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Magnetic flux through a plane of area dA placed in a uniform magnetic field B
If the surface is closed, then
This is because magnetic lines of force are closed lines and free magnetic poles do not exist.
Electromagnetic Induction: Faraday’s Law
The direction of the induced emf or current in the circuit is such that it opposes the cause by which it is produced, so that
N → Number of turns in the coil
Lenz’s law is based on energy conservation
Fleming’s right-hand rule: It states that if the thumb and the first two fingers of the right hand are stretched mutually perpendicular to each other and if the forefinger gives the direction of the magnetic field and the thumb gives the direction of motion of the conductor, then the central finger gives the direction of the induced current.
Emf, current and charge induced in the circuit
Charge depends only on net change in flux and does not depend on time.
Emf induced due to linear motion of a conducting rod in a uniform magnetic field
If are perpendicular to each other, then
Induced emf due to rotation of a conducting rod in a uniform magnetic field
where n is the frequency of rotation of the conducting rod.
Direction of emf is given by the right-hand thumb rule.
Induced emf due to rotation of a metallic disc in a uniform magnetic field
Induced emf, current and energy conservation in a rectangular loop moving in a non-uniform magnetic field with a constant velocity
Rotation of a rectangular coil in a uniform magnetic field
Self-induction and self-inductance (L)
When an induced emf is produced in the coil on changing the current in a coil, the phenomenon is called self-induction.
Mutual induction and mutual inductance
For two coils wound in the opposite direction and connected in series,
L = L1 + L2 – 2M
For two coils in parallel,
Energy stored in an inductor
Energy stored in the form of Magnetic Field
Magnetic energy density
Growth of current in series LR circuit
If a circuit consists of a cell, an inductor L, a resistor R and a switch S connected in series, and the switch is closed at t = 0, the current in the circuit I will increase as
The quantity L/R is called the time constant of the circuit and is denoted by τ. The variation of current with time is as shown.
Generator or dynamo
It is a device by which mechanical energy is converted to electrical energy. It is based on the principle of EMI.
In an AC generator, mechanical energy is converted to electrical energy by virtue of electromagnetic induction. It consists of a field magnet, armature, slip rings and brushes. If the coil of N turns and area A is rotated at revolution per second in a uniform magnetic field B, then the motional emf produced is , where we have assumed that at time t = 0 s, the coil is perpendicular to the field.
Alternating current (AC)
The current whose magnitude changes with time and direction reverses periodically is called alternating current.
Values of alternating current and voltage
Phase difference between emf (Voltage) and current in an AC circuit
The reciprocal of resistance is called conductance.
Power in an AC circuit
The component of current whose contribution to the average power is nil is called wattless current.
RMS value of wattless current:
An inductive coil used for controlling the alternating current whose self-inductance is high and resistance in negligible is called a choke coil.
The power factor of this coil is given by
Now, as we know that , the power factor is small, and hence, the power absorbed will be very small. Also, on account of its large impedance (large inductance), current passing through the
coil is very small. Hence, such a coil is preferred in electrical circuits for the purpose of adjusting the current to any desired value without significant energy waste.
Series resonant circuit
Frequencies f1 and f2 at which the power is half of the maximum power (power at resonance), i.e.
and are called half-power frequencies.
The frequency interval between half-power frequencies is called bandwidth.
∴ Bandwidth Δf = f2 – f1
For a series LCR resonant circuit,
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