1. Electromagnetic induction is the phenomenon of production of
electric emf (or current) in a circuit whenever the magnetic flux linked with
the circuit changes. In other words, electromagnetic induction is the phenomenon in which electric current is generated in a circuit by varying
magnetic fields in that region.
2. The magnetic flux linked with a surface held in a magnetic
field is defined as the total number of magnetic field lines crossing the
surface normally. Mathematically, magnetic flux linked with a surface \[\Phi_B = \vec{B}.\vec{s} = \int \vec{B}.\hat{n}ds = \int Bcos\theta ds\]
3. Magnetic flux is a scalar quantity and its SI unit is 1
weber (1 Wb). One weber is the magnetic flux linked with a surface area of $ 1 m^2 $ when held normally inside a uniform magnetic field of 1 tesla. (Thus, $ 1 Wb = 1
T m^2 $).
4. On the basis of his experimental studies, Faraday gave
following laws of electromagnetic induction: Whenever there is a change of
magnetic flux through a circuit, there will be an induced emf and this emf will
last as long as the change persists. The magnitude of the induced emf is equal
to the time rate of change of magnetic flux through the circuit.
Mathematically, the induced emf e is given by \[\varepsilon = - \frac{d\Phi_B}{dt} = \frac{\phi_i - \phi_f}{t}\]Here -ve sign indicates the direction of induce emf and
hence the direction of induced current in a closed loop.
5. For a coil of N-turns of same cross-sectional area, we have \[\varepsilon = - N\frac{d\phi_B}{dt} = - \frac{d}{dt}(N\phi_B)\]
6. If an electrical circuit is complete, an induced current
flow in the circuit on account of the induced emf. Magnitude of induced current
is given by \[I = \frac{\varepsilon}{R} = - N\frac{d\phi_B}{Rdt}\]
7. As magnetic flux $\phi_B $ can be varied by changing either the
magnetic field B or by changing the area of the coil/circuit or by changing the
shape of a coil or rotating a coil in a magnetic field such that angle $\theta $ between B and s changes, hence induced emf can be set up by any of these
methods.
8. Lenz’s law gives the direction of induced emf/current.
According to it, the polarity of the induced emf is such that it tends to
produced a current which opposes the change in magnetic flux that produced it.
9. The phenomenon of electromagnetic induction and Lenz’s law are
strictly in accordance with the principle of conservation of energy.
10. Lenz’s law is usually applied to know the direction of
induced currents for closed circuits
11. Motional emf is the emf induced in a conductor, when it is
moving in the direction perpendicular to its length and a uniform time-independent magnetic field is present which
is perpendicular to the plane of the conductor as well as the direction of
motion. If a conducting rod of length ‘l’ moves with a constant speed ‘v’ in a
normal uniform magnetic field ‘B’, the magnitude of motional emf is given by Motional emf \[|\varepsilon | = Blv\]
12. Whenever magnetic flux linked with a bulk metallic conductor
changes, induced currents are set up in the conductor in the form of closed
loops and are, thus, known as eddy currents. Hence, eddy currents are the
currents induced in a bulk conductor when placed in a changing magnetic field.
13. Eddy currents are undersirable since they dissipate electric
energy in the form of heat. To reduce eddy currents (i) slots are made in the
conductor, and (ii) the conducting parts are built in the form of laminated
metal sheets separated by an insulating lacquer. The plane of the laminations
are arranged parallel to the magnetic field.
14. Eddy currents cause electromagnetic damping which may be
used in (i) electric brake system, (ii) induction furnace, (iii) speedometer,
(iv) electromagnetic damping, (v) moving coil galvanometer to make it dead beat,
(iv) a.c. induction motor etc.
15. Self -induction is the phenomenon according to which an
opposing induced emf is produced in a coil, as a result of change in current flowing through it. Self-induction is also
referred to as the “electrical inertia”.
16. The coefficient of self-induction or self-inductance (L) of
a coil is numerically equal to the magnetic flux linked with the coil, when a
unit current flows through it.
17. The self-inductance of a coil depends only on the geometry
of the coil and intrinsic material properties. Moreover, inductance is a scalar
quantity.
18. Value of induced emf due to self-induction phenomenon is
given by\[\varepsilon = -L\frac{dI}{dt}\]
19. Self-inductance of a coil is numerically equal to the
induced emf produced in the coil, when rate of change of current in the coil is
unity.
20. SI unit of self-inductance is 1 henry (1 H). Inductance of a
coil is said to be 1 henry, if a rate of change of current of $A s^{-1}$ induces an
emf of 1 volt in it.
21. The self-inductance of an air core long solenoid of length
l, total number of turns N and cross-section area A is given by \[L = \frac{\mu_0 N^2 A}{l} = \mu_0 n^2 lA\]where n is the number of turns per unit length.
22. Mutual induction is the phenomenon according to which an
opposing emf is induced in a coil, as a result of change in current or magnetic
flux linked with a neighboring coil.
23. Coefficient of mutual induction or mutual inductance (M) of
a pair of two neighboring coils is numerically equal to the magnetic flux
linked with one coil when a unit current flows through the neighbouring coil.
24. Induced emf due to mutual induction phenomenon is given by \[\varepsilon _1 = - M\frac{dI_2}{dt}\]Hence, mutual inductance for a given pair of two coils is
numerically equal to the induced enf produced in one coil when the rate of
change of current in the other coil is unity.
25. SI unit of mutual inductance too is 1 henry (1H).
26. Mutual induction of the pair of coaxial long solenoids is
given by \[M = \mu_0 \mu_r \frac{N_1N_2A}{l} = \mu_0 \mu_r n_1n_2lA\]where $ n_1 , n_2 $ is the number of the turns per unit length. The mutual inductance of a pair of coils also depends on
their separation as well as their relative orientation.
27. If there are two solenoids $S_1$ and $ S_2 $ then it can be easily
proved that mutual inductance $ M_{21} $ of
solenoid $ S_2 $ with respect to $ S_1 $ is exactly equal to mutual inductance $ M_{12} $ of
solenoid $ S_1 $ with respect to $ S_2 $ i.e., \[M_{12} = M_{21} = M\]
28. The self-induced emf is also called back emf as it opposes
any change in the current in a circuit. Thus, work is to be done against the
back emf in establishing a current in the coil. The work done in establishing a
current I in a coil of inductance L is given by \[W = \frac{1}{2} LI^2\]
29. Energy stored (in the form of magnetic energy) in an
inductor L, while a current I is established in it, is given by \[U = \frac{1}{2} LI^2\]
30. The magnetic energy stored per unit volume (or the magnetic
energy density) in a region of uniform magnetic field ‘B’ is usually given \[u = \frac{B^2}{2\mu_0}\]
31. When a conducting rod of length l kept perpendicular to a
uniform magnetic field B is rotating about one of its ends with a uniform
angular velocity $ \omega $, the emf induced between its ends has a magnitude \[\varepsilon = \frac{1}{2}B\omega l^2\]However, when the rod is rotating about its centre, there is
no emf induced between its ends.
32. If a flat rectangular coil of N-turns each of area A is
rotating in a uniform magnetic field B with a uniform angular velocity $\omega $ so
that its axis of rotation is in the plane of the loop and is at right angles to
the magnetic field, the induced emf at any instant t is given by the relation, Induced emf \[\varepsilon = N B A\omega sin(\omega t)\]
33. Thus, whenever the coil is perpendicular to the magnetic field,
magnitude of induced emf is zero and whenever the coil is parallel to the
magnetic field, magnitude of induced emf is zero and whenever the coil is
parallel to the magnetic field, the magnitude of induced emf is maximum having
a value\[\varepsilon_{max} = N B A\omega \]
34. The direction of induced emf is (current) in a coil when
rotated in a uniform magnetic field may be easily obtained by Fleming’s right-hand rule. According to it, stretch the central finger, the fore-finger point and
the thumb of your right hand mutually perpendicular to each other such as the
for-finger points in the direction of magnetic field and thumb toward the
motion of conductor then the central
figure points in the direction of induced current (emf) in the conductor.
Fleming’s right-hand rule is in accordance with Lenz’s law.
35. An a.c. generator is a device that converts machinal energy
into electric energy on the basis of electromagnetic induction.
36. An a.c. generator is based on the principle of production of
induced emf in a rectangular coil, being rotated about its axis with a uniform
angular velocity when a uniform magnetic field is present in a perpendicular
plane. The induced emf \[\varepsilon = N B A\omega sin(\omega t)\] changes both in magnitude as well as
direction with time and is, therefore, known as alternating emf (or induced
current is known as an alternating current).
Watch Video Lectures
4.1 Magnetic Flux and Faraday's Law watch video
4.2 Lenz Law and Motional EMF watch video
4.3 Eddy current and self-induction watch video
4.4 Self Induction of solenoid and Grouping of the Inductor coils watch video
4.5 Mutual Induction and Mutual Induction of the solenoids watch video
