b) If current in two coils are in opposite directions, then
L = L1 + L2 – 2M
(ii) In parallel
(a) If current in two coils are in same direction, then
L = L1 L2 – M2 / L1 + L2 + 2M
(b) If current in two coils are in opposite directions, then
L = L1 L2 – M2 / L1 + L2 – 2M
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