44Electromagnetic induction

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ELECTROMAGNETIC INDUCTION
▢ Magnetic Flux:
❖ Symbol: \(\phi\)
❖ Definition: Magnetic field lines passing normally through a surface

Table 1: Flux Formulae

Quantity
Formula
Flux through one turn
\(\phi=BA\cos\theta\)
Flux linkage for \(N\) turns
\(N\phi=NBA\cos\theta\)
Flux linkage and self-inductance
\(N\phi=LI\)
❖ Symbols:
  • •
    \(B\) = magnetic field
  • •
    \(A\) = area
  • •
    \(\theta\) = angle between \(B\) and area vector
  • •
    \(N\) = number of turns
  • •
    \(L\) = self inductance
  • •
    \(I\) = current
▢ Induced EMF:
❖ Law: Faraday-Lenz law

Table 1: Induced EMF Formulae

Condition
Formula
General
\(E=-\frac{d\phi}{dt}\)
Using flux linkage
\(E=-\frac{d(N\phi)}{dt}\)
Using self inductance
\(E=-L\frac{dI}{dt}\)
Changing area / field / angle
\(E=-\frac{d(NBA\cos\theta)}{dt}\)
Rotating coil, \(\theta=\omega t\)
\(E=-\frac{d(NBA\cos\omega t)}{dt}\)
Rotating coil result
\(E=NBA\omega\sin\omega t\)
Maximum induced emf
\(E_0=NBA\omega\)
❖ Negative Sign: Shows Lenz's law: induced emf opposes change in magnetic flux
▢ Induced EMF in Various Cases:
❖ Conducting Rod in Uniform Magnetic Field:

Table 1: Motional EMF in Straight Rod

Condition
Formula
Vector form
\(E=\vec B\cdot(\vec l\times\vec v)\)
General
\(E=Blv\sin\theta\)
\(B,l,v\) mutually perpendicular
\(E=Blv\)
◉ Symbols:
  • •
    \(l\) = length of rod
  • •
    \(v\) = velocity of rod
  • •
    \(B\) = magnetic field
  • •
    \(\theta\) = angle between \(\vec B\) and \((\vec l\times\vec v)\)
❖ Conducting Rod Rotating in Uniform Magnetic Field:

Table 1: Rotating Rod EMF

Condition
Formula
Rod of length \(l\), angular velocity \(\omega\)
\(E=\frac{1}{2}Bl^2\omega\)
Using \(v=l\omega\)
\(E=\frac{Blv}{2}\)
Using frequency \(f\)
\(E=B\pi l^2f\)
Using swept area \(A=\pi l^2\)
\(E=BAf\)
▢ Self Inductance:
❖ Definition: Property of a coil by which change of current in it induces emf in the same coil

Table 1: Self Inductance Formulae

System
Self inductance
General
\(L=\frac{N\phi}{I}\)
Circular coil
\(L=\frac{\mu_0N^2A}{2r}\)
Solenoid of length \(l\)
\(L=\frac{\mu_0N^2A}{l}\)
❖ Induced EMF: \(E=-L\frac{dI}{dt}\)
▢ Mutual Inductance:
❖ Definition: Property by which change of current in one coil induces emf in nearby coil
❖ _*table:
    ❖ Induced EMF: \(E=-M\frac{dI}{dt}\)
    ▢ Energy Stored in Inductor:

    Table 1: Energy Stored

    Condition
    Formula
    In self-inductor
    \(U=\frac{1}{2}LI^2\)
    Using mutual inductance notation
    \(U=\frac{1}{2}MI^2\)
    ▢ Transformer:
    ❖ Definition: Device used to change AC voltage using mutual induction

    Table 1: Ideal Transformer Relations

    Relation
    Formula
    Voltage ratio
    \(\frac{E_s}{E_p}=\frac{N_s}{N_p}\)
    Inductance ratio
    \(\frac{E_s}{E_p}=\sqrt{\frac{L_s}{L_p}}\)
    Current ratio
    \(\frac{E_s}{E_p}=\frac{I_p}{I_s}\)
    Transformation ratio
    \(K=\frac{E_s}{E_p}=\frac{N_s}{N_p}=\sqrt{\frac{L_s}{L_p}}=\frac{I_p}{I_s}\)
    ❖ Symbols:
    • •
      \(E_p\) = primary emf / voltage
    • •
      \(E_s\) = secondary emf / voltage
    • •
      \(N_p\) = number of primary turns
    • •
      \(N_s\) = number of secondary turns
    • •
      \(I_p\) = primary current
    • •
      \(I_s\) = secondary current
    • •
      \(L_p\) = self inductance of primary coil
    • •
      \(L_s\) = self inductance of secondary coil
    • •
      \(K\) = transformation ratio
    ❖ Types:

    Table 1: Transformer Types

    Type
    Condition
    Effect
    Step-up transformer
    \(N_s>N_p\), \(K>1\)
    Increases voltage, decreases current
    Step-down transformer
    \(N_s
    Decreases voltage, increases current
    ▢ High-Yield Recall:

    Table 1: Electromagnetic Induction One-Liners

    Fact
    Formula
    Flux through one turn
    \(\phi=BA\cos\theta\)
    Flux linkage
    \(N\phi=NBA\cos\theta=LI\)
    Faraday's law
    \(E=-\frac{d\phi}{dt}\)
    Self-induced emf
    \(E=-L\frac{dI}{dt}\)
    Rotating coil emf
    \(E=NBA\omega\sin\omega t\)
    Maximum emf
    \(E_0=NBA\omega\)
    Moving rod emf
    \(E=Blv\sin\theta\)
    Moving rod perpendicular case
    \(E=Blv\)
    Rotating rod emf
    \(E=\frac{1}{2}Bl^2\omega\)
    Rotating rod using frequency
    \(E=BAf\)
    Circular coil self inductance
    \(L=\frac{\mu_0N^2A}{2r}\)
    Solenoid self inductance
    \(L=\frac{\mu_0N^2A}{l}\)
    Circular coil mutual inductance
    \(M=\frac{\mu_0N_1N_2A}{2r}\)
    Solenoid mutual inductance
    \(M=\frac{\mu_0N_1N_2A}{l}\)
    Energy in inductor
    \(U=\frac{1}{2}LI^2\)
    Transformer ratio
    \(K=\frac{E_s}{E_p}=\frac{N_s}{N_p}=\frac{I_p}{I_s}\)
    Transformer inductance relation
    \(\frac{E_s}{E_p}=\sqrt{\frac{L_s}{L_p}}\)
    Q1.
    Mutual inductance of two coils is 5mH. The current changes from 3A to 2 A in 0.01 sec, then the emf produced is [BP 2014]
    Q2.
    Eddy current is produced in [BP 2013]
    Q3.
    An iron is placed horizontally, what will be the acceleration of falling magnet [BP 2013/ 2016]
    Q4.
    The core of transformer is made up to: [BP 2011]
    Q5.
    Two coils are at fixed locations. When coil I has no current in coil 2 increases at the 15.0 A/s, the emf in coil I is 25 mV, when coil 2 has no current and coil I has a current of 3.6 A, the flux linkage in coil 2 is ?
    Q6.
    In a dynamo, voltage is 6V. current 0.5A. What is the power generated? [IOM 2013]
    Q7.
    A uniform metal disc of radius R lies in XY - plane and rotates with uniform angular velocity ω about the Z - axis. If a constant field B is parallel to the Z- axis, the total induced EMF between the center and the rim of the disc is equal to: [KU 2011]
    Q8.
    A transformer is designed to work from 220 volts AC mains to give a supply at 11 volts AC to run a device. The primary coil has 480 turns. The number of turns in the secondary coil should be-
    Q9.
    A straight conducting rod of length 0.4 m is moving perpendicular to magnetic flux density of 0.9 Wb/m² with a speed of 7m/s. The emf developed in the rod is : [IE 2014]
    Q10.
    Which of the following has zero average value in electromagnetic field? [IE 2013]
    Q11.
    A coil of metal wire is kept stationary in non uniform magnetic field [BP 2014]
    Q12.
    When a motor car starts, the brightness of bulb is decreased then after some time its brightness increases due to: [BP 2014]
    Q13.
    Lenz's law is a consequence of the law of conservation of [MOE 09/ KU 2017]
    Q14.
    Two parallel conductors carrying current in same direction. [KU 08]
    Q15.
    A transformer is employed to reduce 220 V to 11 V. The primary coil draws a current of 5 A and secondary draws 90 A. The efficiency of transformer is
    Q16.
    The two parallel wires carrying current in opposite direction [Bangladesh Emb]
    Q17.
    An electron enters a field of 1 Tesla with a velocity of 1000 m/s parallel the field. The force experienced by it is[ Bangladesh Emb]
    Q18.
    A freely falling magnet passes lengthwise through a horizontal circular coil of conductor. The correct expression relating the acceleration of the falling magnet a and the acceleration due to gravity (g) is [MOE 2054, BP 2014, KU 2013]
    Q19.
    Laminated cores are used in a transformer in order to: [MOE]
    Q20.
    If the flux associated with a coil varies a the rate of 1 Weber per minute, the induced emf is [MOE 2000]
    Q21.
    The two conductors carrying current in same direction attract each other due to the existence of: [MOE 2063]
    Q22.
    Transformer is a device to [IOM 1995]
    Q23.
    A rectangular loop carrying a current 'i' is situated near a long straight wire such that the wire is in the plane of the loop and parallel to one of its sides. If steady current 'I' is established in the wire than the loop will: [IE-08]
    Q24.
    If a wire of resistance 10Ω is moving in a non-uniform magnetic field in such a way that decrease in K.E. is 10J/s. What is current induced in wire ? [IE-06]
    Q25.
    A rectangular loop carrying a current 'i' is situated near a long straight wire such that the wire is in the plane of the loop and parallel to one of its sides. If steady current 'I' is established in the wire than the loop will:
    Q26.
    The magnetic resonance imaging (MRI) based on; the phenomenon of: [BPKIHS-08]
    Q27.
    When an electron moves in the influence of magnetic field with velocity 'v' in a circle of radius 'r', given that magnetic field = B and mass of electron is m. Then'r is equal to [BPKIHS 02]
    Q28.
    A circular loop of radius 0.1 m and carrying a current of 10 A is placed parallel to a magnetic field of 0.1T. The torque acting on the loop in Nm is: [BPKIHS 05]
    Q29.
    The magnetic field at a point at a large distance r at the axis of a current carrying circular coil of small radius is proportional to [BPKIHS-06]
    Q30.
    A circular coil is moving parallel to magnetic field then value of emf is [IE-02]
    Q31.
    The phenomenon of acquiring temporary electrification under the influence of a charged body is called [IOM 98]
    Q32.
    The energy per unit volume in a magnetic field is given by: [MOE 2060]
    Q33.
    A cylindrical bar magnet is kept along the axis of a circular coil. The magnet is rotated about its axis such that north pole faces the coil. The induced current in the coil.
    Q34.
    A helicopter rises vertically with a speed of 10m/s. If helicopter has length of 10m and horizontal component of earth's mag. field is 1.5 x 10-3 Wb/m2, the e.m.f induced between tip of nose and tail of helicopter is
    Q35.
    A magnetic flux 5 x 10-5 Wb is passing through a coil of 1000 turns. If the e.m.f induced through coil is 5 volt, the time interval will be
    Q36.
    A coil of cross-sectional area 10-2m2 is placed in the magnetic field, which changes to 4x10-3Wb/m2 within 5 sec. What will be induced emf across the coil?
    Q37.
    A coil having 500 square loops each of side 10cm is placed normal to a magnetic field which increases at the rate of 1 Wb/m2s. The induced e.m.f is
    Q38.
    The current from A to B is increasing in magnitude. What is the direction of induced current, if any, in the loop shown?
    Q39.
    A coil of area 100cm2 having 50 turns is perpendicular to a magnetic field of intensity 0.02 weber/m². The resistance of the coil is 2ohm. If it is removed in 1sec from the magnetic field, the induced charge produced is
    Q40.
    The current passing through a choke coil of 5H is decreasing at the rate of 2As-1. The emf developed across the coil is
    Q41.
    A coil of mean area 500cm2 and having 1000 turns is held perpendicular to a uniform field of 0.4 gauss. The coil is turned through 180° in 1/10 second, the induced e.m.f. is
    Q42.
    A circular coil and a bar magnet placed nearby are mode to move in the same direction. If the coil covers a distance of 1m in 0.5sec and the magnet a distance of 2m in 1sec, the induced e.m.f. produced in the coil is
    Q43.
    Two solenoids of same cross-sectional area have their lengths and number of turns in the ratio of 1:2. The ratio of self-inductance of two solenoids is
    Q44.
    Two coils, a primary of 100 turns and a secondary of 20 turns are wound over an iron core of length 20cm and cross-section of 2cm radius. If μr = 800, then coefficient of mutual induction is approximately
    Q45.
    Two coils A and B have 200 and 400 turns respectively. A current of 1A in coil A causes a flux per turn of 10-3 Wb to link with A and a flux per turn of 0.8x10-3 Wb through B. The ratio of mutual inductance of A and B is
    Q46.
    Magnetic flux φ in Weber in a closed circuit of resistance 10Ω varies with time t (sec) as φ = 6t2 - 5t + 1. The magnitude of induced current at t = 0.25s is
    Q47.
    When a wire loop is rotated in a magnetic field, the direction of induced e.m.f. changes in every [IE 2008]
    Q48.
    A coil is wound on an iron core and looped back on itself so that core has two sets of closely wound coils carrying current in opposite directions. The self-inductance is
    Q49.
    A magnet is moved towards a coil through a certain distance slowly in one case and quickly in second case. The emf induced is
    Q50.
    A coil of area A and resistance R is kept inside a uniform magnetic field B normal to plane of coil, then charge induced in the coil if it is revolved end for end is
    Q51.
    Current in the coil start from 10A decreases linearly with time and become zero at 2sec, then change in flux through the coil if inductance of coil is 5H is
    Q52.
    A circular coil of radius r is kept inside magnetic field B normal to plane of coil then emf induced in the coil if it is converted in square in time t is
    Q53.
    A coil and an electric bulb are connected in series with a DC source. A soft iron is now inserted in the coil, then
    Q54.
    The number of turns of primary and secondary coils of transformer are 5 and 10 respectively and mutual inductance of the transformer is 25H. Now number of turns in primary and secondary are 10 and 5 respectively. Mutual inductance of the transformer will be
    Q55.
    A coil is wound on a core of rectangular cross-section. If all the linear dimensions of core are increased by a factor 2 and number of turns per unit length of coll remains same, then self-inductance increases by a factor of
    Q56.
    A coil of wire of a certain radius has 10 turns and a self inductance of 15mH. The self inductance of second similar coil of 500 turns will be
    Q57.
    A d.c. circuit contains 10Ω of resistance in series with 10H coil. The impedence of the circuit is
    Q58.
    The charge which will flow through a 20Ω galvanometer connected to a 400Ω circular coil of 100 turns wound on a wooden stick 20mm in diameter, if B = 0.012T parallel to the axis of the stick decreased suddenly to zero is
    Q59.
    The induced emf of a generator when the flux of poles is doubled and speed is doubled
    Q60.
    The armature of a shunt wound motor can withstand current upto 8A before it overheats and is damaged. If the armature resistance is 0.5Ω, minimum back e.m.f that must be generated to avoid motor damage, when the motor is connected to a 120V line is [BPKIHS]
    Q61.
    When an electric motor is run at 120 volt, 10A current . flows through it and the Induced back emf is 115 volt. What will be the current flowing in the coil at the time of switch off?
    Q62.
    A circuit has a self inductance of 1H and carries a current of 2A. To prevent sparking, when the circuit is broken, a capacitor which can withstand 400 V is used. The least capacitance of the capacitor is
    Q63.
    A 5cm long solenoid having 10ohm resistance and 5mH inductance is joined to a 10V battery. At steady state, current through solenoid is
    Q64.
    In a noiseless transformer, an a.c. of 2A is flowing in the primary coil. The number of turns in the primary and secondary coils are 100 and 20 respectively. The current in the secondary coil is
    Q65.
    The number of turns in a secondary coil is twice the number of turns in primary. A Leclanche cell of 1.5 V is connected across the primary. The voltage across the secondary is
    Q66.
    The number of turns in primary and secondary coils are 500 and 5000 respectively. Input voltage is 20V and frequency is 50Hz. Then in output we have
    Q67.
    A transformer is used to light 140W, 24V lamp from 240V a.c. mains. The current it the mains is 0.7A. The efficiency of the transformer is nearest to
    Q68.
    In a step down transformer, use of 120V line, provides a pot. diff. of 2400 V. If the primary coil has 75 turns, number of turns in secondary coil is:
    Q69.
    The primary winding of a transformer has 200 turns and its secondary winding has 50 turns. If the current in the secondary winding is 40A, the current in the primary is
    Q70.
    The power loss in transformer working on 220V a.c. supply is 30%. The ratio of primary to secondary current when output voltage is 110V is
    Q71.
    A transformer is used to reduce, the mains supply of 220V to 11V. If the current in the primary and secondary coil are 5A and 90A respectively, efficiency of transformer is
    Q72.
    The self inductance of a straight conductor is
    Q73.
    If the length of a solenoid is doubled and no. of turns doubled, keeping the area constant, then inductance becomes [IOM 2015]
    Q74.
    A circular coil with its horizontal axis is perpendicular to the magnetic field. What is the angle between magnetic field and the plane of coil when the induced emf is maximum? [IOM 2016]