Magnetic flux : The number of magnetic lines of force crossing a surface is known as magnetic flux linked with that surface.
It is given by, Φ= B. A = BA cosθ, where B is the strength of magnetic field, A is the area of surface and θ is the angle between normal to area and field direction.
- When B is perpendicular to the surface i.e. θ = 0°, Φ = NBA (maximum value).
- When B is parallel to the surface i.e. θ = 90°, Φ = 0 (minimum value).
- In case of a coil of area A having N turns, Φ = NBA cosθ
- Magnetic flux is a scalar quantity. It can be positive, negative or zero.
- The dimensional formula of magnetic flux is [ML2T–2A–1].
- The SI unit of magnetic flux is weber.
- The CGS unit of magnetic flux is maxwell.
- 1 weber = 108 maxwell
Electromagnetic induction: It is the phenomenon of generating an emf by changing the number of magnetic lines of force (i.e. magnetic flux) associated with the circuit.
The emf so generated is known as induced emf. If the circuit is closed the current which flows in it due to induced emf is known as induced current.
Faraday’s law of electromagnetic induction
🎯 First law : Whenever the amount of magnetic flux linked with a circuit changes, an emf is induced in the circuit. This induced emf persists as long as the change in magnetic flux continues.
🎯 Second law : The magnitude of the induced emf is equal to the time rate of change of magnetic flux. Mathematically, induced emf is given by, ℰ = – dΦ/dt, where negative sign indicates the direction of e.
🎯 Lenz’s law : This law gives us the direction of induced emf. According to this law, the direction of induced emf in a circuit is such that it opposes the change in magnetic flux responsible for its production. Lenz’s law is in accordance with the principle of conservation of energy.
🎯 Fleming’s right hand rule : Fleming’s right hand rule also gives us the direction of induced emf or current, in a conductor moving in a magnetic field. According to this rule, if we stretch the fore finger, central finger and thumb of our right hand in mutually perpendicular directions such that fore finger points along the direction of the field and thumb is along the direction of motion of the conductor, then the central finger would give us the direction of induced current or emf.
🎯 Applications of Lenz’s law
– When a north pole of a bar magnet is moved towards a coil, the current induced in the coil will be in anticlockwise direction as shown in the figure.
– When a north pole of a bar magnet is moved away from the coil, the current induced in the coil will be in clockwise direction as shown in the figure.
When a current carrying coil is moved towards a stationary coil, the direction of current induced in stationary coil is as shown in figure.
When a current carrying coil is moved away from a stationary coil, the direction of current induced in stationary coil is as shown in figure.
When two coils A and B are arranged as shown in figure, then on pressing K, current in A increases in clockwise direction. Therefore, induced current in B will be in anticlockwise direction.
However, when key K is released, current in A decreases in clockwise direction. Therefore, induced current in B will be in clockwise direction.
– When current in a straight conductor AB is increased, induced current in loop will be in clockwise direction as shown in the figure. If current in AB is decreasing, the induced current in the loop will be in anticlockwise direction.
Motional emf: When a conducting rod of length l, moves with a velocity v perpendicular to a uniform magnetic field B, the induced emf across its ends is |ℰ| =Blv. This emf is known as motional emf.
If the rod makes an angle q with the direction of the field, then induced emf is |ℰ| =Blv sinθ
When a conducting rod of length l is rotated perpendicular to a uniform magnetic field B, then induced emf between the ends of the rod is
where, ω is angular frequency and ν is frequency of rod, A = πl2.
When a conducting solid disc of radius r is rotating with a uniform angular velocity w with its plane perpendicular to a uniform magnetic field B, the emf induced between the centre and rim of disc is
Eddy currents : Eddy currents are basically the currents induced in the body of a conductor due to change in magnetic flux linked with the conductor.
- The direction of eddy currents is given by Lenz’s law, or Fleming’s right hand rule.
- According to Lenz’s law, eddy currents set up in a metallic conductor ow in such a direction so as to oppose the change in magnetic flux linked with it.
- Eddy currents cannot be eliminated but can be minimised by
– laminating the core
– by taking the metallic core in the form of thin laminated sheets attached together.
- Eddy currents are useful in
– Electromagnetic damping
– Induction furnace
– Electric brakes
Inductor : An inductor is a device for storing energy in a magnetic field. An inductor is generally called as inductance. In usual practice a coil or solenoid is treated as inductor. It is denoted by symbol
Self induction : Whenever the current passing through a coil or circuit changes, the magnetic flux linked with it will also change. As a result of this, an emf is induced in the coil or the circuit which opposes the change that causes it. This phenomenon is known as self induction and the emf induced is known as self induced emf or back emf.
– When a current I flows through a coil and Φ is the magnetic flux linked with the coil, then Φ ∝ I or Φ=LI where L is coefficient of self induction or self inductance of the coil.
– The self induced emf is ℰ = – dΦ/dt = – L dI/dt
– The SI unit of L is henry (H) and its dimensional formula is [ML2T–2A–2].
– Self inductance of a solenoid is L = μ0l2A where l is length of the solenoid, N is number of turns per unit length of a solenoid and A is area of cross section of the solenoid.
– Self inductance of a circular coil is L = μ0N2πR/2, where R is the radius of a coil and N is the number of turns.
Mutual induction : Whenever the current passing through a coil or circuit changes, the magnetic flux linked with a neighbouring coil or circuit will also change. Hence an emf will be induced in the neighbouring coil or circuit. This phenomenon is known as mutual induction.
The coil or circuit in which the current changes is known as primary while the other in which emf is set up is known as secondary.
– Let IP be the current flowing through primary coil at any instant. If ΦS is the flux linked with secondary coil then ΦS ∝ IP or ΦS = MIP , where M is coefficient of mutual inductance of the two coils.
– The emf induced in the secondary coil is given by, ℰs= – M dIp/dt
– The SI unit of M is henry (H) and its dimensional formula is [ML2T–2A–2].
Coefficient of coupling (K) : Coefficient of coupling of two coils is a measure of the coupling between the two coils and is given by, K=M/√L1L2, where L1 and L2 are coefficients of self inductance of the two coils and M is coefficient of mutual inductance of the two coils.
The coefficient of mutual inductance of two long co-axial solenoids, each of length l, area of cross section A, wound on air core is, M=μ0N1N2A/l, where N1, N2 are total number of turns of the two solenoid.
🎯 Combination of inductances
– Two inductors of self-inductances L1 and L2 are kept so far apart that their mutual inductance is zero. These are connected in series. Then the equivalent inductance is
L = L1 + L2
– Two inductors of self-inductance L1 and L2 are connected in series and they have mutual inductance M. Then the equivalent inductance of the combination is
L = L1 + L2 ± 2M
– The plus sign occurs if windings in the two coils are in the same sense, while minus sign occurs if windings are in opposite sense.
– Two inductors of self-inductors L1 and L2 are connected in parallel. The inductors are so far apart that their mutual inductance is negligible. Then their equivalent inductance is
Energy stored in an inductor : When a current I flows through an inductor, the energy stored in it is given by
The energy stored in an inductor is in the form of magnetic energy.
AC Generator : A generator produces electrical energy from mechanical work, just the opposite of what a motor does.
🎯 An AC generator is based on the phenomena of electromagnetic induction, which states that whenever magnetic flux linked with a conductor (or coil) changes, an emf is induced in the coil. Here ℰ is the emf induced in the coil, then ℰ = NBAω sinωt or, ℰ = ℰ0 sinωt, where ℰ0 = NBAω is the maximum or peak value of induced EMF.
The instantaneous EMF ℰ produced in coil varies sinusoidally with time and hence is also known as alternating EMF.
where, N = number of turns in the coil, B = strength of magnetic field, A = Area of each turns of the coil and ω = angular velocity of rotation.